Daily Papers

In this paper, we introduce analytic federated learning (AFL), a new training paradigm that brings analytical (i.e., closed-form) solutions to the federated learning (FL) community. Our AFL draws inspiration from analytic learning -- a gradient-free technique that trains neural networks with analytical solutions in one epoch. In the local client training stage, the AFL facilitates a one-epoch training, eliminating the necessity for multi-epoch updates. In the aggregation stage, we derive an absolute aggregation (AA) law. This AA law allows a single-round aggregation, removing the need for multiple aggregation rounds. More importantly, the AFL exhibits a weight-invariant property, meaning that regardless of how the full dataset is distributed among clients, the aggregated result remains identical. This could spawn various potentials, such as data heterogeneity invariance, client-number invariance, absolute convergence, and being hyperparameter-free (our AFL is the first hyperparameter-free method in FL history). We conduct experiments across various FL settings including extremely non-IID ones, and scenarios with a large number of clients (e.g., ge 1000). In all these settings, our AFL constantly performs competitively while existing FL techniques encounter various obstacles. Code is available at https://github.com/ZHUANGHP/Analytic-federated-learning

Class incremental learning (CIL) trains a network on sequential tasks with separated categories but suffers from catastrophic forgetting, where models quickly lose previously learned knowledge when acquiring new tasks. The generalized CIL (GCIL) aims to address the CIL problem in a more real-world scenario, where incoming data have mixed data categories and unknown sample size distribution, leading to intensified forgetting. Existing attempts for the GCIL either have poor performance, or invade data privacy by saving historical exemplars. To address this, in this paper, we propose an exemplar-free generalized analytic class incremental learning (G-ACIL). The G-ACIL adopts analytic learning (a gradient-free training technique), and delivers an analytical solution (i.e., closed-form) to the GCIL scenario. This solution is derived via decomposing the incoming data into exposed and unexposed classes, allowing an equivalence between the incremental learning and its joint training, i.e., the weight-invariant property. Such an equivalence is theoretically validated through matrix analysis tools, and hence contributes interpretability in GCIL. It is also empirically evidenced by experiments on various datasets and settings of GCIL. The results show that the G-ACIL exhibits leading performance with high robustness compared with existing competitive GCIL methods. Codes will be ready at https://github.com/ZHUANGHP/Analytic-continual-learning.

In the field of autonomous driving, even a meticulously trained model can encounter failures when faced with unfamiliar sceanrios. One of these scenarios can be formulated as an online continual learning (OCL) problem. That is, data come in an online fashion, and models are updated according to these streaming data. Two major OCL challenges are catastrophic forgetting and data imbalance. To address these challenges, in this paper, we propose an Analytic Exemplar-Free Online Continual Learning (AEF-OCL). The AEF-OCL leverages analytic continual learning principles and employs ridge regression as a classifier for features extracted by a large backbone network. It solves the OCL problem by recursively calculating the analytical solution, ensuring an equalization between the continual learning and its joint-learning counterpart, and works without the need to save any used samples (i.e., exemplar-free). Additionally, we introduce a Pseudo-Features Generator (PFG) module that recursively estimates the deviation of real features. The PFG generates offset pseudo-features following a normal distribution, thereby addressing the data imbalance issue. Experimental results demonstrate that despite being an exemplar-free strategy, our method outperforms various methods on the autonomous driving SODA10M dataset. Source code is available at https://github.com/ZHUANGHP/Analytic-continual-learning.

Reviewing the progress in artificial intelligence over the past decade, various significant advances (e.g. object detection, image generation, large language models) have enabled AI systems to produce more semantically meaningful outputs and achieve widespread adoption in internet scenarios. Nevertheless, AI systems still struggle when it comes to understanding and interacting with the physical world. This reveals an important issue: relying solely on semantic-level concepts learned from internet data (e.g. texts, images) to understand the physical world is far from sufficient -- machine intelligence currently lacks an effective way to learn about the physical world. This research introduces the idea of analytic concept -- representing the concepts related to the physical world through programs of mathematical procedures, providing machine intelligence a portal to perceive, reason about, and interact with the physical world. Except for detailing the design philosophy and providing guidelines for the application of analytic concepts, this research also introduce about the infrastructure that has been built around analytic concepts. I aim for my research to contribute to addressing these questions: What is a proper abstraction of general concepts in the physical world for machine intelligence? How to systematically integrate structured priors with neural networks to constrain AI systems to comply with physical laws?

We examine the text-free speech representations of raw audio obtained from a self-supervised learning (SSL) model by analyzing the synthesized speech using the SSL representations instead of conventional text representations. Since raw audio does not have paired speech representations as transcribed texts do, obtaining speech representations from unpaired speech is crucial for augmenting available datasets for speech synthesis. Specifically, the proposed speech synthesis is conducted using discrete symbol representations from the SSL model in comparison with text representations, and analytical examinations of the synthesized speech have been carried out. The results empirically show that using text representations is advantageous for preserving semantic information, while using discrete symbol representations is superior for preserving acoustic content, including prosodic and intonational information.

Motivated by the large progress made by large language models (LLMs), we introduce the framework of verbalized machine learning (VML). In contrast to conventional machine learning models that are typically optimized over a continuous parameter space, VML constrains the parameter space to be human-interpretable natural language. Such a constraint leads to a new perspective of function approximation, where an LLM with a text prompt can be viewed as a function parameterized by the text prompt. Guided by this perspective, we revisit classical machine learning problems, such as regression and classification, and find that these problems can be solved by an LLM-parameterized learner and optimizer. The major advantages of VML include (1) easy encoding of inductive bias: prior knowledge about the problem and hypothesis class can be encoded in natural language and fed into the LLM-parameterized learner; (2) automatic model class selection: the optimizer can automatically select a concrete model class based on data and verbalized prior knowledge, and it can update the model class during training; and (3) interpretable learner updates: the LLM-parameterized optimizer can provide explanations for why each learner update is performed. We conduct several studies to empirically evaluate the effectiveness of VML, and hope that VML can serve as a stepping stone to stronger interpretability and trustworthiness in ML.

A significant body of recent research in the field of Learning Analytics has focused on leveraging machine learning approaches for predicting at-risk students in order to initiate timely interventions and thereby elevate retention and completion rates. The overarching feature of the majority of these research studies has been on the science of prediction only. The component of predictive analytics concerned with interpreting the internals of the models and explaining their predictions for individual cases to stakeholders has largely been neglected. Additionally, works that attempt to employ data-driven prescriptive analytics to automatically generate evidence-based remedial advice for at-risk learners are in their infancy. eXplainable AI is a field that has recently emerged providing cutting-edge tools which support transparent predictive analytics and techniques for generating tailored advice for at-risk students. This study proposes a novel framework that unifies both transparent machine learning as well as techniques for enabling prescriptive analytics, while integrating the latest advances in large language models. This work practically demonstrates the proposed framework using predictive models for identifying at-risk learners of programme non-completion. The study then further demonstrates how predictive modelling can be augmented with prescriptive analytics on two case studies in order to generate human-readable prescriptive feedback for those who are at risk using ChatGPT.

This work studies the general principles of improving the learning of language models (LMs), which aims at reducing the necessary training steps for achieving superior performance. Specifically, we present a theory for the optimal learning of LMs. We first propose an objective that optimizes LM learning by maximizing the data compression ratio in an "LM-training-as-lossless-compression" view. Then, we derive a theorem, named Learning Law, to reveal the properties of the dynamics in the optimal learning process under our objective. The theorem is then validated by experiments on a linear classification and a real-world language modeling task. Finally, we empirically verify that the optimal learning of LMs essentially stems from the improvement of the coefficients in the scaling law of LMs, indicating great promise and significance for designing practical learning acceleration methods. Our code can be found at https://aka.ms/LearningLaw.

Analytical reasoning is an essential and challenging task that requires a system to analyze a scenario involving a set of particular circumstances and perform reasoning over it to make conclusions. In this paper, we study the challenge of analytical reasoning of text and introduce a new dataset consisting of questions from the Law School Admission Test from 1991 to 2016. We analyze what knowledge understanding and reasoning abilities are required to do well on this task. Furthermore, to address this reasoning challenge, we design two different baselines: (1) a Transformer-based method which leverages the state-of-the-art pre-trained language models and (2) Analytical Reasoning Machine (ARM), a logical-level reasoning framework extracting symbolic knowledge (e.g, participants, facts, logical functions) to deduce legitimate solutions. In our experiments, we find that the Transformer-based models struggle to solve this task as their performance is close to random guess and ARM achieves better performance by leveraging symbolic knowledge and interpretable reasoning steps. Results show that both methods still lag far behind human performance, which leave further space for future research.

Despite the success of neural networks (NNs), there is still a concern among many over their "black box" nature. Why do they work? Here we present a simple analytic argument that NNs are in fact essentially polynomial regression models. This view will have various implications for NNs, e.g. providing an explanation for why convergence problems arise in NNs, and it gives rough guidance on avoiding overfitting. In addition, we use this phenomenon to predict and confirm a multicollinearity property of NNs not previously reported in the literature. Most importantly, given this loose correspondence, one may choose to routinely use polynomial models instead of NNs, thus avoiding some major problems of the latter, such as having to set many tuning parameters and dealing with convergence issues. We present a number of empirical results; in each case, the accuracy of the polynomial approach matches or exceeds that of NN approaches. A many-featured, open-source software package, polyreg, is available.

Realistic simulation is critical for applications ranging from robotics to animation. Traditional analytic simulators sometimes struggle to capture sufficiently realistic simulation which can lead to problems including the well known "sim-to-real" gap in robotics. Learned simulators have emerged as an alternative for better capturing real-world physical dynamics, but require access to privileged ground truth physics information such as precise object geometry or particle tracks. Here we propose a method for learning simulators directly from observations. Visual Particle Dynamics (VPD) jointly learns a latent particle-based representation of 3D scenes, a neural simulator of the latent particle dynamics, and a renderer that can produce images of the scene from arbitrary views. VPD learns end to end from posed RGB-D videos and does not require access to privileged information. Unlike existing 2D video prediction models, we show that VPD's 3D structure enables scene editing and long-term predictions. These results pave the way for downstream applications ranging from video editing to robotic planning.

Unsupervised pretraining, which learns a useful representation using a large amount of unlabeled data to facilitate the learning of downstream tasks, is a critical component of modern large-scale machine learning systems. Despite its tremendous empirical success, the rigorous theoretical understanding of why unsupervised pretraining generally helps remains rather limited -- most existing results are restricted to particular methods or approaches for unsupervised pretraining with specialized structural assumptions. This paper studies a generic framework, where the unsupervised representation learning task is specified by an abstract class of latent variable models Phi and the downstream task is specified by a class of prediction functions Psi. We consider a natural approach of using Maximum Likelihood Estimation (MLE) for unsupervised pretraining and Empirical Risk Minimization (ERM) for learning downstream tasks. We prove that, under a mild ''informative'' condition, our algorithm achieves an excess risk of mathcal{O}(mathcal{C_Phi/m} + mathcal{C_Psi/n}) for downstream tasks, where C_Phi, C_Psi are complexity measures of function classes Phi, Psi, and m, n are the number of unlabeled and labeled data respectively. Comparing to the baseline of mathcal{O}(mathcal{C_{Phi circ Psi}/n}) achieved by performing supervised learning using only the labeled data, our result rigorously shows the benefit of unsupervised pretraining when m gg n and C_{Phicirc Psi} > C_Psi. This paper further shows that our generic framework covers a wide range of approaches for unsupervised pretraining, including factor models, Gaussian mixture models, and contrastive learning.

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging. In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.

The success of deep learning is frequently described as the ability to train all parameters of a network on a specific application in an end-to-end fashion. Yet, several design choices on the camera level, including the pixel layout of the sensor, are considered as pre-defined and fixed, and high resolution, regular pixel layouts are considered to be the most generic ones in computer vision and graphics, treating all regions of an image as equally important. While several works have considered non-uniform, \eg, hexagonal or foveated, pixel layouts in hardware and image processing, the layout has not been integrated into the end-to-end learning paradigm so far. In this work, we present the first truly end-to-end trained imaging pipeline that optimizes the size and distribution of pixels on the imaging sensor jointly with the parameters of a given neural network on a specific task. We derive an analytic, differentiable approach for the sensor layout parameterization that allows for task-specific, local varying pixel resolutions. We present two pixel layout parameterization functions: rectangular and curvilinear grid shapes that retain a regular topology. We provide a drop-in module that approximates sensor simulation given existing high-resolution images to directly connect our method with existing deep learning models. We show that network predictions benefit from learnable pixel layouts for two different downstream tasks, classification and semantic segmentation.

Two main challenges in Reinforcement Learning (RL) are designing appropriate reward functions and ensuring the safety of the learned policy. To address these challenges, we present a theoretical framework for Inverse Reinforcement Learning (IRL) in constrained Markov decision processes. From a convex-analytic perspective, we extend prior results on reward identifiability and generalizability to both the constrained setting and a more general class of regularizations. In particular, we show that identifiability up to potential shaping (Cao et al., 2021) is a consequence of entropy regularization and may generally no longer hold for other regularizations or in the presence of safety constraints. We also show that to ensure generalizability to new transition laws and constraints, the true reward must be identified up to a constant. Additionally, we derive a finite sample guarantee for the suboptimality of the learned rewards, and validate our results in a gridworld environment.

Learned world models summarize an agent's experience to facilitate learning complex behaviors. While learning world models from high-dimensional sensory inputs is becoming feasible through deep learning, there are many potential ways for deriving behaviors from them. We present Dreamer, a reinforcement learning agent that solves long-horizon tasks from images purely by latent imagination. We efficiently learn behaviors by propagating analytic gradients of learned state values back through trajectories imagined in the compact state space of a learned world model. On 20 challenging visual control tasks, Dreamer exceeds existing approaches in data-efficiency, computation time, and final performance.

The development of artificial intelligence demands that models incrementally update knowledge by Continual Learning (CL) to adapt to open-world environments. To meet privacy and security requirements, Continual Unlearning (CU) emerges as an important problem, aiming to sequentially forget particular knowledge acquired during the CL phase. However, existing unlearning methods primarily focus on single-shot joint forgetting and face significant limitations when applied to CU. First, most existing methods require access to the retained dataset for re-training or fine-tuning, violating the inherent constraint in CL that historical data cannot be revisited. Second, these methods often suffer from a poor trade-off between system efficiency and model fidelity, making them vulnerable to being overwhelmed or degraded by adversaries through deliberately frequent requests. In this paper, we identify that the limitations of existing unlearning methods stem fundamentally from their reliance on gradient-based updates. To bridge the research gap at its root, we propose a novel gradient-free method for CU, named Analytic Continual Unlearning (ACU), for efficient and exact forgetting with historical data privacy preservation. In response to each unlearning request, our ACU recursively derives an analytical (i.e., closed-form) solution in an interpretable manner using the least squares method. Theoretical and experimental evaluations validate the superiority of our ACU on unlearning effectiveness, model fidelity, and system efficiency.

Pre-trained language models (PLMs) have shown impressive performance in various language tasks. However, they are prone to spurious correlations, and often generate illusory information. In real-world applications, PLMs should justify decisions with formalized, coherent reasoning chains, but this challenge remains under-explored. Cognitive psychology theorizes that humans are capable of utilizing fast and intuitive heuristic thinking to make decisions based on past experience, then rationalizing the decisions through slower and deliberative analytic reasoning. We incorporate these interlinked dual processes in fine-tuning and in-context learning with PLMs, applying them to two language understanding tasks that require coherent physical commonsense reasoning. We show that our proposed Heuristic-Analytic Reasoning (HAR) strategies drastically improve the coherence of rationalizations for model decisions, yielding state-of-the-art results on Tiered Reasoning for Intuitive Physics (TRIP). We also find that this improved coherence is a direct result of more faithful attention to relevant language context in each step of reasoning. Our findings suggest that human-like reasoning strategies can effectively improve the coherence and reliability of PLM reasoning.

Training general robotic policies from heterogeneous data for different tasks is a significant challenge. Existing robotic datasets vary in different modalities such as color, depth, tactile, and proprioceptive information, and collected in different domains such as simulation, real robots, and human videos. Current methods usually collect and pool all data from one domain to train a single policy to handle such heterogeneity in tasks and domains, which is prohibitively expensive and difficult. In this work, we present a flexible approach, dubbed Policy Composition, to combine information across such diverse modalities and domains for learning scene-level and task-level generalized manipulation skills, by composing different data distributions represented with diffusion models. Our method can use task-level composition for multi-task manipulation and be composed with analytic cost functions to adapt policy behaviors at inference time. We train our method on simulation, human, and real robot data and evaluate in tool-use tasks. The composed policy achieves robust and dexterous performance under varying scenes and tasks and outperforms baselines from a single data source in both simulation and real-world experiments. See https://liruiw.github.io/policycomp for more details .

We develop a general approach to distill symbolic representations of a learned deep model by introducing strong inductive biases. We focus on Graph Neural Networks (GNNs). The technique works as follows: we first encourage sparse latent representations when we train a GNN in a supervised setting, then we apply symbolic regression to components of the learned model to extract explicit physical relations. We find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example-a detailed dark matter simulation-and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.

Objective: To transform heterogeneous clinical data from electronic health records into clinically meaningful constructed features using data driven method that rely, in part, on temporal relations among data. Materials and Methods: The clinically meaningful representations of medical concepts and patients are the key for health analytic applications. Most of existing approaches directly construct features mapped to raw data (e.g., ICD or CPT codes), or utilize some ontology mapping such as SNOMED codes. However, none of the existing approaches leverage EHR data directly for learning such concept representation. We propose a new way to represent heterogeneous medical concepts (e.g., diagnoses, medications and procedures) based on co-occurrence patterns in longitudinal electronic health records. The intuition behind the method is to map medical concepts that are co-occuring closely in time to similar concept vectors so that their distance will be small. We also derive a simple method to construct patient vectors from the related medical concept vectors. Results: For qualitative evaluation, we study similar medical concepts across diagnosis, medication and procedure. In quantitative evaluation, our proposed representation significantly improves the predictive modeling performance for onset of heart failure (HF), where classification methods (e.g. logistic regression, neural network, support vector machine and K-nearest neighbors) achieve up to 23% improvement in area under the ROC curve (AUC) using this proposed representation. Conclusion: We proposed an effective method for patient and medical concept representation learning. The resulting representation can map relevant concepts together and also improves predictive modeling performance.

Large Language Models (LLMs) have been successful in mathematical reasoning tasks such as formal theorem proving when integrated with interactive proof assistants like Lean. Existing approaches involve training or fine-tuning an LLM on a specific dataset to perform well on particular domains, such as undergraduate-level mathematics. These methods struggle with generalizability to advanced mathematics. A fundamental limitation is that these approaches operate on static domains, failing to capture how mathematicians often work across multiple domains and projects simultaneously or cyclically. We present LeanAgent, a novel lifelong learning framework for theorem proving that continuously generalizes to and improves on ever-expanding mathematical knowledge without forgetting previously learned knowledge. LeanAgent introduces several key innovations, including a curriculum learning strategy that optimizes the learning trajectory in terms of mathematical difficulty, a dynamic database for efficient management of evolving mathematical knowledge, and progressive training to balance stability and plasticity. LeanAgent successfully proves 162 theorems previously unproved by humans across 23 diverse Lean repositories, many from advanced mathematics. It performs up to 11times better than the static LLM baseline, proving challenging theorems in domains like abstract algebra and algebraic topology while showcasing a clear progression of learning from basic concepts to advanced topics. In addition, we analyze LeanAgent's superior performance on key lifelong learning metrics. LeanAgent achieves exceptional scores in stability and backward transfer, where learning new tasks improves performance on previously learned tasks. This emphasizes LeanAgent's continuous generalizability and improvement, explaining its superior theorem proving performance.

We consider dense, associative neural-networks trained by a teacher (i.e., with supervision) and we investigate their computational capabilities analytically, via statistical-mechanics of spin glasses, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as quality and quantity of the training dataset, network storage and noise, that is valid in the limit of large network size and structureless datasets: these networks may work in a ultra-storage regime (where they can handle a huge amount of patterns, if compared with shallow neural networks) or in a ultra-detection regime (where they can perform pattern recognition at prohibitive signal-to-noise ratios, if compared with shallow neural networks). Guided by the random theory as a reference framework, we also test numerically learning, storing and retrieval capabilities shown by these networks on structured datasets as MNist and Fashion MNist. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate supervised learning in neural networks, beyond the shallow limit, in general.

We consider dense, associative neural-networks trained with no supervision and we investigate their computational capabilities analytically, via a statistical-mechanics approach, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as the quality and quantity of the training dataset and the network storage, valid in the limit of large network size and structureless datasets. Moreover, we establish a bridge between macroscopic observables standardly used in statistical mechanics and loss functions typically used in the machine learning. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate neural networks in general.

We address the problem of video representation learning without human-annotated labels. While previous efforts address the problem by designing novel self-supervised tasks using video data, the learned features are merely on a frame-by-frame basis, which are not applicable to many video analytic tasks where spatio-temporal features are prevailing. In this paper we propose a novel self-supervised approach to learn spatio-temporal features for video representation. Inspired by the success of two-stream approaches in video classification, we propose to learn visual features by regressing both motion and appearance statistics along spatial and temporal dimensions, given only the input video data. Specifically, we extract statistical concepts (fast-motion region and the corresponding dominant direction, spatio-temporal color diversity, dominant color, etc.) from simple patterns in both spatial and temporal domains. Unlike prior puzzles that are even hard for humans to solve, the proposed approach is consistent with human inherent visual habits and therefore easy to answer. We conduct extensive experiments with C3D to validate the effectiveness of our proposed approach. The experiments show that our approach can significantly improve the performance of C3D when applied to video classification tasks. Code is available at https://github.com/laura-wang/video_repres_mas.

Reinforcement learning (RL) has become the dominant paradigm for endowing language models with advanced reasoning capabilities. Despite the substantial empirical gains demonstrated by RL-based training methods like GRPO, a granular understanding of their advantages is still lacking. To address this gap, we introduce a fine-grained analytic framework to dissect the impact of RL on reasoning. Our framework specifically investigates key elements that have been hypothesized to benefit from RL training: (1) plan-following and execution, (2) problem decomposition, and (3) improved reasoning and knowledge utilization. Using this framework, we gain insights beyond mere accuracy. For instance, providing models with explicit step-by-step plans surprisingly degrades performance on the most challenging benchmarks, yet RL-tuned models exhibit greater robustness, experiencing markedly smaller performance drops than their base counterparts. This suggests that RL may not primarily enhance the execution of external plans but rather empower models to formulate and follow internal strategies better suited to their reasoning processes. Conversely, we observe that RL enhances the model's capacity to integrate provided knowledge into its reasoning process, leading to performance improvements across diverse tasks. We also study difficulty, showing improved training by developing new ways to exploit hard problems. Our findings lay a foundation for more principled training and evaluation of reasoning models.

We introduce Feasible Learning (FL), a sample-centric learning paradigm where models are trained by solving a feasibility problem that bounds the loss for each training sample. In contrast to the ubiquitous Empirical Risk Minimization (ERM) framework, which optimizes for average performance, FL demands satisfactory performance on every individual data point. Since any model that meets the prescribed performance threshold is a valid FL solution, the choice of optimization algorithm and its dynamics play a crucial role in shaping the properties of the resulting solutions. In particular, we study a primal-dual approach which dynamically re-weights the importance of each sample during training. To address the challenge of setting a meaningful threshold in practice, we introduce a relaxation of FL that incorporates slack variables of minimal norm. Our empirical analysis, spanning image classification, age regression, and preference optimization in large language models, demonstrates that models trained via FL can learn from data while displaying improved tail behavior compared to ERM, with only a marginal impact on average performance.

Symbolic equations are at the core of scientific discovery. The task of discovering the underlying equation from a set of input-output pairs is called symbolic regression. Traditionally, symbolic regression methods use hand-designed strategies that do not improve with experience. In this paper, we introduce the first symbolic regression method that leverages large scale pre-training. We procedurally generate an unbounded set of equations, and simultaneously pre-train a Transformer to predict the symbolic equation from a corresponding set of input-output-pairs. At test time, we query the model on a new set of points and use its output to guide the search for the equation. We show empirically that this approach can re-discover a set of well-known physical equations, and that it improves over time with more data and compute.

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust learning, it's surprising that we still lack a unified theory; traditional proofs of the equivalence tend to be disparate, and rely on strong model-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining the equivalence of realizable and agnostic learnability: a three-line blackbox reduction that simplifies, unifies, and extends our understanding across a wide variety of settings. This includes models with no known characterization of learnability such as learning with arbitrary distributional assumptions and more general loss functions, as well as a host of other popular settings such as robust learning, partial learning, fair learning, and the statistical query model. More generally, we argue that the equivalence of realizable and agnostic learning is actually a special case of a broader phenomenon we call property generalization: any desirable property of a learning algorithm (e.g. noise tolerance, privacy, stability) that can be satisfied over finite hypothesis classes extends (possibly in some variation) to any learnable hypothesis class.

This paper surveys and organizes research works in a new paradigm in natural language processing, which we dub "prompt-based learning". Unlike traditional supervised learning, which trains a model to take in an input x and predict an output y as P(y|x), prompt-based learning is based on language models that model the probability of text directly. To use these models to perform prediction tasks, the original input x is modified using a template into a textual string prompt x' that has some unfilled slots, and then the language model is used to probabilistically fill the unfilled information to obtain a final string x, from which the final output y can be derived. This framework is powerful and attractive for a number of reasons: it allows the language model to be pre-trained on massive amounts of raw text, and by defining a new prompting function the model is able to perform few-shot or even zero-shot learning, adapting to new scenarios with few or no labeled data. In this paper we introduce the basics of this promising paradigm, describe a unified set of mathematical notations that can cover a wide variety of existing work, and organize existing work along several dimensions, e.g.the choice of pre-trained models, prompts, and tuning strategies. To make the field more accessible to interested beginners, we not only make a systematic review of existing works and a highly structured typology of prompt-based concepts, but also release other resources, e.g., a website http://pretrain.nlpedia.ai/ including constantly-updated survey, and paperlist.

Symbolic regression (SR) is a challenging task in machine learning that involves finding a mathematical expression for a function based on its values. Recent advancements in SR have demonstrated the effectiveness of pretrained transformer-based models in generating equations as sequences, leveraging large-scale pretraining on synthetic datasets and offering notable advantages in terms of inference time over GP-based methods. However, these models primarily rely on supervised pretraining goals borrowed from text generation and overlook equation-specific objectives like accuracy and complexity. To address this, we propose TPSR, a Transformer-based Planning strategy for Symbolic Regression that incorporates Monte Carlo Tree Search into the transformer decoding process. Unlike conventional decoding strategies, TPSR enables the integration of non-differentiable feedback, such as fitting accuracy and complexity, as external sources of knowledge into the transformer-based equation generation process. Extensive experiments on various datasets show that our approach outperforms state-of-the-art methods, enhancing the model's fitting-complexity trade-off, extrapolation abilities, and robustness to noise

Conventional supervised learning typically assumes that the learning task can be solved by learning a single function since the data is sampled from a fixed distribution. However, this assumption is invalid in open-ended environments where no task-level data partitioning is available. In this paper, we present a novel supervised learning framework of learning from open-ended data, which is modeled as data implicitly sampled from multiple domains with the data in each domain obeying a domain-specific target function. Since different domains may possess distinct target functions, open-ended data inherently requires multiple functions to capture all its input-output relations, rendering training a single global model problematic. To address this issue, we devise an Open-ended Supervised Learning (OSL) framework, of which the key component is a subjective function that allocates the data among multiple candidate models to resolve the "conflict" between the data from different domains, exhibiting a natural hierarchy. We theoretically analyze the learnability and the generalization error of OSL, and empirically validate its efficacy in both open-ended regression and classification tasks.

We initiate a principled study of algorithmic collective action on digital platforms that deploy machine learning algorithms. We propose a simple theoretical model of a collective interacting with a firm's learning algorithm. The collective pools the data of participating individuals and executes an algorithmic strategy by instructing participants how to modify their own data to achieve a collective goal. We investigate the consequences of this model in three fundamental learning-theoretic settings: the case of a nonparametric optimal learning algorithm, a parametric risk minimizer, and gradient-based optimization. In each setting, we come up with coordinated algorithmic strategies and characterize natural success criteria as a function of the collective's size. Complementing our theory, we conduct systematic experiments on a skill classification task involving tens of thousands of resumes from a gig platform for freelancers. Through more than two thousand model training runs of a BERT-like language model, we see a striking correspondence emerge between our empirical observations and the predictions made by our theory. Taken together, our theory and experiments broadly support the conclusion that algorithmic collectives of exceedingly small fractional size can exert significant control over a platform's learning algorithm.

Neural sequence models, especially transformers, exhibit a remarkable capacity for in-context learning. They can construct new predictors from sequences of labeled examples (x, f(x)) presented in the input without further parameter updates. We investigate the hypothesis that transformer-based in-context learners implement standard learning algorithms implicitly, by encoding smaller models in their activations, and updating these implicit models as new examples appear in the context. Using linear regression as a prototypical problem, we offer three sources of evidence for this hypothesis. First, we prove by construction that transformers can implement learning algorithms for linear models based on gradient descent and closed-form ridge regression. Second, we show that trained in-context learners closely match the predictors computed by gradient descent, ridge regression, and exact least-squares regression, transitioning between different predictors as transformer depth and dataset noise vary, and converging to Bayesian estimators for large widths and depths. Third, we present preliminary evidence that in-context learners share algorithmic features with these predictors: learners' late layers non-linearly encode weight vectors and moment matrices. These results suggest that in-context learning is understandable in algorithmic terms, and that (at least in the linear case) learners may rediscover standard estimation algorithms. Code and reference implementations are released at https://github.com/ekinakyurek/google-research/blob/master/incontext.

Reasoning in mathematical domains remains a significant challenge for relatively small language models (LMs). Many current methods focus on specializing LMs in mathematical reasoning and rely heavily on knowledge distillation from powerful but inefficient large LMs (LLMs). In this work, we explore a new direction that avoids over-reliance on LLM teachers, introducing a multi-view fine-tuning method that efficiently exploits existing mathematical problem datasets with diverse annotation styles. Our approach uniquely considers the various annotation formats as different "views" and leverages them in training the model. By postpending distinct instructions to input questions, models can learn to generate solutions in diverse formats in a flexible manner. Experimental results show that our strategy enables a LLaMA-7B model to outperform prior approaches that utilize knowledge distillation, as well as carefully established baselines. Additionally, the proposed method grants the models promising generalization ability across various views and datasets, and the capability to learn from inaccurate or incomplete noisy data. We hope our multi-view training paradigm could inspire future studies in other machine reasoning domains.

We introduce Explanatory Learning (EL), a framework to let machines use existing knowledge buried in symbolic sequences -- e.g. explanations written in hieroglyphic -- by autonomously learning to interpret them. In EL, the burden of interpreting symbols is not left to humans or rigid human-coded compilers, as done in Program Synthesis. Rather, EL calls for a learned interpreter, built upon a limited collection of symbolic sequences paired with observations of several phenomena. This interpreter can be used to make predictions on a novel phenomenon given its explanation, and even to find that explanation using only a handful of observations, like human scientists do. We formulate the EL problem as a simple binary classification task, so that common end-to-end approaches aligned with the dominant empiricist view of machine learning could, in principle, solve it. To these models, we oppose Critical Rationalist Networks (CRNs), which instead embrace a rationalist view on the acquisition of knowledge. CRNs express several desired properties by construction, they are truly explainable, can adjust their processing at test-time for harder inferences, and can offer strong confidence guarantees on their predictions. As a final contribution, we introduce Odeen, a basic EL environment that simulates a small flatland-style universe full of phenomena to explain. Using Odeen as a testbed, we show how CRNs outperform empiricist end-to-end approaches of similar size and architecture (Transformers) in discovering explanations for novel phenomena.

Mathematical problem-solving is a key field in artificial intelligence (AI) and a critical benchmark for evaluating the capabilities of large language models (LLMs). While extensive research has focused on mathematical problem-solving, most existing work and datasets concentrate on computational tasks, leaving gaps in areas like mathematical analysis, which demands rigorous proofs and formal reasoning. We developed the DEMI-MathAnalysis dataset, comprising proof-based problems from mathematical analysis topics such as Sequences and Limits, Infinite Series, and Convex Functions. We also designed a guiding framework to rigorously enhance LLMs' ability to solve these problems. Through fine-tuning LLMs on this dataset and employing our framework, we observed significant improvements in their capability to generate logical, complete, and elegant proofs. This work addresses critical gaps in mathematical reasoning and contributes to advancing trustworthy AI capable of handling formalized mathematical language. The code is publicly accessible at LLMs for Mathematical Analysis.

When do machine learning systems fail to generalize, and what mechanisms could improve their generalization? Here, we draw inspiration from cognitive science to argue that one weakness of machine learning systems is their failure to exhibit latent learning -- learning information that is not relevant to the task at hand, but that might be useful in a future task. We show how this perspective links failures ranging from the reversal curse in language modeling to new findings on agent-based navigation. We then highlight how cognitive science points to episodic memory as a potential part of the solution to these issues. Correspondingly, we show that a system with an oracle retrieval mechanism can use learning experiences more flexibly to generalize better across many of these challenges. We also identify some of the essential components for effectively using retrieval, including the importance of within-example in-context learning for acquiring the ability to use information across retrieved examples. In summary, our results illustrate one possible contributor to the relative data inefficiency of current machine learning systems compared to natural intelligence, and help to understand how retrieval methods can complement parametric learning to improve generalization.

Large Language Models (LLMs) have shown remarkable performance in various natural language processing tasks but face challenges in mathematical reasoning, where complex problem-solving requires both linguistic understanding and mathematical reasoning skills. Existing approaches to address this challenge often rely on ensemble methods and suffer from the problem of data scarcity in target domains. In this work, we present a novel method to enhance LLMs' capabilities in mathematical reasoning tasks. Motivated by the need to bridge this gap, our approach incorporates a question paraphrase strategy, which aims at diversifying the linguistic forms of mathematical questions to improve generalization. Additionally, specialized training objectives are employed to guide the model's learning process, focusing on enhancing its understanding of mathematical concepts and reasoning processes. We conduct experiments on four datasets using different LLMs, and demonstrate the effectiveness of our approach in improving LLMs' performance on mathematical reasoning tasks. Our findings underscore the significance of our methodology in the advancement of large language models and its potential implications for real-world applications that require mathematical reasoning abilities.

While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are optimal for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that achieve optimality. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and Neural Tangent Kernels, we provide explicit activation functions that can be used to construct networks that achieve optimality. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: (1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); (2) majority vote (model predictions are given by the label of the class with greatest representation in the training set); or (3) singular kernel classifiers (a set of classifiers containing those that achieve optimality). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful.

Self-improvement methods enable large language models (LLMs) to generate solutions themselves and iteratively train on filtered, high-quality rationales. This process proves effective and reduces the reliance on human supervision in LLMs' reasoning, but the performance soon plateaus. We delve into the process and find that models tend to over-sample on easy queries and under-sample on queries they have yet to master. As iterations proceed, this imbalance in sampling is exacerbated, leading to a long-tail distribution where solutions to difficult queries almost diminish. This phenomenon limits the performance gain of self-improving models. A straightforward solution is brute-force sampling to balance the distribution, which significantly raises computational costs. In this paper, we introduce Guided Self-Improvement (GSI), a strategy aimed at improving the efficiency of sampling challenging heavy-tailed data. It leverages Socratic-style guidance signals to help LLM reasoning with complex queries, reducing the exploration effort and minimizing computational overhead. Experiments on four models across diverse mathematical tasks show that GSI strikes a balance between performance and efficiency, while also being effective on held-out tasks.

According to the forgetting curve theory, we can enhance memory retention by learning extensive data and taking adequate rest. This means that in order to effectively retain new knowledge, it is essential to learn it thoroughly and ensure sufficient rest so that our brain can memorize without forgetting. The main takeaway from this theory is that learning extensive data at once necessitates sufficient rest before learning the same data again. This aspect of human long-term memory retention can be effectively utilized to address the continual learning of neural networks. Retaining new knowledge for a long period of time without catastrophic forgetting is the critical problem of continual learning. Therefore, based on Ebbinghaus' theory, we introduce the view-batch model that adjusts the learning schedules to optimize the recall interval between retraining the same samples. The proposed view-batch model allows the network to get enough rest to learn extensive knowledge from the same samples with a recall interval of sufficient length. To this end, we specifically present two approaches: 1) a replay method that guarantees the optimal recall interval, and 2) a self-supervised learning that acquires extensive knowledge from a single training sample at a time. We empirically show that these approaches of our method are aligned with the forgetting curve theory, which can enhance long-term memory. In our experiments, we also demonstrate that our method significantly improves many state-of-the-art continual learning methods in various protocols and scenarios. We open-source this project at https://github.com/hankyul2/ViewBatchModel.

Large language models (LLMs) have demonstrated remarkable capabilities in various natural language processing tasks. However, achieving strong performance in specialized domains like mathematical reasoning and non-English languages often requires extensive training on massive datasets. This paper investigates a contrasting approach: strategic fine-tuning on a small, high-quality, bilingual (English-French) dataset to enhance both the reasoning capabilities and French language proficiency of a large language model. Rather than relying on scale, we explore the hypothesis that targeted data curation and optimized training can achieve competitive, or even superior, performance. We demonstrate, through targeted supervised fine-tuning (SFT) on only 2,000 carefully selected samples, significant improvements in mathematical reasoning. Specifically, Pensez 7B exhibits an increase in accuracy of the base model up to 20% on the AIME25 and a 12% increase on a French MATH level 5 benchmark. These results challenge the prevailing assumption that massive datasets are aprerequisite for strong reasoning performance in LLMs, highlighting the potential of strategic data curation and optimized fine-tuning for enhancing both specialized skills and multilingual capabilities. Our findings have implications for the efficient development of high-performing, multilingual LLMs, especially in resource-constrained scenarios.

Understanding a student's problem-solving strategy can have a significant impact on effective math learning using Intelligent Tutoring Systems (ITSs) and Adaptive Instructional Systems (AISs). For instance, the ITS/AIS can better personalize itself to correct specific misconceptions that are indicated by incorrect strategies, specific problems can be designed to improve strategies and frustration can be minimized by adapting to a student's natural way of thinking rather than trying to fit a standard strategy for all. While it may be possible for human experts to identify strategies manually in classroom settings with sufficient student interaction, it is not possible to scale this up to big data. Therefore, we leverage advances in Machine Learning and AI methods to perform scalable strategy prediction that is also fair to students at all skill levels. Specifically, we develop an embedding called MVec where we learn a representation based on the mastery of students. We then cluster these embeddings with a non-parametric clustering method where we progressively learn clusters such that we group together instances that have approximately symmetrical strategies. The strategy prediction model is trained on instances sampled from these clusters. This ensures that we train the model over diverse strategies and also that strategies from a particular group do not bias the DNN model, thus allowing it to optimize its parameters over all groups. Using real world large-scale student interaction datasets from MATHia, we implement our approach using transformers and Node2Vec for learning the mastery embeddings and LSTMs for predicting strategies. We show that our approach can scale up to achieve high accuracy by training on a small sample of a large dataset and also has predictive equality, i.e., it can predict strategies equally well for learners at diverse skill levels.

Mixture models arise in many regression problems, but most methods have seen limited adoption partly due to these algorithms' highly-tailored and model-specific nature. On the other hand, transformers are flexible, neural sequence models that present the intriguing possibility of providing general-purpose prediction methods, even in this mixture setting. In this work, we investigate the hypothesis that transformers can learn an optimal predictor for mixtures of regressions. We construct a generative process for a mixture of linear regressions for which the decision-theoretic optimal procedure is given by data-driven exponential weights on a finite set of parameters. We observe that transformers achieve low mean-squared error on data generated via this process. By probing the transformer's output at inference time, we also show that transformers typically make predictions that are close to the optimal predictor. Our experiments also demonstrate that transformers can learn mixtures of regressions in a sample-efficient fashion and are somewhat robust to distribution shifts. We complement our experimental observations by proving constructively that the decision-theoretic optimal procedure is indeed implementable by a transformer.

Large Language Models (LLMs) are thought to struggle with arithmetic learning due to the inherent differences between language modeling and numerical computation, but concrete evidence has been lacking. This work responds to this claim through a two-side experiment. We first investigate whether LLMs leverage partial products during arithmetic learning. We find that although LLMs can identify some partial products after learning, they fail to leverage them for arithmetic tasks, conversely. We then explore how LLMs approach arithmetic symbolically by breaking tasks into subgroups, hypothesizing that difficulties arise from subgroup complexity and selection. Our results show that when subgroup complexity is fixed, LLMs treat a collection of different arithmetic operations similarly. By analyzing position-level accuracy across different training sizes, we further observe that it follows a U-shaped pattern: LLMs quickly learn the easiest patterns at the first and last positions, while progressively learning the more difficult patterns in the middle positions. This suggests that LLMs select subgroup following an easy-to-hard paradigm during learning. Our work confirms that LLMs are pure symbolic learners in arithmetic tasks and underscores the importance of understanding them deeply through subgroup-level quantification.

We study the process through which reasoning models trained with reinforcement learning on verifiable rewards (RLVR) can learn to solve new problems. We find that RLVR drives performance in two main ways: (1) by compressing pass@k into pass@1 and (2) via "capability gain" in which models learn to solve new problems that they previously could not solve even at high k. We find that while capability gain exists across model scales, learning to solve new problems is primarily driven through self-distillation. We demonstrate these findings across model scales ranging from 0.5B to 72B parameters on >500,000 reasoning problems with prompts and verifiable final answers across math, science, and code domains. We further show that we can significantly improve pass@k rates by leveraging natural language guidance for the model to consider within context while still requiring the model to derive a solution chain from scratch. Based of these insights, we derive Guide -- a new class of online training algorithms. Guide adaptively incorporates hints into the model's context on problems for which all rollouts were initially incorrect and adjusts the importance sampling ratio for the "off-policy" trajectories in order to optimize the policy for contexts in which the hints are no longer present. We describe variants of Guide for GRPO and PPO and empirically show that Guide-GRPO on 7B and 32B parameter models improves generalization over its vanilla counterpart with up to 4% macro-average improvement across math benchmarks. We include careful ablations to analyze Guide's components and theoretically analyze Guide's learning efficiency.

Zero-shot learning (ZSL) which aims at predicting classes that have never appeared during the training using external knowledge (a.k.a. side information) has been widely investigated. In this paper we present a literature review towards ZSL in the perspective of external knowledge, where we categorize the external knowledge, review their methods and compare different external knowledge. With the literature review, we further discuss and outlook the role of symbolic knowledge in addressing ZSL and other machine learning sample shortage issues.

There is growing evidence that pretraining on high quality, carefully thought-out tokens such as code or mathematics plays an important role in improving the reasoning abilities of large language models. For example, Minerva, a PaLM model finetuned on billions of tokens of mathematical documents from arXiv and the web, reported dramatically improved performance on problems that require quantitative reasoning. However, because all known open source web datasets employ preprocessing that does not faithfully preserve mathematical notation, the benefits of large scale training on quantitive web documents are unavailable to the research community. We introduce OpenWebMath, an open dataset inspired by these works containing 14.7B tokens of mathematical webpages from Common Crawl. We describe in detail our method for extracting text and LaTeX content and removing boilerplate from HTML documents, as well as our methods for quality filtering and deduplication. Additionally, we run small-scale experiments by training 1.4B parameter language models on OpenWebMath, showing that models trained on 14.7B tokens of our dataset surpass the performance of models trained on over 20x the amount of general language data. We hope that our dataset, openly released on the Hugging Face Hub, will help spur advances in the reasoning abilities of large language models.

This paper formalizes an emerging learning paradigm that uses a trained model as a reference to guide and enhance the training of a target model through strategic data selection or weighting, named model steering. While ad-hoc methods have been used in various contexts, including the training of large foundation models, its underlying principles remain insufficiently understood, leading to sub-optimal performance. In this work, we propose a theory-driven framework for model steering called DRRho risk minimization, which is rooted in Distributionally Robust Optimization (DRO). Through a generalization analysis, we provide theoretical insights into why this approach improves generalization and data efficiency compared to training without a reference model. To the best of our knowledge, this is the first time such theoretical insights are provided for the new learning paradigm, which significantly enhance our understanding and practice of model steering. Building on these insights and the connection between contrastive learning and DRO, we introduce a novel method for Contrastive Language-Image Pretraining (CLIP) with a reference model, termed DRRho-CLIP. Extensive experiments validate the theoretical insights, reveal a superior scaling law compared to CLIP without a reference model, and demonstrate its strength over existing heuristic approaches.

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

Despite the success of deep neural networks, there are still many challenges in deep representation learning due to the data scarcity issues such as data imbalance, unseen distribution, and domain shift. To address the above-mentioned issues, a variety of methods have been devised to explore the sample relationships in a vanilla way (i.e., from the perspectives of either the input or the loss function), failing to explore the internal structure of deep neural networks for learning with sample relationships. Inspired by this, we propose to enable deep neural networks themselves with the ability to learn the sample relationships from each mini-batch. Specifically, we introduce a batch transformer module or BatchFormer, which is then applied into the batch dimension of each mini-batch to implicitly explore sample relationships during training. By doing this, the proposed method enables the collaboration of different samples, e.g., the head-class samples can also contribute to the learning of the tail classes for long-tailed recognition. Furthermore, to mitigate the gap between training and testing, we share the classifier between with or without the BatchFormer during training, which can thus be removed during testing. We perform extensive experiments on over ten datasets and the proposed method achieves significant improvements on different data scarcity applications without any bells and whistles, including the tasks of long-tailed recognition, compositional zero-shot learning, domain generalization, and contrastive learning. Code will be made publicly available at https://github.com/zhihou7/BatchFormer.

In-context learning is one of the surprising and useful features of large language models. How it works is an active area of research. Recently, stylized meta-learning-like setups have been devised that train these models on a sequence of input-output pairs (x, f(x)) from a function class using the language modeling loss and observe generalization to unseen functions from the same class. One of the main discoveries in this line of research has been that for several problems such as linear regression, trained transformers learn algorithms for learning functions in context. However, the inductive biases of these models resulting in this behavior are not clearly understood. A model with unlimited training data and compute is a Bayesian predictor: it learns the pretraining distribution. It has been shown that high-capacity transformers mimic the Bayesian predictor for linear regression. In this paper, we show empirical evidence of transformers exhibiting the behavior of this ideal learner across different linear and non-linear function classes. We also extend the previous setups to work in the multitask setting and verify that transformers can do in-context learning in this setup as well and the Bayesian perspective sheds light on this setting also. Finally, via the example of learning Fourier series, we study the inductive bias for in-context learning. We find that in-context learning may or may not have simplicity bias depending on the pretraining data distribution.

The use of large pretrained neural networks to create contextualized word embeddings has drastically improved performance on several natural language processing (NLP) tasks. These computationally expensive models have begun to be applied to domain-specific NLP tasks such as re-hospitalization prediction from clinical notes. This paper demonstrates that using large pretrained models produces excellent results on common learning analytics tasks. Pre-training deep language models using student forum data from a wide array of online courses improves performance beyond the state of the art on three text classification tasks. We also show that a smaller, distilled version of our model produces the best results on two of the three tasks while limiting computational cost. We make both models available to the research community at large.

Despite achieving state-of-the-art results in nearly all Natural Language Processing applications, fine-tuning Transformer-based language models still requires a significant amount of labeled data to work. A well known technique to reduce the amount of human effort in acquiring a labeled dataset is Active Learning (AL): an iterative process in which only the minimal amount of samples is labeled. AL strategies require access to a quantified confidence measure of the model predictions. A common choice is the softmax activation function for the final layer. As the softmax function provides misleading probabilities, this paper compares eight alternatives on seven datasets. Our almost paradoxical finding is that most of the methods are too good at identifying the true most uncertain samples (outliers), and that labeling therefore exclusively outliers results in worse performance. As a heuristic we propose to systematically ignore samples, which results in improvements of various methods compared to the softmax function.

Advanced applied mathematics problems are underrepresented in existing Large Language Model (LLM) benchmark datasets. To address this, we introduce HARDMath, a dataset inspired by a graduate course on asymptotic methods, featuring challenging applied mathematics problems that require analytical approximation techniques. These problems demand a combination of mathematical reasoning, computational tools, and subjective judgment, making them difficult for LLMs. Our framework auto-generates a large number of problems with solutions validated against numerical ground truths. We evaluate both open- and closed-source LLMs on HARDMath-mini, a sub-sampled test set of 366 problems, as well as on 40 word problems formulated in applied science contexts. Even leading closed-source models like GPT-4 achieve only 43.8% overall accuracy with few-shot Chain-of-Thought prompting, and all models demonstrate significantly lower performance compared to results on existing mathematics benchmark datasets. We additionally conduct a detailed error analysis to gain insights into the failure cases of LLMs. These results demonstrate limitations of current LLM performance on advanced graduate-level applied math problems and underscore the importance of datasets like HARDMath to advance mathematical abilities of LLMs.

Traditional machine learning systems are deployed under the closed-world setting, which requires the entire training data before the offline training process. However, real-world applications often face the incoming new classes, and a model should incorporate them continually. The learning paradigm is called Class-Incremental Learning (CIL). We propose a Python toolbox that implements several key algorithms for class-incremental learning to ease the burden of researchers in the machine learning community. The toolbox contains implementations of a number of founding works of CIL such as EWC and iCaRL, but also provides current state-of-the-art algorithms that can be used for conducting novel fundamental research. This toolbox, named PyCIL for Python Class-Incremental Learning, is available at https://github.com/G-U-N/PyCIL

Large Language Models (LLMs) have highlighted the necessity of effective unlearning mechanisms to comply with data regulations and ethical AI practices. LLM unlearning aims at removing undesired data influences and associated model capabilities without compromising utility out of the scope of unlearning. While interest in studying LLM unlearning is growing,the impact of the optimizer choice for LLM unlearning remains under-explored. In this work, we shed light on the significance of optimizer selection in LLM unlearning for the first time, establishing a clear connection between {second-order optimization} and influence unlearning (a classical approach using influence functions to update the model for data influence removal). This insight propels us to develop a second-order unlearning framework, termed SOUL, built upon the second-order clipped stochastic optimization (Sophia)-based LLM training method. SOUL extends the static, one-shot model update using influence unlearning to a dynamic, iterative unlearning process. Our extensive experiments show that SOUL consistently outperforms conventional first-order methods across various unlearning tasks, models, and metrics, suggesting the promise of second-order optimization in providing a scalable and easily implementable solution for LLM unlearning.

Humanities scholars commonly provide evidence for claims that they make about a work of literature (e.g., a novel) in the form of quotations from the work. We collect a large-scale dataset (RELiC) of 78K literary quotations and surrounding critical analysis and use it to formulate the novel task of literary evidence retrieval, in which models are given an excerpt of literary analysis surrounding a masked quotation and asked to retrieve the quoted passage from the set of all passages in the work. Solving this retrieval task requires a deep understanding of complex literary and linguistic phenomena, which proves challenging to methods that overwhelmingly rely on lexical and semantic similarity matching. We implement a RoBERTa-based dense passage retriever for this task that outperforms existing pretrained information retrieval baselines; however, experiments and analysis by human domain experts indicate that there is substantial room for improvement over our dense retriever.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps R to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between R^{C-1} and the projected probability simplex Delta^{C-1} by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.

We present a fundamental discovery that challenges our understanding of how complex reasoning emerges in large language models. While conventional wisdom suggests that sophisticated reasoning tasks demand extensive training data (>100,000 examples), we demonstrate that complex mathematical reasoning abilities can be effectively elicited with surprisingly few examples. Through comprehensive experiments, our proposed model LIMO demonstrates unprecedented performance in mathematical reasoning. With merely 817 curated training samples, LIMO achieves 57.1% accuracy on AIME and 94.8% on MATH, improving from previous SFT-based models' 6.5% and 59.2% respectively, while only using 1% of the training data required by previous approaches. LIMO demonstrates exceptional out-of-distribution generalization, achieving 40.5% absolute improvement across 10 diverse benchmarks, outperforming models trained on 100x more data, challenging the notion that SFT leads to memorization rather than generalization. Based on these results, we propose the Less-Is-More Reasoning Hypothesis (LIMO Hypothesis): In foundation models where domain knowledge has been comprehensively encoded during pre-training, sophisticated reasoning capabilities can emerge through minimal but precisely orchestrated demonstrations of cognitive processes. This hypothesis posits that the elicitation threshold for complex reasoning is determined by two key factors: (1) the completeness of the model's encoded knowledge foundation during pre-training, and (2) the effectiveness of post-training examples as "cognitive templates" that show the model how to utilize its knowledge base to solve complex reasoning tasks. To facilitate reproducibility and future research in data-efficient reasoning, we release LIMO as a comprehensive open-source suite at https://github.com/GAIR-NLP/LIMO.

This paper proposes a method for the automatic creation of variables (in the case of regression) that complement the information contained in the initial input vector. The method works as a pre-processing step in which the continuous values of the variable to be regressed are discretized into a set of intervals which are then used to define value thresholds. Then classifiers are trained to predict whether the value to be regressed is less than or equal to each of these thresholds. The different outputs of the classifiers are then concatenated in the form of an additional vector of variables that enriches the initial vector of the regression problem. The implemented system can thus be considered as a generic pre-processing tool. We tested the proposed enrichment method with 5 types of regressors and evaluated it in 33 regression datasets. Our experimental results confirm the interest of the approach.

Reasoning abilities, especially those for solving complex math problems, are crucial components of general intelligence. Recent advances by proprietary companies, such as o-series models of OpenAI, have made remarkable progress on reasoning tasks. However, the complete technical details remain unrevealed, and the techniques that are believed certainly to be adopted are only reinforcement learning (RL) and the long chain of thoughts. This paper proposes a new RL framework, termed OREAL, to pursue the performance limit that can be achieved through Outcome REwArd-based reinforcement Learning for mathematical reasoning tasks, where only binary outcome rewards are easily accessible. We theoretically prove that behavior cloning on positive trajectories from best-of-N (BoN) sampling is sufficient to learn the KL-regularized optimal policy in binary feedback environments. This formulation further implies that the rewards of negative samples should be reshaped to ensure the gradient consistency between positive and negative samples. To alleviate the long-existing difficulties brought by sparse rewards in RL, which are even exacerbated by the partial correctness of the long chain of thought for reasoning tasks, we further apply a token-level reward model to sample important tokens in reasoning trajectories for learning. With OREAL, for the first time, a 7B model can obtain 94.0 pass@1 accuracy on MATH-500 through RL, being on par with 32B models. OREAL-32B also surpasses previous 32B models trained by distillation with 95.0 pass@1 accuracy on MATH-500. Our investigation also indicates the importance of initial policy models and training queries for RL. Code, models, and data will be released to benefit future researchhttps://github.com/InternLM/OREAL.

In this paper, we introduce the idea of decomposing the residuals of regression with respect to the data instances instead of features. This allows us to determine the effects of each individual instance on the model and each other, and in doing so makes for a model-agnostic method of identifying instances of interest. In doing so, we can also determine the appropriateness of the model and data in the wider context of a given study. The paper focuses on the possible applications that such a framework brings to the relatively unexplored field of instance analysis in the context of Explainable AI tasks.

With the increasing numbers of publicly available models, there are probably pretrained, online models for most tasks users require. However, current model search methods are rudimentary, essentially a text-based search in the documentation, thus users cannot find the relevant models. This paper presents ProbeLog, a method for retrieving classification models that can recognize a target concept, such as "Dog", without access to model metadata or training data. Differently from previous probing methods, ProbeLog computes a descriptor for each output dimension (logit) of each model, by observing its responses on a fixed set of inputs (probes). Our method supports both logit-based retrieval ("find more logits like this") and zero-shot, text-based retrieval ("find all logits corresponding to dogs"). As probing-based representations require multiple costly feedforward passes through the model, we develop a method, based on collaborative filtering, that reduces the cost of encoding repositories by 3x. We demonstrate that ProbeLog achieves high retrieval accuracy, both in real-world and fine-grained search tasks and is scalable to full-size repositories.

Multimodal large language models (MLLMs) have demonstrated remarkable capabilities in vision-language answering tasks. Despite their strengths, these models often encounter challenges in achieving complex reasoning tasks such as mathematical problem-solving. Previous works have focused on fine-tuning on specialized mathematical datasets. However, these datasets are typically distilled directly from teacher models, which capture only static reasoning patterns and leaving substantial gaps compared to student models. This reliance on fixed teacher-derived datasets not only restricts the model's ability to adapt to novel or more intricate questions that extend beyond the confines of the training data, but also lacks the iterative depth needed for robust generalization. To overcome these limitations, we propose \method, a Mathematical Self-Evolving framework for MLLMs. In contrast to traditional one-shot fine-tuning paradigms, \method iteratively refines the model through cycles of inference, reflection, and reward-based feedback. Specifically, we leverage iterative fine-tuning by incorporating correct reasoning paths derived from previous-stage inference and integrating reflections from a specialized Outcome Reward Model (ORM). To verify the effectiveness of \method, we evaluate it on a suite of challenging benchmarks, demonstrating significant performance gains over backbone models. Notably, our experimental results on MathVL-test surpass the leading open-source multimodal mathematical reasoning model QVQ. Our code and models are available at https://zheny2751\allowbreak-dotcom.github.io/\allowbreak MathSE.github.io/.

While the successes of transformers across many domains are indisputable, accurate understanding of the learning mechanics is still largely lacking. Their capabilities have been probed on benchmarks which include a variety of structured and reasoning tasks -- but mathematical understanding is lagging substantially behind. Recent lines of work have begun studying representational aspects of this question: that is, the size/depth/complexity of attention-based networks to perform certain tasks. However, there is no guarantee the learning dynamics will converge to the constructions proposed. In our paper, we provide fine-grained mechanistic understanding of how transformers learn "semantic structure", understood as capturing co-occurrence structure of words. Precisely, we show, through a combination of experiments on synthetic data modeled by Latent Dirichlet Allocation (LDA), Wikipedia data, and mathematical analysis that the embedding layer and the self-attention layer encode the topical structure. In the former case, this manifests as higher average inner product of embeddings between same-topic words. In the latter, it manifests as higher average pairwise attention between same-topic words. The mathematical results involve several assumptions to make the analysis tractable, which we verify on data, and might be of independent interest as well.

Mathematical programming -- the task of expressing operations and decision-making problems in precise mathematical language -- is fundamental across domains, yet remains a skill-intensive process requiring operations research expertise. Recent advances in large language models for complex reasoning have spurred interest in automating this task, translating natural language into executable optimization models. Current approaches, however, achieve limited accuracy, hindered by scarce and noisy training data without leveraging domain knowledge. In this work, we systematically integrate optimization expertise to improve formulation accuracy for mixed-integer linear programming, a key family of mathematical programs. Our approach first cleans training data through class-based error analysis to explicitly prevent common mistakes within each optimization class. We then develop multi-turn inference strategies that guide LLMs with class-specific error summaries and solver feedback, enabling iterative refinement. Experiments across multiple base LLMs demonstrate that combining cleaned data with domain-informed prompting and feedback improves formulation accuracy by 14 percentage points on average, enabling further progress toward robust LLM-assisted optimization formulation.

The key components of machine learning are data samples for training, model for learning patterns, and loss function for optimizing accuracy. Analogously, unlearning can potentially be achieved through anti-data samples (or anti-samples), unlearning method, and reversed loss function. While prior research has explored unlearning methods and reversed loss functions, the potential of anti-samples remains largely untapped. In this paper, we introduce UnSTAR: Unlearning with Self-Taught Anti-Sample Reasoning for large language models (LLMs). Our contributions are threefold; first, we propose a novel concept of anti-sample-induced unlearning; second, we generate anti-samples by leveraging misleading rationales, which help reverse learned associations and accelerate the unlearning process; and third, we enable fine-grained targeted unlearning, allowing for the selective removal of specific associations without impacting related knowledge - something not achievable by previous works. Results demonstrate that anti-samples offer an efficient, targeted unlearning strategy for LLMs, opening new avenues for privacy-preserving machine learning and model modification.

Meta learning have achieved promising performance in low-resource text classification which aims to identify target classes with knowledge transferred from source classes with sets of small tasks named episodes. However, due to the limited training data in the meta-learning scenario and the inherent properties of parameterized neural networks, poor generalization performance has become a pressing problem that needs to be addressed. To deal with this issue, we propose a meta-learning based method called Retrieval-Augmented Meta Learning(RAML). It not only uses parameterization for inference but also retrieves non-parametric knowledge from an external corpus to make inferences, which greatly alleviates the problem of poor generalization performance caused by the lack of diverse training data in meta-learning. This method differs from previous models that solely rely on parameters, as it explicitly emphasizes the importance of non-parametric knowledge, aiming to strike a balance between parameterized neural networks and non-parametric knowledge. The model is required to determine which knowledge to access and utilize during inference. Additionally, our multi-view passages fusion network module can effectively and efficiently integrate the retrieved information into low-resource classification task. The extensive experiments demonstrate that RAML significantly outperforms current SOTA low-resource text classification models.

Recent research has shown that smaller language models can acquire substantial reasoning abilities when fine-tuned with reasoning exemplars crafted by a significantly larger teacher model. We explore this paradigm for the financial domain, focusing on the challenge of answering questions that require multi-hop numerical reasoning over financial texts. We assess the performance of several smaller models that have been fine-tuned to generate programs that encode the required financial reasoning and calculations. Our findings demonstrate that these fine-tuned smaller models approach the performance of the teacher model. To provide a granular analysis of model performance, we propose an approach to investigate the specific student model capabilities that are enhanced by fine-tuning. Our empirical analysis indicates that fine-tuning refines the student models ability to express and apply the required financial concepts along with adapting the entity extraction for the specific data format. In addition, we hypothesize and demonstrate that comparable financial reasoning capability can be induced using relatively smaller datasets.

Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are formalized by categories of predictor and target spaces, whose morphisms represent the algorithm's invariances, and an index category whose morphisms represent permutations of the training examples. An invariant learning algorithm is a natural transformation between two functors from the product of these categories to the category of sets, representing training datasets and learned functions respectively. We illustrate the framework by characterizing and contrasting the invariances of linear regression and ridge regression.

Enhancing the intelligibility and interpretability of machine learning is a crucial task in responding to the demand for Explicability as an AI principle, and in promoting the better social implementation of AI. The aim of our research is to contribute to this improvement by reformulating machine learning models through the lens of category theory, thereby developing a semantic framework for structuring and understanding AI systems. Our categorical modeling in this paper clarifies and formalizes the structural interplay between residuals and parameters in supervised learning. The present paper focuses on the multiple linear regression model, which represents the most basic form of supervised learning. By defining two concrete categories corresponding to parameters and data, along with an adjoint pair of functors between them, we introduce our categorical formulation of supervised learning. We show that the essential structure of this framework is captured by what we call the Gauss-Markov Adjunction. Within this setting, the dual flow of information can be explicitly described as a correspondence between variations in parameters and residuals. The ordinary least squares estimator for the parameters and the minimum residual are related via the preservation of limits by the right adjoint functor. Furthermore, we position this formulation as an instance of extended denotational semantics for supervised learning, and propose applying a semantic perspective developed in theoretical computer science as a formal foundation for Explicability in AI.

We propose sparsemax, a new activation function similar to the traditional softmax, but able to output sparse probabilities. After deriving its properties, we show how its Jacobian can be efficiently computed, enabling its use in a network trained with backpropagation. Then, we propose a new smooth and convex loss function which is the sparsemax analogue of the logistic loss. We reveal an unexpected connection between this new loss and the Huber classification loss. We obtain promising empirical results in multi-label classification problems and in attention-based neural networks for natural language inference. For the latter, we achieve a similar performance as the traditional softmax, but with a selective, more compact, attention focus.

To improve language models' proficiency in mathematical reasoning via continual pretraining, we introduce a novel strategy that leverages base language models for autonomous data selection. Departing from conventional supervised fine-tuning or trained classifiers with human-annotated data, our approach utilizes meta-prompted language models as zero-shot verifiers to autonomously evaluate and select high-quality mathematical content, and we release the curated open-source AutoMathText dataset encompassing over 200GB of data. To demonstrate the efficacy of our method, we continuously pretrained a 7B-parameter Mistral language model on the AutoMathText dataset, achieving substantial improvements in downstream performance on the MATH dataset with a token amount reduced by orders of magnitude compared to previous continuous pretraining works. Our method showcases a 2 times increase in pretraining token efficiency compared to baselines, underscoring the potential of our approach in enhancing models' mathematical reasoning capabilities. The AutoMathText dataset is available at https://huggingface.co/datasets/math-ai/AutoMathText. The code is available at https://github.com/yifanzhang-pro/AutoMathText.

No free lunch theorems for supervised learning state that no learner can solve all problems or that all learners achieve exactly the same accuracy on average over a uniform distribution on learning problems. Accordingly, these theorems are often referenced in support of the notion that individual problems require specially tailored inductive biases. While virtually all uniformly sampled datasets have high complexity, real-world problems disproportionately generate low-complexity data, and we argue that neural network models share this same preference, formalized using Kolmogorov complexity. Notably, we show that architectures designed for a particular domain, such as computer vision, can compress datasets on a variety of seemingly unrelated domains. Our experiments show that pre-trained and even randomly initialized language models prefer to generate low-complexity sequences. Whereas no free lunch theorems seemingly indicate that individual problems require specialized learners, we explain how tasks that often require human intervention such as picking an appropriately sized model when labeled data is scarce or plentiful can be automated into a single learning algorithm. These observations justify the trend in deep learning of unifying seemingly disparate problems with an increasingly small set of machine learning models.

Models of language trained on very large corpora have been demonstrated useful for NLP. As fixed artifacts, they have become the object of intense study, with many researchers "probing" the extent to which linguistic abstractions, factual and commonsense knowledge, and reasoning abilities they acquire and readily demonstrate. Building on this line of work, we consider a new question: for types of knowledge a language model learns, when during (pre)training are they acquired? We plot probing performance across iterations, using RoBERTa as a case study. Among our findings: linguistic knowledge is acquired fast, stably, and robustly across domains. Facts and commonsense are slower and more domain-sensitive. Reasoning abilities are, in general, not stably acquired. As new datasets, pretraining protocols, and probes emerge, we believe that probing-across-time analyses can help researchers understand the complex, intermingled learning that these models undergo and guide us toward more efficient approaches that accomplish necessary learning faster.

The goal of continual learning is to find a model that solves multiple learning tasks which are presented sequentially to the learner. A key challenge in this setting is that the learner may forget how to solve a previous task when learning a new task, a phenomenon known as catastrophic forgetting. To address this challenge, many practical methods have been proposed, including memory-based, regularization-based, and expansion-based methods. However, a rigorous theoretical understanding of these methods remains elusive. This paper aims to bridge this gap between theory and practice by proposing a new continual learning framework called Ideal Continual Learner (ICL), which is guaranteed to avoid catastrophic forgetting by construction. We show that ICL unifies multiple well-established continual learning methods and gives new theoretical insights into the strengths and weaknesses of these methods. We also derive generalization bounds for ICL which allow us to theoretically quantify how rehearsal affects generalization. Finally, we connect ICL to several classic subjects and research topics of modern interest, which allows us to make historical remarks and inspire future directions.

Despite being the standard loss function to train multi-class neural networks, the log-softmax has two potential limitations. First, it involves computations that scale linearly with the number of output classes, which can restrict the size of problems we are able to tackle with current hardware. Second, it remains unclear how close it matches the task loss such as the top-k error rate or other non-differentiable evaluation metrics which we aim to optimize ultimately. In this paper, we introduce an alternative classification loss function, the Z-loss, which is designed to address these two issues. Unlike the log-softmax, it has the desirable property of belonging to the spherical loss family (Vincent et al., 2015), a class of loss functions for which training can be performed very efficiently with a complexity independent of the number of output classes. We show experimentally that it significantly outperforms the other spherical loss functions previously investigated. Furthermore, we show on a word language modeling task that it also outperforms the log-softmax with respect to certain ranking scores, such as top-k scores, suggesting that the Z-loss has the flexibility to better match the task loss. These qualities thus makes the Z-loss an appealing candidate to train very efficiently large output networks such as word-language models or other extreme classification problems. On the One Billion Word (Chelba et al., 2014) dataset, we are able to train a model with the Z-loss 40 times faster than the log-softmax and more than 4 times faster than the hierarchical softmax.

Leveraging mathematical Large Language Models (LLMs) for proof generation is a fundamental topic in LLMs research. We argue that the ability of current LLMs to prove statements largely depends on whether they have encountered the relevant proof process during training. This reliance limits their deeper understanding of mathematical theorems and related concepts. Inspired by the pedagogical method of "proof by counterexamples" commonly used in human mathematics education, our work aims to enhance LLMs' ability to conduct mathematical reasoning and proof through counterexamples. Specifically, we manually create a high-quality, university-level mathematical benchmark, CounterMATH, which requires LLMs to prove mathematical statements by providing counterexamples, thereby assessing their grasp of mathematical concepts. Additionally, we develop a data engineering framework to automatically obtain training data for further model improvement. Extensive experiments and detailed analyses demonstrate that CounterMATH is challenging, indicating that LLMs, such as OpenAI o1, have insufficient counterexample-driven proof capabilities. Moreover, our exploration into model training reveals that strengthening LLMs' counterexample-driven conceptual reasoning abilities is crucial for improving their overall mathematical capabilities. We believe that our work offers new perspectives on the community of mathematical LLMs.

Traditional language model-based theorem proving assumes that by training on a sufficient amount of formal proof data, a model will learn to prove theorems. Our key observation is that a wealth of informal information that is not present in formal proofs can be useful for learning to prove theorems. For instance, humans think through steps of a proof, but this thought process is not visible in the resulting code. We present Lean-STaR, a framework for training language models to produce informal thoughts prior to each step of a proof, thereby boosting the model's theorem-proving capabilities. Lean-STaR uses retrospective ground-truth tactics to generate synthetic thoughts for training the language model. At inference time, the trained model directly generates the thoughts prior to the prediction of the tactics in each proof step. Building on the self-taught reasoner framework, we then apply expert iteration to further fine-tune the model on the correct proofs it samples and verifies using the Lean solver. Lean-STaR achieves state-of-the-art results on the miniF2F-test benchmark within the Lean theorem proving environment, significantly outperforming base models (43.4% rightarrow 46.3%, Pass@64). We also analyze the impact of the augmented thoughts on various aspects of the theorem proving process, providing insights into their effectiveness.

Active Learning (AL) has been a powerful paradigm for improving model efficiency and performance by selecting the most informative data points for labeling and training. In recent active learning frameworks, Large Language Models (LLMs) have been employed not only for selection but also for generating entirely new data instances and providing more cost-effective annotations. Motivated by the increasing importance of high-quality data and efficient model training in the era of LLMs, we present a comprehensive survey on LLM-based Active Learning. We introduce an intuitive taxonomy that categorizes these techniques and discuss the transformative roles LLMs can play in the active learning loop. We further examine the impact of AL on LLM learning paradigms and its applications across various domains. Finally, we identify open challenges and propose future research directions. This survey aims to serve as an up-to-date resource for researchers and practitioners seeking to gain an intuitive understanding of LLM-based AL techniques and deploy them to new applications.

1. The popularity of Machine learning (ML), Deep learning (DL), and Artificial intelligence (AI) has risen sharply in recent years. Despite this spike in popularity, the inner workings of ML and DL algorithms are often perceived as opaque, and their relationship to classical data analysis tools remains debated. 2. Although it is often assumed that ML and DL excel primarily at making predictions, ML and DL can also be used for analytical tasks traditionally addressed with statistical models. Moreover, most recent discussions and reviews on ML focus mainly on DL, missing out on synthesizing the wealth of ML algorithms with different advantages and general principles. 3. Here, we provide a comprehensive overview of the field of ML and DL, starting by summarizing its historical developments, existing algorithm families, differences to traditional statistical tools, and universal ML principles. We then discuss why and when ML and DL models excel at prediction tasks and where they could offer alternatives to traditional statistical methods for inference, highlighting current and emerging applications for ecological problems. Finally, we summarize emerging trends such as scientific and causal ML, explainable AI, and responsible AI that may significantly impact ecological data analysis in the future. 4. We conclude that ML and DL are powerful new tools for predictive modeling and data analysis. The superior performance of ML and DL algorithms compared to statistical models can be explained by their higher flexibility and automatic data-dependent complexity optimization. However, their use for causal inference is still disputed as the focus of ML and DL methods on predictions creates challenges for the interpretation of these models. Nevertheless, we expect ML and DL to become an indispensable tool in E&E, comparable to other traditional statistical tools.

Contrastive learning -- a modern approach to extract useful representations from unlabeled data by training models to distinguish similar samples from dissimilar ones -- has driven significant progress in foundation models. In this work, we develop a new theoretical framework for analyzing data augmentation-based contrastive learning, with a focus on SimCLR as a representative example. Our approach is based on the concept of approximate sufficient statistics, which we extend beyond its original definition in oko2025statistical for contrastive language-image pretraining (CLIP) using KL-divergence. We generalize it to equivalent forms and general f-divergences, and show that minimizing SimCLR and other contrastive losses yields encoders that are approximately sufficient. Furthermore, we demonstrate that these near-sufficient encoders can be effectively adapted to downstream regression and classification tasks, with performance depending on their sufficiency and the error induced by data augmentation in contrastive learning. Concrete examples in linear regression and topic classification are provided to illustrate the broad applicability of our results.

AI holds promise for transforming scientific processes, including hypothesis generation. Prior work on hypothesis generation can be broadly categorized into theory-driven and data-driven approaches. While both have proven effective in generating novel and plausible hypotheses, it remains an open question whether they can complement each other. To address this, we develop the first method that combines literature-based insights with data to perform LLM-powered hypothesis generation. We apply our method on five different datasets and demonstrate that integrating literature and data outperforms other baselines (8.97\% over few-shot, 15.75\% over literature-based alone, and 3.37\% over data-driven alone). Additionally, we conduct the first human evaluation to assess the utility of LLM-generated hypotheses in assisting human decision-making on two challenging tasks: deception detection and AI generated content detection. Our results show that human accuracy improves significantly by 7.44\% and 14.19\% on these tasks, respectively. These findings suggest that integrating literature-based and data-driven approaches provides a comprehensive and nuanced framework for hypothesis generation and could open new avenues for scientific inquiry.

Language models typically need to be trained or finetuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this work, we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate kNN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. On benchmarks including code and mathematics, we find that the model is capable of making use of newly defined functions and theorems during test time.

With the rise of large language models (LLMs) for flexibly processing information as strings, a natural application is regression, specifically by preprocessing string representations into LLM embeddings as downstream features for metric prediction. In this paper, we provide one of the first comprehensive investigations into embedding-based regression and demonstrate that LLM embeddings as features can be better for high-dimensional regression tasks than using traditional feature engineering. This regression performance can be explained in part due to LLM embeddings over numeric data inherently preserving Lipschitz continuity over the feature space. Furthermore, we quantify the contribution of different model effects, most notably model size and language understanding, which we find surprisingly do not always improve regression performance.

This paper investigates Who's Harry Potter (WHP), a pioneering yet insufficiently understood method for LLM unlearning. We explore it in two steps. First, we introduce a new task of LLM targeted unlearning, where given an unlearning target (e.g., a person) and some unlearning documents, we aim to unlearn only the information about the target, rather than everything in the unlearning documents. We further argue that a successful unlearning should satisfy criteria such as not outputting gibberish, not fabricating facts about the unlearning target, and not releasing factual information under jailbreak attacks. Second, we construct a causal intervention framework for targeted unlearning, where the knowledge of the unlearning target is modeled as a confounder between LLM input and output, and the unlearning process as a deconfounding process. This framework justifies and extends WHP, deriving a simple unlearning algorithm that includes WHP as a special case. Experiments on existing and new datasets show that our approach, without explicitly optimizing for the aforementioned criteria, achieves competitive performance in all of them. Our code is available at https://github.com/UCSB-NLP-Chang/causal_unlearn.git.

State of the art Symbolic Regression (SR) methods currently build specialized models, while the application of Large Language Models (LLMs) remains largely unexplored. In this work, we introduce the first comprehensive framework that utilizes LLMs for the task of SR. We propose In-Context Symbolic Regression (ICSR), an SR method which iteratively refines a functional form with an LLM and determines its coefficients with an external optimizer. ICSR leverages LLMs' strong mathematical prior both to propose an initial set of possible functions given the observations and to refine them based on their errors. Our findings reveal that LLMs are able to successfully find symbolic equations that fit the given data, matching or outperforming the overall performance of the best SR baselines on four popular benchmarks, while yielding simpler equations with better out of distribution generalization.

Reinforcement learning with AI feedback (RLAIF) is a popular paradigm for improving the instruction-following abilities of powerful pre-trained language models. RLAIF first performs supervised fine-tuning (SFT) using demonstrations from a teacher model and then further fine-tunes the model with reinforcement learning (RL), using feedback from a critic model. While recent popular open-source models have demonstrated substantial improvements in performance from the RL step, in this paper we question whether the complexity of this RL step is truly warranted for AI feedback. We show that the improvements of the RL step are virtually entirely due to the widespread practice of using a weaker teacher model (e.g. GPT-3.5) for SFT data collection than the critic (e.g., GPT-4) used for AI feedback generation. Specifically, we show that simple supervised fine-tuning with GPT-4 as the teacher outperforms existing RLAIF pipelines. More generally, we find that the gains from RLAIF vary substantially across base model families, test-time evaluation protocols, and critic models. Finally, we provide a mechanistic explanation for when SFT may outperform the full two-step RLAIF pipeline as well as suggestions for making RLAIF maximally useful in practice.

As language models become more powerful, training and evaluation are increasingly bottlenecked by the data and metrics used for a particular task. For example, summarization models are often trained to predict human reference summaries and evaluated using ROUGE, but both of these metrics are rough proxies for what we really care about -- summary quality. In this work, we show that it is possible to significantly improve summary quality by training a model to optimize for human preferences. We collect a large, high-quality dataset of human comparisons between summaries, train a model to predict the human-preferred summary, and use that model as a reward function to fine-tune a summarization policy using reinforcement learning. We apply our method to a version of the TL;DR dataset of Reddit posts and find that our models significantly outperform both human reference summaries and much larger models fine-tuned with supervised learning alone. Our models also transfer to CNN/DM news articles, producing summaries nearly as good as the human reference without any news-specific fine-tuning. We conduct extensive analyses to understand our human feedback dataset and fine-tuned models We establish that our reward model generalizes to new datasets, and that optimizing our reward model results in better summaries than optimizing ROUGE according to humans. We hope the evidence from our paper motivates machine learning researchers to pay closer attention to how their training loss affects the model behavior they actually want.

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.

Understanding the internal representations learned by neural networks is a cornerstone challenge in the science of machine learning. While there have been significant recent strides in some cases towards understanding how neural networks implement specific target functions, this paper explores a complementary question -- why do networks arrive at particular computational strategies? Our inquiry focuses on the algebraic learning tasks of modular addition, sparse parities, and finite group operations. Our primary theoretical findings analytically characterize the features learned by stylized neural networks for these algebraic tasks. Notably, our main technique demonstrates how the principle of margin maximization alone can be used to fully specify the features learned by the network. Specifically, we prove that the trained networks utilize Fourier features to perform modular addition and employ features corresponding to irreducible group-theoretic representations to perform compositions in general groups, aligning closely with the empirical observations of Nanda et al. and Chughtai et al. More generally, we hope our techniques can help to foster a deeper understanding of why neural networks adopt specific computational strategies.

Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.

A central notion in practical and theoretical machine learning is that of a weak learner, classifiers that achieve better-than-random performance (on any given distribution over data), even by a small margin. Such weak learners form the practical basis for canonical machine learning methods such as boosting. In this work, we illustrate that prompt-based large language models can operate effectively as said weak learners. Specifically, we illustrate the use of a large language model (LLM) as a weak learner in a boosting algorithm applied to tabular data. We show that by providing (properly sampled according to the distribution of interest) text descriptions of tabular data samples, LLMs can produce a summary of the samples that serves as a template for classification and achieves the aim of acting as a weak learner on this task. We incorporate these models into a boosting approach, which in some settings can leverage the knowledge within the LLM to outperform traditional tree-based boosting. The model outperforms both few-shot learning and occasionally even more involved fine-tuning procedures, particularly for tasks involving small numbers of data points. The results illustrate the potential for prompt-based LLMs to function not just as few-shot learners themselves, but as components of larger machine learning pipelines.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for pi, ln(2), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.

Large language models (LLMs) offer promise in generating educational content, providing instructor feedback, and reducing teacher workload on assessments. While prior studies have focused on studying LLM-powered learning analytics, limited research has examined how effective LLMs are in a bilingual context. In this paper, we study the effectiveness of multilingual large language models (MLLMs) across monolingual (English-only, Spanish-only) and bilingual (Spanglish) student writing. We present a learning analytics use case that details LLM performance in assessing acceptable and unacceptable explanations of Science and Social Science concepts. Our findings reveal a significant bias in the grading performance of pre-trained models for bilingual writing compared to English-only and Spanish-only writing. Following this, we fine-tune open-source MLLMs including Llama 3.1 and Mistral NeMo using synthetic datasets generated in English, Spanish, and Spanglish. Our experiments indicate that the models perform significantly better for all three languages after fine-tuning with bilingual data. This study highlights the potential of enhancing MLLM effectiveness to support authentic language practices amongst bilingual learners. It also aims to illustrate the value of incorporating non-English languages into the design and implementation of language models in education.

Active learning promises to alleviate the massive data needs of supervised machine learning: it has successfully improved sample efficiency by an order of magnitude on traditional tasks like topic classification and object recognition. However, we uncover a striking contrast to this promise: across 5 models and 4 datasets on the task of visual question answering, a wide variety of active learning approaches fail to outperform random selection. To understand this discrepancy, we profile 8 active learning methods on a per-example basis, and identify the problem as collective outliers -- groups of examples that active learning methods prefer to acquire but models fail to learn (e.g., questions that ask about text in images or require external knowledge). Through systematic ablation experiments and qualitative visualizations, we verify that collective outliers are a general phenomenon responsible for degrading pool-based active learning. Notably, we show that active learning sample efficiency increases significantly as the number of collective outliers in the active learning pool decreases. We conclude with a discussion and prescriptive recommendations for mitigating the effects of these outliers in future work.

Alpha factor mining is pivotal in quantitative investment for identifying predictive signals from complex financial data. While traditional formulaic alpha mining relies on human expertise, contemporary automated methods, such as those based on genetic programming or reinforcement learning, often struggle with search inefficiency or yield alpha factors that are difficult to interpret. This paper introduces a novel framework that integrates Large Language Models (LLMs) with Monte Carlo Tree Search (MCTS) to overcome these limitations. Our framework leverages the LLM's instruction-following and reasoning capability to iteratively generate and refine symbolic alpha formulas within an MCTS-driven exploration. A key innovation is the guidance of MCTS exploration by rich, quantitative feedback from financial backtesting of each candidate factor, enabling efficient navigation of the vast search space. Furthermore, a frequent subtree avoidance mechanism is introduced to enhance search diversity and prevent formulaic homogenization, further improving performance. Experimental results on real-world stock market data demonstrate that our LLM-based framework outperforms existing methods by mining alphas with superior predictive accuracy and trading performance. The resulting formulas are also more amenable to human interpretation, establishing a more effective and efficient paradigm for formulaic alpha mining.

Mathematical understanding and reasoning are crucial tasks for assessing the capabilities of artificial intelligence (AI). However, existing benchmarks either require just a few steps of reasoning, or only contain a small amount of data in one specific topic, making it hard to analyse AI's behaviour with reference to different problems within a specific topic in detail. In this work, we propose Conic10K, a challenging math problem dataset on conic sections in Chinese senior high school education. Our dataset contains various problems with different reasoning depths, while only the knowledge from conic sections is required. Since the dataset only involves a narrow range of knowledge, it is easy to separately analyse the knowledge a model possesses and the reasoning ability it has. For each problem, we provide a high-quality formal representation, the reasoning steps, and the final solution. Experiments show that existing large language models, including GPT-4, exhibit weak performance on complex reasoning. We hope that our findings could inspire more advanced techniques for precise natural language understanding and reasoning. Our dataset and codes are available at https://github.com/whyNLP/Conic10K.

Even though large language models are becoming increasingly capable, it is still unreasonable to expect them to excel at tasks that are under-represented on the Internet. Leveraging LLMs for specialized applications, particularly in niche programming languages and private domains, remains challenging and largely unsolved. In this work, we address this gap by presenting a comprehensive, open-source approach for adapting LLMs to the Q programming language, a popular tool in quantitative finance that is much less present on the Internet compared to Python, C, Java, and other ``mainstream" languages and is therefore not a strong suit of general-purpose AI models. We introduce a new Leetcode style evaluation dataset for Q, benchmark major frontier models on the dataset, then do pretraining, supervised fine tuning, and reinforcement learning to train a suite of reasoning and non-reasoning models based on the Qwen-2.5 series, spanning five parameter sizes (1.5B, 3B, 7B, 14B, 32B). Our best model achieves a pass@1 accuracy of 59 percent on our Q benchmark, surpassing the best-performing frontier model, Claude Opus-4 by 29.5 percent. Additionally, all models, even our 1.5B model, outperform GPT-4.1 on this task. In addition to releasing models, code, and data, we provide a detailed blueprint for dataset construction, model pretraining, supervised fine-tuning, and reinforcement learning. Our methodology is broadly applicable, and we discuss how these techniques can be extended to other tasks, including those where evaluation may rely on soft or subjective signals.

A fundamental question in modern machine learning is why large, over-parameterized models, such as deep neural networks and transformers, tend to generalize well, even when their number of parameters far exceeds the number of training samples. We investigate this phenomenon through the lens of information theory, grounded in universal learning theory. Specifically, we study a Bayesian mixture learner with log-loss and (almost) uniform prior over an expansive hypothesis class. Our key result shows that the learner's regret is not determined by the overall size of the hypothesis class, but rather by the cumulative probability of all models that are close, in Kullback-Leibler divergence distance, to the true data-generating process. We refer to this cumulative probability as the weight of the hypothesis. This leads to a natural notion of model simplicity: simple models are those with large weight and thus require fewer samples to generalize, while complex models have small weight and need more data. This perspective provides a rigorous and intuitive explanation for why over-parameterized models often avoid overfitting: the presence of simple hypotheses allows the posterior to concentrate on them when supported by the data. We further bridge theory and practice by recalling that stochastic gradient descent with Langevin dynamics samples from the correct posterior distribution, enabling our theoretical learner to be approximated using standard machine learning methods combined with ensemble learning. Our analysis yields non-uniform regret bounds and aligns with key practical concepts such as flat minima and model distillation. The results apply broadly across online, batch, and supervised learning settings, offering a unified and principled understanding of the generalization behavior of modern AI systems.

In order to understand the in-context learning phenomenon, recent works have adopted a stylized experimental framework and demonstrated that Transformers can learn gradient-based learning algorithms for various classes of real-valued functions. However, the limitations of Transformers in implementing learning algorithms, and their ability to learn other forms of algorithms are not well understood. Additionally, the degree to which these capabilities are confined to attention-based models is unclear. Furthermore, it remains to be seen whether the insights derived from these stylized settings can be extrapolated to pretrained Large Language Models (LLMs). In this work, we take a step towards answering these questions by demonstrating the following: (a) On a test-bed with a variety of Boolean function classes, we find that Transformers can nearly match the optimal learning algorithm for 'simpler' tasks, while their performance deteriorates on more 'complex' tasks. Additionally, we find that certain attention-free models perform (almost) identically to Transformers on a range of tasks. (b) When provided a teaching sequence, i.e. a set of examples that uniquely identifies a function in a class, we show that Transformers learn more sample-efficiently. Interestingly, our results show that Transformers can learn to implement two distinct algorithms to solve a single task, and can adaptively select the more sample-efficient algorithm depending on the sequence of in-context examples. (c) Lastly, we show that extant LLMs, e.g. LLaMA-2, GPT-4, can compete with nearest-neighbor baselines on prediction tasks that are guaranteed to not be in their training set.

In recent years, active learning has been successfully applied to an array of NLP tasks. However, prior work often assumes that training and test data are drawn from the same distribution. This is problematic, as in real-life settings data may stem from several sources of varying relevance and quality. We show that four popular active learning schemes fail to outperform random selection when applied to unlabelled pools comprised of multiple data sources on the task of natural language inference. We reveal that uncertainty-based strategies perform poorly due to the acquisition of collective outliers, i.e., hard-to-learn instances that hamper learning and generalization. When outliers are removed, strategies are found to recover and outperform random baselines. In further analysis, we find that collective outliers vary in form between sources, and show that hard-to-learn data is not always categorically harmful. Lastly, we leverage dataset cartography to introduce difficulty-stratified testing and find that different strategies are affected differently by example learnability and difficulty.

Recently, large language models (LLMs) have demonstrated outstanding reasoning capabilities on mathematical and coding tasks. However, their application to financial tasks-especially the most fundamental task of stock movement prediction-remains underexplored. We study a three-class classification problem (up, hold, down) and, by analyzing existing reasoning responses, observe that: (1) LLMs follow analysts' opinions rather than exhibit a systematic, independent analytical logic (CoTs). (2) LLMs list summaries from different sources without weighing adversarial evidence, yet such counterevidence is crucial for reliable prediction. It shows that the model does not make good use of its reasoning ability to complete the task. To address this, we propose Reflective Evidence Tuning (RETuning), a cold-start method prior to reinforcement learning, to enhance prediction ability. While generating CoT, RETuning encourages dynamically constructing an analytical framework from diverse information sources, organizing and scoring evidence for price up or down based on that framework-rather than on contextual viewpoints-and finally reflecting to derive the prediction. This approach maximally aligns the model with its learned analytical framework, ensuring independent logical reasoning and reducing undue influence from context. We also build a large-scale dataset spanning all of 2024 for 5,123 A-share stocks, with long contexts (32K tokens) and over 200K samples. In addition to price and news, it incorporates analysts' opinions, quantitative reports, fundamental data, macroeconomic indicators, and similar stocks. Experiments show that RETuning successfully unlocks the model's reasoning ability in the financial domain. Inference-time scaling still works even after 6 months or on out-of-distribution stocks, since the models gain valuable insights about stock movement prediction.

Symbolic regression (SR) aims to discover closed-form mathematical expressions that accurately describe data, offering interpretability and analytical insight beyond standard black-box models. Existing SR methods often rely on population-based search or autoregressive modeling, which struggle with scalability and symbolic consistency. We introduce LIES (Logarithm, Identity, Exponential, Sine), a fixed neural network architecture with interpretable primitive activations that are optimized to model symbolic expressions. We develop a framework to extract compact formulae from LIES networks by training with an appropriate oversampling strategy and a tailored loss function to promote sparsity and to prevent gradient instability. After training, it applies additional pruning strategies to further simplify the learned expressions into compact formulae. Our experiments on SR benchmarks show that the LIES framework consistently produces sparse and accurate symbolic formulae outperforming all baselines. We also demonstrate the importance of each design component through ablation studies.

In nature, the behaviors of many complex systems can be described by parsimonious math equations. Automatically distilling these equations from limited data is cast as a symbolic regression process which hitherto remains a grand challenge. Keen efforts in recent years have been placed on tackling this issue and demonstrated success in symbolic regression. However, there still exist bottlenecks that current methods struggle to break when the discrete search space tends toward infinity and especially when the underlying math formula is intricate. To this end, we propose a novel Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data. The RSRM model is composed of three key modules: (1) a Monte Carlo tree search (MCTS) agent that explores optimal math expression trees consisting of pre-defined math operators and variables, (2) a Double Q-learning block that helps reduce the feasible search space of MCTS via properly understanding the distribution of reward, and (3) a modulated sub-tree discovery block that heuristically learns and defines new math operators to improve representation ability of math expression trees. Biding of these modules yields the state-of-the-art performance of RSRM in symbolic regression as demonstrated by multiple sets of benchmark examples. The RSRM model shows clear superiority over several representative baseline models.

In deep metric learning, the Triplet Loss has emerged as a popular method to learn many computer vision and natural language processing tasks such as facial recognition, object detection, and visual-semantic embeddings. One issue that plagues the Triplet Loss is network collapse, an undesirable phenomenon where the network projects the embeddings of all data onto a single point. Researchers predominately solve this problem by using triplet mining strategies. While hard negative mining is the most effective of these strategies, existing formulations lack strong theoretical justification for their empirical success. In this paper, we utilize the mathematical theory of isometric approximation to show an equivalence between the Triplet Loss sampled by hard negative mining and an optimization problem that minimizes a Hausdorff-like distance between the neural network and its ideal counterpart function. This provides the theoretical justifications for hard negative mining's empirical efficacy. In addition, our novel application of the isometric approximation theorem provides the groundwork for future forms of hard negative mining that avoid network collapse. Our theory can also be extended to analyze other Euclidean space-based metric learning methods like Ladder Loss or Contrastive Learning.

Generative models form the backbone of modern machine learning, underpinning state-of-the-art systems in text, vision, and multimodal applications. While Maximum Likelihood Estimation has traditionally served as the dominant training paradigm, recent work have highlighted its limitations, particularly in generalization and susceptibility to catastrophic forgetting compared to Reinforcement Learning techniques, such as Policy Gradient methods. However, these approaches depend on explicit reward signals, which are often unavailable in practice, leaving open the fundamental problem of how to align generative models when only high-quality datasets are accessible. In this work, we address this challenge via a Bilevel Optimization framework, where the reward function is treated as the optimization variable of an outer-level problem, while a policy gradient objective defines the inner-level. We then conduct a theoretical analysis of this optimization problem in a tractable setting and extract insights that, as we demonstrate, generalize to applications such as tabular classification and model-based reinforcement learning. We release the code at https://github.com/abenechehab/nll_to_po .

Large language models (LLMs) have demonstrated remarkable reasoning capability in solving mathematical problems. However, existing approaches primarily focus on improving the quality of correct training data, e.g., distilling high-quality correct solutions from advanced models, neglecting the value contained in error data, potentially hindering the model's reflective ability. Though some studies attempt to leverage error data, they often involve complex mechanisms, such as Monte Carlo Tree Search (MCTS) to explore error nodes. In this work, we propose to enhance LLMs' reasoning ability by Learning from Errors for Mathematical Advancement (LEMMA). LEMMA constructs data consisting of an incorrect solution with an erroneous step and a reflection connection to a correct solution for fine-tuning. Specifically, we systematically analyze the model-generated error types and introduce an error-type grounded mistake augmentation method to collect diverse and representative errors. Correct solutions are either from fixing the errors or generating a fresh start. Through a model-aware smooth reflection connection, the erroneous solution is transferred to the correct one. By fine-tuning on the constructed dataset, the model is able to self-correct errors autonomously within the generation process without relying on external critique models. Experimental results demonstrate that LEMMA achieves significant performance improvements over other strong baselines.

MDPs with low-rank transitions -- that is, the transition matrix can be factored into the product of two matrices, left and right -- is a highly representative structure that enables tractable learning. The left matrix enables expressive function approximation for value-based learning and has been studied extensively. In this work, we instead investigate sample-efficient learning with density features, i.e., the right matrix, which induce powerful models for state-occupancy distributions. This setting not only sheds light on leveraging unsupervised learning in RL, but also enables plug-in solutions for convex RL. In the offline setting, we propose an algorithm for off-policy estimation of occupancies that can handle non-exploratory data. Using this as a subroutine, we further devise an online algorithm that constructs exploratory data distributions in a level-by-level manner. As a central technical challenge, the additive error of occupancy estimation is incompatible with the multiplicative definition of data coverage. In the absence of strong assumptions like reachability, this incompatibility easily leads to exponential error blow-up, which we overcome via novel technical tools. Our results also readily extend to the representation learning setting, when the density features are unknown and must be learned from an exponentially large candidate set.

Curriculum learning has shown promise in improving training efficiency and generalization in various machine learning domains, yet its potential in pretraining language models remains underexplored, prompting our work as the first systematic investigation in this area. We experimented with different settings, including vanilla curriculum learning, pacing-based sampling, and interleaved curricula-guided by six difficulty metrics spanning linguistic and information-theoretic perspectives. We train models under these settings and evaluate their performance on eight diverse benchmarks. Our experiments reveal that curriculum learning consistently improves convergence in early and mid-training phases, and can yield lasting gains when used as a warmup strategy with up to 3.5% improvement. Notably, we identify compression ratio, lexical diversity, and readability as effective difficulty signals across settings. Our findings highlight the importance of data ordering in large-scale pretraining and provide actionable insights for scalable, data-efficient model development under realistic training scenarios.

With the advent of DeepSeek-R1, a new wave of reinforcement learning (RL) methods has emerged that seem to unlock stronger mathematical reasoning. However, a closer look at the open-source ecosystem reveals a critical limitation: with sufficiently many draws (e.g., pass@1024), many existing base models already solve nearly all questions on widely used math benchmarks such as MATH-500 and AIME 2024. This suggests that the RL fine-tuning methods prevalent in the LLM reasoning literature largely sharpen existing solution modes rather than discovering entirely new ones. Such sharpening stands in contrast to the broader promise of RL: to foster exploration and to acquire new skills. To move beyond this plateau, we introduce MATH-Beyond (MATH-B), a benchmark deliberately constructed to defeat common open-source models of up to 8B parameters even under large sampling budgets. Improving performance on our benchmark via RL requires methods that learn to reason in ways that go beyond base model capabilities in repeated sampling. Since the problems are drawn from subsets of DAPO-Math-17K and DeepScaleR datasets, they remain topically equivalent to standard high-school math. Validating our premise, RL fine-tuned models such as Nemotron-Research-Reasoning-Qwen-1.5B and DeepScaleR-1.5B-Preview perform poorly on MATH-B at pass@1024, showing how existing approaches fall short on tackling harder instances. We hope MATH-B will catalyze exploration-driven RL approaches that elicit deeper reasoning capabilities. We release MATH-B at https://huggingface.co/datasets/brendel-group/MATH-Beyond.

We propose a novel framework for comprehending the reasoning capabilities of large language models (LLMs) through the perspective of meta-learning. By conceptualizing reasoning trajectories as pseudo-gradient descent updates to the LLM's parameters, we identify parallels between LLM reasoning and various meta-learning paradigms. We formalize the training process for reasoning tasks as a meta-learning setup, with each question treated as an individual task, and reasoning trajectories serving as the inner loop optimization for adapting model parameters. Once trained on a diverse set of questions, the LLM develops fundamental reasoning capabilities that can generalize to previously unseen questions. Extensive empirical evaluations substantiate the strong connection between LLM reasoning and meta-learning, exploring several issues of significant interest from a meta-learning standpoint. Our work not only enhances the understanding of LLM reasoning but also provides practical insights for improving these models through established meta-learning techniques.

Social scientists are now using large language models to create "silicon samples" - synthetic datasets intended to stand in for human respondents, aimed at revolutionising human subjects research. However, there are many analytic choices which must be made to produce these samples. Though many of these choices are defensible, their impact on sample quality is poorly understood. I map out these analytic choices and demonstrate how a very small number of decisions can dramatically change the correspondence between silicon samples and human data. Configurations (N = 252) varied substantially in their capacity to estimate (i) rank ordering of participants, (ii) response distributions, and (iii) between-scale correlations. Most critically, configurations were not consistent in quality: those that performed well on one dimension often performed poorly on another, implying that there is no "one-size-fits-all" configuration that optimises the accuracy of these samples. I call for greater attention to the threat of analytic flexibility in using silicon samples.

Knowledge Distillation (KD) has proven effective for compressing large teacher models into smaller student models. While it is well known that student models can achieve similar accuracies as the teachers, it has also been shown that they nonetheless often do not learn the same function. It is, however, often highly desirable that the student's and teacher's functions share similar properties such as basing the prediction on the same input features, as this ensures that students learn the 'right features' from the teachers. In this work, we explore whether this can be achieved by not only optimizing the classic KD loss but also the similarity of the explanations generated by the teacher and the student. Despite the idea being simple and intuitive, we find that our proposed 'explanation-enhanced' KD (e^2KD) (1) consistently provides large gains in terms of accuracy and student-teacher agreement, (2) ensures that the student learns from the teacher to be right for the right reasons and to give similar explanations, and (3) is robust with respect to the model architectures, the amount of training data, and even works with 'approximate', pre-computed explanations.

We demonstrate that LLMs may learn indicators of document usefulness and modulate their updates accordingly. We introduce random strings ("tags") as indicators of usefulness in a synthetic fine-tuning dataset. Fine-tuning on this dataset leads to implicit meta-learning (IML): in further fine-tuning, the model updates to make more use of text that is tagged as useful. We perform a thorough empirical investigation of this phenomenon, finding (among other things) that (i) it occurs in both pretrained LLMs and those trained from scratch, as well as on a vision task, and (ii) larger models and smaller batch sizes tend to give more IML. We also use probing to examine how IML changes the way models store knowledge in their parameters. Finally, we reflect on what our results might imply about capabilities, risks, and controllability of future AI systems. Our code can be found at https://github.com/krasheninnikov/internalization.

Supervised learning relies on annotated data, which is expensive to obtain. A longstanding strategy to reduce annotation costs is active learning, an iterative process, in which a human annotates only data instances deemed informative by a model. Large language models (LLMs) have pushed the effectiveness of active learning, but have also improved methods such as few- or zero-shot learning, and text synthesis - thereby introducing potential alternatives. This raises the question: has active learning become obsolete? To answer this fully, we must look beyond literature to practical experiences. We conduct an online survey in the NLP community to collect previously intangible insights on the perceived relevance of data annotation, particularly focusing on active learning, including best practices, obstacles and expected future developments. Our findings show that annotated data remains a key factor, and active learning continues to be relevant. While the majority of active learning users find it effective, a comparison with a community survey from over a decade ago reveals persistent challenges: setup complexity, estimation of cost reduction, and tooling. We publish an anonymized version of the collected dataset

Large Language Models (LLMs) solely trained on next-token prediction learn to solve a wide range of problems involving mathematical reasoning. But how does this ability evolve during training? We show the first analysis of how mathematical reasoning abilities of several open-weight LLMs develop during pre-training and post-training. To this end, we construct MathCAMPS, a synthetic dataset of novel mathematical reasoning problems grounded in 44 fine-grained skills taken from the Common Core curriculum from K to 8th grades. In one experiment, we show that mathematical skills are learned during pre-training in an order that measurably correlates with the human-designed curriculum, even though training data are randomly ordered. We also show a detailed analysis of which mathematical abilities benefit from instruction tuning, a widely used post-training method and, in contrast, which skills suffer. Our work paves the way for an empirical understanding of LLM training dynamics in relation to reasoning.

Suppose Alice trains an open-weight language model and Bob uses a blackbox derivative of Alice's model to produce text. Can Alice prove that Bob is using her model, either by querying Bob's derivative model (query setting) or from the text alone (observational setting)? We formulate this question as an independence testing problem--in which the null hypothesis is that Bob's model or text is independent of Alice's randomized training run--and investigate it through the lens of palimpsestic memorization in language models: models are more likely to memorize data seen later in training, so we can test whether Bob is using Alice's model using test statistics that capture correlation between Bob's model or text and the ordering of training examples in Alice's training run. If Alice has randomly shuffled her training data, then any significant correlation amounts to exactly quantifiable statistical evidence against the null hypothesis, regardless of the composition of Alice's training data. In the query setting, we directly estimate (via prompting) the likelihood Bob's model gives to Alice's training examples and order; we correlate the likelihoods of over 40 fine-tunes of various Pythia and OLMo base models ranging from 1B to 12B parameters with the base model's training data order, achieving a p-value on the order of at most 1e-8 in all but six cases. In the observational setting, we try two approaches based on estimating 1) the likelihood of Bob's text overlapping with spans of Alice's training examples and 2) the likelihood of Bob's text with respect to different versions of Alice's model we obtain by repeating the last phase (e.g., 1%) of her training run on reshuffled data. The second approach can reliably distinguish Bob's text from as little as a few hundred tokens; the first does not involve any retraining but requires many more tokens (several hundred thousand) to achieve high power.

Deep learning sometimes appears to work in unexpected ways. In pursuit of a deeper understanding of its surprising behaviors, we investigate the utility of a simple yet accurate model of a trained neural network consisting of a sequence of first-order approximations telescoping out into a single empirically operational tool for practical analysis. Across three case studies, we illustrate how it can be applied to derive new empirical insights on a diverse range of prominent phenomena in the literature -- including double descent, grokking, linear mode connectivity, and the challenges of applying deep learning on tabular data -- highlighting that this model allows us to construct and extract metrics that help predict and understand the a priori unexpected performance of neural networks. We also demonstrate that this model presents a pedagogical formalism allowing us to isolate components of the training process even in complex contemporary settings, providing a lens to reason about the effects of design choices such as architecture & optimization strategy, and reveals surprising parallels between neural network learning and gradient boosting.

The fine-tuning of pre-trained large language models (LLMs) using reinforcement learning (RL) is generally formulated as direct policy optimization. This approach was naturally favored as it efficiently improves a pretrained LLM, seen as an initial policy. Another RL paradigm, Q-learning methods, has received far less attention in the LLM community while demonstrating major success in various non-LLM RL tasks. In particular, Q-learning effectiveness comes from its sample efficiency and ability to learn offline, which is particularly valuable given the high computational cost of sampling with LLMs. However, naively applying a Q-learning-style update to the model's logits is ineffective due to the specificity of LLMs. Our core contribution is to derive theoretically grounded loss functions from Bellman equations to adapt Q-learning methods to LLMs. To do so, we carefully adapt insights from the RL literature to account for LLM-specific characteristics, ensuring that the logits become reliable Q-value estimates. We then use this loss to build a practical algorithm, ShiQ for Shifted-Q, that supports off-policy, token-wise learning while remaining simple to implement. Finally, we evaluate ShiQ on both synthetic data and real-world benchmarks, e.g., UltraFeedback and BFCL-V3, demonstrating its effectiveness in both single-turn and multi-turn LLM settings

We develop a rigorous mathematical analysis of zero-shot learning with attributes. In this setting, the goal is to label novel classes with no training data, only detectors for attributes and a description of how those attributes are correlated with the target classes, called the class-attribute matrix. We develop the first non-trivial lower bound on the worst-case error of the best map from attributes to classes for this setting, even with perfect attribute detectors. The lower bound characterizes the theoretical intrinsic difficulty of the zero-shot problem based on the available information -- the class-attribute matrix -- and the bound is practically computable from it. Our lower bound is tight, as we show that we can always find a randomized map from attributes to classes whose expected error is upper bounded by the value of the lower bound. We show that our analysis can be predictive of how standard zero-shot methods behave in practice, including which classes will likely be confused with others.

The recent success of machine learning models, especially large-scale classifiers and language models, relies heavily on training with massive data. These data are often collected from online sources. This raises serious concerns about the protection of user data, as individuals may not have given consent for their data to be used in training. To address this concern, recent studies introduce the concept of unlearnable examples, i.e., data instances that appear natural but are intentionally altered to prevent models from effectively learning from them. While existing methods demonstrate empirical effectiveness, they typically rely on heuristic trials and lack formal guarantees. Besides, when unlearnable examples are mixed with clean data, as is often the case in practice, their unlearnability disappears. In this work, we propose a novel approach to constructing unlearnable examples by systematically maximising the Bayes error, a measurement of irreducible classification error. We develop an optimisation-based approach and provide an efficient solution using projected gradient ascent. Our method provably increases the Bayes error and remains effective when the unlearning examples are mixed with clean samples. Experimental results across multiple datasets and model architectures are consistent with our theoretical analysis and show that our approach can restrict data learnability, effectively in practice.

Mathematical reasoning has long represented one of the most fundamental and challenging frontiers in artificial intelligence research. In recent years, large language models (LLMs) have achieved significant advances in this area. This survey examines the development of mathematical reasoning abilities in LLMs through two high-level cognitive phases: comprehension, where models gain mathematical understanding via diverse pretraining strategies, and answer generation, which has progressed from direct prediction to step-by-step Chain-of-Thought (CoT) reasoning. We review methods for enhancing mathematical reasoning, ranging from training-free prompting to fine-tuning approaches such as supervised fine-tuning and reinforcement learning, and discuss recent work on extended CoT and "test-time scaling". Despite notable progress, fundamental challenges remain in terms of capacity, efficiency, and generalization. To address these issues, we highlight promising research directions, including advanced pretraining and knowledge augmentation techniques, formal reasoning frameworks, and meta-generalization through principled learning paradigms. This survey tries to provide some insights for researchers interested in enhancing reasoning capabilities of LLMs and for those seeking to apply these techniques to other domains.

Machine learning has achieved remarkable success in many applications. However, existing studies are largely based on the closed-world assumption, which assumes that the environment is stationary, and the model is fixed once deployed. In many real-world applications, this fundamental and rather naive assumption may not hold because an open environment is complex, dynamic, and full of unknowns. In such cases, rejecting unknowns, discovering novelties, and then incrementally learning them, could enable models to be safe and evolve continually as biological systems do. This paper provides a holistic view of open-world machine learning by investigating unknown rejection, novel class discovery, and class-incremental learning in a unified paradigm. The challenges, principles, and limitations of current methodologies are discussed in detail. Finally, we discuss several potential directions for future research. This paper aims to provide a comprehensive introduction to the emerging open-world machine learning paradigm, to help researchers build more powerful AI systems in their respective fields, and to promote the development of artificial general intelligence.

Recent large-scale language models (LLMs) with long Chain-of-Thought reasoning-such as DeepSeek-R1-have achieved impressive results on Olympiad-level mathematics benchmarks. However, they often rely on a narrow set of strategies and struggle with problems that require a novel way of thinking. To systematically investigate these limitations, we introduce OMEGA-Out-of-distribution Math Problems Evaluation with 3 Generalization Axes-a controlled yet diverse benchmark designed to evaluate three axes of out-of-distribution generalization, inspired by Boden's typology of creativity: (1) Exploratory-applying known problem solving skills to more complex instances within the same problem domain; (2) Compositional-combining distinct reasoning skills, previously learned in isolation, to solve novel problems that require integrating these skills in new and coherent ways; and (3) Transformative-adopting novel, often unconventional strategies by moving beyond familiar approaches to solve problems more effectively. OMEGA consists of programmatically generated training-test pairs derived from templated problem generators across geometry, number theory, algebra, combinatorics, logic, and puzzles, with solutions verified using symbolic, numerical, or graphical methods. We evaluate frontier (or top-tier) LLMs and observe sharp performance degradation as problem complexity increases. Moreover, we fine-tune the Qwen-series models across all generalization settings and observe notable improvements in exploratory generalization, while compositional generalization remains limited and transformative reasoning shows little to no improvement. By isolating and quantifying these fine-grained failures, OMEGA lays the groundwork for advancing LLMs toward genuine mathematical creativity beyond mechanical proficiency.

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator (MDI) is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky MDIs. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.

We analyze how well pre-trained large language models (e.g., Llama2, GPT-4, Claude 3, etc) can do linear and non-linear regression when given in-context examples, without any additional training or gradient updates. Our findings reveal that several large language models (e.g., GPT-4, Claude 3) are able to perform regression tasks with a performance rivaling (or even outperforming) that of traditional supervised methods such as Random Forest, Bagging, or Gradient Boosting. For example, on the challenging Friedman #2 regression dataset, Claude 3 outperforms many supervised methods such as AdaBoost, SVM, Random Forest, KNN, or Gradient Boosting. We then investigate how well the performance of large language models scales with the number of in-context exemplars. We borrow from the notion of regret from online learning and empirically show that LLMs are capable of obtaining a sub-linear regret.

Language models are increasingly capable, yet still fail at a seemingly simple task of multi-digit multiplication. In this work, we study why, by reverse-engineering a model that successfully learns multiplication via implicit chain-of-thought, and report three findings: (1) Evidence of long-range structure: Logit attributions and linear probes indicate that the model encodes the necessary long-range dependencies for multi-digit multiplication. (2) Mechanism: the model encodes long-range dependencies using attention to construct a directed acyclic graph to ``cache'' and ``retrieve'' pairwise partial products. (3) Geometry: the model implements partial products in attention heads by forming Minkowski sums between pairs of digits, and digits are represented using a Fourier basis, both of which are intuitive and efficient representations that the standard fine-tuning model lacks. With these insights, we revisit the learning dynamics of standard fine-tuning and find that the model converges to a local optimum that lacks the required long-range dependencies. We further validate this understanding by introducing an auxiliary loss that predicts the ``running sum'' via a linear regression probe, which provides an inductive bias that enables the model to successfully learn multi-digit multiplication. In summary, by reverse-engineering the mechanisms of an implicit chain-of-thought model we uncover a pitfall for learning long-range dependencies in Transformers and provide an example of how the correct inductive bias can address this issue.

While LLMs can provide reasoned explanations along with their answers, the nature and quality of those explanations are still poorly understood. In response, our goal is to define a detailed way of characterizing the explanation capabilities of modern models and to create a nuanced, interpretable explanation evaluation tool that can generate such characterizations automatically, without relying on expensive API calls or human annotations. Our approach is to (a) define the new task of explanation critiquing - identifying and categorizing any main flaw in an explanation and providing suggestions to address the flaw, (b) create a sizeable, human-verified dataset for this task, and (c) train an open-source, automatic critiquing model (called Digital Socrates) using this data. Through quantitative and qualitative analysis, we demonstrate how Digital Socrates is useful for revealing insights about student models by examining their reasoning chains, and how it can provide high-quality, nuanced, automatic evaluation of those model explanations for the first time. Digital Socrates thus fills an important gap in evaluation tools for understanding and improving the explanation behavior of models.

Offline optimization has recently emerged as an increasingly popular approach to mitigate the prohibitively expensive cost of online experimentation. The key idea is to learn a surrogate of the black-box function that underlines the target experiment using a static (offline) dataset of its previous input-output queries. Such an approach is, however, fraught with an out-of-distribution issue where the learned surrogate becomes inaccurate outside the offline data regimes. To mitigate this, existing offline optimizers have proposed numerous conditioning techniques to prevent the learned surrogate from being too erratic. Nonetheless, such conditioning strategies are often specific to particular surrogate or search models, which might not generalize to a different model choice. This motivates us to develop a model-agnostic approach instead, which incorporates a notion of model sharpness into the training loss of the surrogate as a regularizer. Our approach is supported by a new theoretical analysis demonstrating that reducing surrogate sharpness on the offline dataset provably reduces its generalized sharpness on unseen data. Our analysis extends existing theories from bounding generalized prediction loss (on unseen data) with loss sharpness to bounding the worst-case generalized surrogate sharpness with its empirical estimate on training data, providing a new perspective on sharpness regularization. Our extensive experimentation on a diverse range of optimization tasks also shows that reducing surrogate sharpness often leads to significant improvement, marking (up to) a noticeable 9.6% performance boost. Our code is publicly available at https://github.com/cuong-dm/IGNITE

As machine learning models become increasingly larger, trained weakly supervised on large, possibly uncurated data sets, it becomes increasingly important to establish mechanisms for inspecting, interacting, and revising models to mitigate learning shortcuts and guarantee their learned knowledge is aligned with human knowledge. The recently proposed XIL framework was developed for this purpose, and several such methods have been introduced, each with individual motivations and methodological details. In this work, we provide a unification of various XIL methods into a single typology by establishing a common set of basic modules. In doing so, we pave the way for a principled comparison of existing, but, importantly, also future XIL approaches. In addition, we discuss existing and introduce novel measures and benchmarks for evaluating the overall abilities of a XIL method. Given this extensive toolbox, including our typology, measures, and benchmarks, we finally compare several recent XIL methods methodologically and quantitatively. In our evaluations, all methods prove to revise a model successfully. However, we found remarkable differences in individual benchmark tasks, revealing valuable application-relevant aspects for integrating these benchmarks in developing future methods.

Auto-regressive language models factorize sequence probabilities and are trained by minimizing the negative log-likelihood (NLL) objective. While empirically powerful, a deep theoretical understanding of why this simple objective yields such versatile representations remains elusive. This work introduces a unifying analytical framework using Markov Categories (MCs) to deconstruct the AR generation process and the NLL objective. We model the single-step generation map as a composition of Markov kernels in the category Stoch. This compositional view, when enriched with statistical divergences, allows us to dissect information flow and learned geometry. Our framework makes three main contributions. First, we provide a formal, information-theoretic rationale for the success of modern speculative decoding methods like EAGLE, quantifying the information surplus in hidden states that these methods exploit. Second, we formalize how NLL minimization forces the model to learn not just the next token, but the data's intrinsic conditional stochasticity, a process we analyze using categorical entropy. Third, and most centrally, we prove that NLL training acts as an implicit form of spectral contrastive learning. By analyzing the information geometry of the model's prediction head, we show that NLL implicitly forces the learned representation space to align with the eigenspectrum of a predictive similarity operator, thereby learning a geometrically structured space without explicit contrastive pairs. This compositional and information-geometric perspective reveals the deep structural principles underlying the effectiveness of modern LMs. Project Page: https://github.com/asiresearch/lm-theory

We find that the cross-entropy loss curves of neural language models empirically adhere to a scaling law with learning rate (LR) annealing over training steps (s): $L(s) = L_0 + Acdot S_1^{-alpha} - Ccdot S_2 Where S_1 is forward area and S_2$ is learning rate annealing area. This formulation takes into account two factors: (1) The forward scaling defined as typical scaling law, and (2) the additional loss drop brought by LR annealing. Therefore, this formulation can describe the full loss curve at each step, rather than the single loss point at the end of training. Applying the scaling law with LR annealing and fitting only one or two training curves, we can accurately predict the loss of language model training at any given step and across any learning rate scheduler (LRS). Furthermore, this equation accurately describes the dynamics during training process, and provides a theoretical verification and explanation for numerous experimental findings of previous studies, particularly those focusing on LR schedule and LR annealing. The resulting insights, also serve as a guide for researchers to select critical LRS in advance by prediction using our equation. Most significantly, since all the points in a full training curve follow the equation, we can achieve accurate loss prediction at any given step across any learning rate scheduler, while expending less than 1\% of the computational cost required by the chinchilla scaling law to fit language modeling loss. This approach extremely democratizes scaling law fitting and predicting in developing large language models.

Reinforcement learning (RL)-based fine-tuning has become a crucial step in post-training language models for advanced mathematical reasoning and coding. Following the success of frontier reasoning models, recent work has demonstrated that RL fine-tuning consistently improves performance, even in smaller-scale models; however, the underlying mechanisms driving these improvements are not well-understood. Understanding the effects of RL fine-tuning requires disentangling its interaction with pretraining data composition, hyperparameters, and model scale, but such problems are exacerbated by the lack of transparency regarding the training data used in many existing models. In this work, we present a systematic end-to-end study of RL fine-tuning for mathematical reasoning by training models entirely from scratch on different mixtures of fully open datasets. We investigate the effects of various RL fine-tuning algorithms (PPO, GRPO, and Expert Iteration) across models of different scales. Our study reveals that RL algorithms consistently converge towards a dominant output distribution, amplifying patterns in the pretraining data. We also find that models of different scales trained on the same data mixture will converge to distinct output distributions, suggesting that there are scale-dependent biases in model generalization. Moreover, we find that RL post-training on simpler questions can lead to performance gains on harder ones, indicating that certain reasoning capabilities generalize across tasks. Our findings show that small-scale proxies in controlled settings can elicit interesting insights regarding the role of RL in shaping language model behavior.

Self-distillation (SD) is the process of first training a teacher model and then using its predictions to train a student model with the same architecture. Specifically, the student's objective function is big(xi*ell(teacher's predictions, student's predictions) + (1-xi)*ell(given labels, student's predictions)big), where ell is some loss function and xi is some parameter in [0,1]. Empirically, SD has been observed to provide performance gains in several settings. In this paper, we theoretically characterize the effect of SD in two supervised learning problems with noisy labels. We first analyze SD for regularized linear regression and show that in the high label noise regime, the optimal value of xi that minimizes the expected error in estimating the ground truth parameter is surprisingly greater than 1. Empirically, we show that xi > 1 works better than xi leq 1 even with the cross-entropy loss for several classification datasets when 50\% or 30\% of the labels are corrupted. Further, we quantify when optimal SD is better than optimal regularization. Next, we analyze SD in the case of logistic regression for binary classification with random label corruption and quantify the range of label corruption in which the student outperforms the teacher in terms of accuracy. To our knowledge, this is the first result of its kind for the cross-entropy loss.

In an era where symbolic mathematical equations are indispensable for modeling complex natural phenomena, scientific inquiry often involves collecting observations and translating them into mathematical expressions. Recently, deep learning has emerged as a powerful tool for extracting insights from data. However, existing models typically specialize in either numeric or symbolic domains, and are usually trained in a supervised manner tailored to specific tasks. This approach neglects the substantial benefits that could arise from a task-agnostic unified understanding between symbolic equations and their numeric counterparts. To bridge the gap, we introduce SNIP, a Symbolic-Numeric Integrated Pre-training, which employs joint contrastive learning between symbolic and numeric domains, enhancing their mutual similarities in the pre-trained embeddings. By performing latent space analysis, we observe that SNIP provides cross-domain insights into the representations, revealing that symbolic supervision enhances the embeddings of numeric data and vice versa. We evaluate SNIP across diverse tasks, including symbolic-to-numeric mathematical property prediction and numeric-to-symbolic equation discovery, commonly known as symbolic regression. Results show that SNIP effectively transfers to various tasks, consistently outperforming fully supervised baselines and competing strongly with established task-specific methods, especially in few-shot learning scenarios where available data is limited.

Recently, contrastive learning has found impressive success in advancing the state of the art in solving various machine learning tasks. However, the existing generalization analysis is very limited or even not meaningful. In particular, the existing generalization error bounds depend linearly on the number k of negative examples while it was widely shown in practice that choosing a large k is necessary to guarantee good generalization of contrastive learning in downstream tasks. In this paper, we establish novel generalization bounds for contrastive learning which do not depend on k, up to logarithmic terms. Our analysis uses structural results on empirical covering numbers and Rademacher complexities to exploit the Lipschitz continuity of loss functions. For self-bounding Lipschitz loss functions, we further improve our results by developing optimistic bounds which imply fast rates in a low noise condition. We apply our results to learning with both linear representation and nonlinear representation by deep neural networks, for both of which we derive Rademacher complexity bounds to get improved generalization bounds.

In the quest for super-human performance, Large Language Models (LLMs) have traditionally been tethered to human-annotated datasets and predefined training objectives-a process that is both labor-intensive and inherently limited. This paper presents a transformative approach: Autonomous Learning for LLMs, a self-sufficient learning paradigm that frees models from the constraints of human supervision. This method endows LLMs with the ability to self-educate through direct interaction with text, akin to a human reading and comprehending literature. Our approach eliminates the reliance on annotated data, fostering an Autonomous Learning environment where the model independently identifies and reinforces its knowledge gaps. Empirical results from our comprehensive experiments, which utilized a diverse array of learning materials and were evaluated against standard public quizzes, reveal that Autonomous Learning outstrips the performance of both Pre-training and Supervised Fine-Tuning (SFT), as well as retrieval-augmented methods. These findings underscore the potential of Autonomous Learning to not only enhance the efficiency and effectiveness of LLM training but also to pave the way for the development of more advanced, self-reliant AI systems.

Language models (LMs) often exhibit undesirable text generation behaviors, including generating false, toxic, or irrelevant outputs. Reinforcement learning from human feedback (RLHF) - where human preference judgments on LM outputs are transformed into a learning signal - has recently shown promise in addressing these issues. However, such holistic feedback conveys limited information on long text outputs; it does not indicate which aspects of the outputs influenced user preference; e.g., which parts contain what type(s) of errors. In this paper, we use fine-grained human feedback (e.g., which sentence is false, which sub-sentence is irrelevant) as an explicit training signal. We introduce Fine-Grained RLHF, a framework that enables training and learning from reward functions that are fine-grained in two respects: (1) density, providing a reward after every segment (e.g., a sentence) is generated; and (2) incorporating multiple reward models associated with different feedback types (e.g., factual incorrectness, irrelevance, and information incompleteness). We conduct experiments on detoxification and long-form question answering to illustrate how learning with such reward functions leads to improved performance, supported by both automatic and human evaluation. Additionally, we show that LM behaviors can be customized using different combinations of fine-grained reward models. We release all data, collected human feedback, and codes at https://FineGrainedRLHF.github.io.

Large language models have been shown to suffer from reasoning inconsistency issues. That is, they fail more in situations unfamiliar to the training data, even though exact or very similar reasoning paths exist in more common cases that they can successfully solve. Such observations motivate us to propose methods that encourage models to understand the high-level and abstract reasoning processes during training instead of only the final answer. This way, models can transfer the exact solution to similar cases, regardless of their relevance to the pre-training data distribution. In this work, we propose SAL, a self-supervised analogical learning framework. SAL mimics the human analogy process and trains models to explicitly transfer high-quality symbolic solutions from cases that they know how to solve to other rare cases in which they tend to fail more. We show that the resulting models after SAL learning outperform base language models on a wide range of reasoning benchmarks, such as StrategyQA, GSM8K, and HotpotQA, by 2% to 20%. At the same time, we show that our model is more generalizable and controllable through analytical studies.

Classification is a core task in machine learning. Recent research has shown that although Multimodal Large Language Models (MLLMs) are initially poor at image classification, fine-tuning them with an adequate amount of data can significantly enhance their performance, making them comparable to SOTA classification models. However, acquiring large-scale labeled data is expensive. In this paper, we explore few-shot MLLM classification fine-tuning. We found that SFT can cause severe overfitting issues and may even degrade performance over the zero-shot approach. To address this challenge, inspired by the recent successes in rule-based reinforcement learning, we propose CLS-RL, which uses verifiable signals as reward to fine-tune MLLMs. We discovered that CLS-RL outperforms SFT in most datasets and has a much higher average accuracy on both base-to-new and few-shot learning setting. Moreover, we observed a free-lunch phenomenon for CLS-RL; when models are fine-tuned on a particular dataset, their performance on other distinct datasets may also improve over zero-shot models, even if those datasets differ in distribution and class names. This suggests that RL-based methods effectively teach models the fundamentals of classification. Lastly, inspired by recent works in inference time thinking, we re-examine the `thinking process' during fine-tuning, a critical aspect of RL-based methods, in the context of visual classification. We question whether such tasks require extensive thinking process during fine-tuning, proposing that this may actually detract from performance. Based on this premise, we introduce the No-Thinking-CLS-RL method, which minimizes thinking processes during training by setting an equality accuracy reward. Our findings indicate that, with much less fine-tuning time, No-Thinking-CLS-RL method achieves superior in-domain performance and generalization capabilities than CLS-RL.

Large language models (LLMs) have become the foundation of many applications, leveraging their extensive capabilities in processing and understanding natural language. While many open-source LLMs have been released with technical reports, the lack of training details hinders further research and development. This paper presents the development of YuLan, a series of open-source LLMs with 12 billion parameters. The base model of YuLan is pre-trained on approximately 1.7T tokens derived from a diverse corpus, including massive English, Chinese, and multilingual texts. We design a three-stage pre-training method to enhance YuLan's overall capabilities. Subsequent phases of training incorporate instruction-tuning and human alignment, employing a substantial volume of high-quality synthesized data. To facilitate the learning of complex and long-tail knowledge, we devise a curriculum-learning framework throughout across these stages, which helps LLMs learn knowledge in an easy-to-hard manner. YuLan's training is finished on Jan, 2024 and has achieved performance on par with state-of-the-art LLMs across various English and Chinese benchmarks. This paper outlines a comprehensive technical roadmap for developing LLMs from scratch. Our model and codes are available at https://github.com/RUC-GSAI/YuLan-Chat.

Beginning with McCarthy's Advice Taker (1959), AI has pursued the goal of providing a system with explicit, general knowledge and having the system reason over that knowledge. However, expressing the knowledge in a formal (logical or probabilistic) representation has been a major obstacle to this research. This paper investigates a modern approach to this problem where the facts and rules are provided as natural language sentences, thus bypassing a formal representation. We train transformers to reason (or emulate reasoning) over these sentences using synthetically generated data. Our models, that we call RuleTakers, provide the first empirical demonstration that this kind of soft reasoning over language is learnable, can achieve high (99%) accuracy, and generalizes to test data requiring substantially deeper chaining than seen during training (95%+ scores). We also demonstrate that the models transfer well to two hand-authored rulebases, and to rulebases paraphrased into more natural language. These findings are significant as it suggests a new role for transformers, namely as limited "soft theorem provers" operating over explicit theories in language. This in turn suggests new possibilities for explainability, correctability, and counterfactual reasoning in question-answering.

Multimodal Large Language Models (MLLMs) have demonstrated impressive capabilities across various tasks, but still struggle with complex mathematical reasoning. Existing research primarily focuses on dataset construction and method optimization, often overlooking two critical aspects: comprehensive knowledge-driven design and model-centric data space modeling. In this paper, we introduce We-Math 2.0, a unified system that integrates a structured mathematical knowledge system, model-centric data space modeling, and a reinforcement learning (RL)-based training paradigm to comprehensively enhance the mathematical reasoning abilities of MLLMs. The key contributions of We-Math 2.0 are fourfold: (1) MathBook Knowledge System: We construct a five-level hierarchical system encompassing 491 knowledge points and 1,819 fundamental principles. (2) MathBook-Standard & Pro: We develop MathBook-Standard, a dataset that ensures broad conceptual coverage and flexibility through dual expansion. Additionally, we define a three-dimensional difficulty space and generate 7 progressive variants per problem to build MathBook-Pro, a challenging dataset for robust training. (3) MathBook-RL: We propose a two-stage RL framework comprising: (i) Cold-Start Fine-tuning, which aligns the model with knowledge-oriented chain-of-thought reasoning; and (ii) Progressive Alignment RL, leveraging average-reward learning and dynamic data scheduling to achieve progressive alignment across difficulty levels. (4) MathBookEval: We introduce a comprehensive benchmark covering all 491 knowledge points with diverse reasoning step distributions. Experimental results show that MathBook-RL performs competitively with existing baselines on four widely-used benchmarks and achieves strong results on MathBookEval, suggesting promising generalization in mathematical reasoning.

Incremental Learning (IL) has been a long-standing problem in both vision and Natural Language Processing (NLP) communities. In recent years, as Pre-trained Language Models (PLMs) have achieved remarkable progress in various NLP downstream tasks, utilizing PLMs as backbones has become a common practice in recent research of IL in NLP. Most assume that catastrophic forgetting is the biggest obstacle to achieving superior IL performance and propose various techniques to overcome this issue. However, we find that this assumption is problematic. Specifically, we revisit more than 20 methods on four classification tasks (Text Classification, Intent Classification, Relation Extraction, and Named Entity Recognition) under the two most popular IL settings (Class-Incremental and Task-Incremental) and reveal that most of them severely underestimate the inherent anti-forgetting ability of PLMs. Based on the observation, we propose a frustratingly easy method called SEQ* for IL with PLMs. The results show that SEQ* has competitive or superior performance compared to state-of-the-art (SOTA) IL methods and requires considerably less trainable parameters and training time. These findings urge us to revisit the IL with PLMs and encourage future studies to have a fundamental understanding of the catastrophic forgetting in PLMs. The data, code and scripts are publicly available at https://github.com/zzz47zzz/codebase-for-incremental-learning-with-llm.

Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({\bf P}roof {\bf A}rtifact {\bf C}o-{\bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32\% to 48\%.

Large language models (LLMs) are revolutionizing education, with LLM-based agents playing a key role in simulating student behavior. A major challenge in student simulation is modeling the diverse learning patterns of students at various cognitive levels. However, current LLMs, typically trained as ``helpful assistants'', target at generating perfect responses. As a result, they struggle to simulate students with diverse cognitive abilities, as they often produce overly advanced answers, missing the natural imperfections that characterize student learning and resulting in unrealistic simulations. To address this issue, we propose a training-free framework for student simulation. We begin by constructing a cognitive prototype for each student using a knowledge graph, which captures their understanding of concepts from past learning records. This prototype is then mapped to new tasks to predict student performance. Next, we simulate student solutions based on these predictions and iteratively refine them using a beam search method to better replicate realistic mistakes. To validate our approach, we construct the Student\_100 dataset, consisting of 100 students working on Python programming and 5,000 learning records. Experimental results show that our method consistently outperforms baseline models, achieving 100% improvement in simulation accuracy.

Long-tailed data distributions pose challenges for a variety of domains like e-commerce, finance, biomedical science, and cyber security, where the performance of machine learning models is often dominated by head categories while tail categories are inadequately learned. This work aims to provide a systematic view of long-tailed learning with regard to three pivotal angles: (A1) the characterization of data long-tailedness, (A2) the data complexity of various domains, and (A3) the heterogeneity of emerging tasks. We develop HeroLT, a comprehensive long-tailed learning benchmark integrating 18 state-of-the-art algorithms, 10 evaluation metrics, and 17 real-world datasets across 6 tasks and 4 data modalities. HeroLT with novel angles and extensive experiments (315 in total) enables effective and fair evaluation of newly proposed methods compared with existing baselines on varying dataset types. Finally, we conclude by highlighting the significant applications of long-tailed learning and identifying several promising future directions. For accessibility and reproducibility, we open-source our benchmark HeroLT and corresponding results at https://github.com/SSSKJ/HeroLT.

In the era of data-driven intelligence, the paradox of data abundance and annotation scarcity has emerged as a critical bottleneck in the advancement of machine learning. This paper gives a detailed overview of Active Learning (AL), which is a strategy in machine learning that helps models achieve better performance using fewer labeled examples. It introduces the basic concepts of AL and discusses how it is used in various fields such as computer vision, natural language processing, transfer learning, and real-world applications. The paper focuses on important research topics such as uncertainty estimation, handling of class imbalance, domain adaptation, fairness, and the creation of strong evaluation metrics and benchmarks. It also shows that learning methods inspired by humans and guided by questions can improve data efficiency and help models learn more effectively. In addition, this paper talks about current challenges in the field, including the need to rebuild trust, ensure reproducibility, and deal with inconsistent methodologies. It points out that AL often gives better results than passive learning, especially when good evaluation measures are used. This work aims to be useful for both researchers and practitioners by providing key insights and proposing directions for future progress in active learning.

In this paper, we delve into several mechanisms employed by Transformer-based language models (LLMs) for factual recall tasks. We outline a pipeline consisting of three major steps: (1) Given a prompt ``The capital of France is,'' task-specific attention heads extract the topic token, such as ``France,'' from the context and pass it to subsequent MLPs. (2) As attention heads' outputs are aggregated with equal weight and added to the residual stream, the subsequent MLP acts as an ``activation,'' which either erases or amplifies the information originating from individual heads. As a result, the topic token ``France'' stands out in the residual stream. (3) A deep MLP takes ``France'' and generates a component that redirects the residual stream towards the direction of the correct answer, i.e., ``Paris.'' This procedure is akin to applying an implicit function such as ``get\_capital(X),'' and the argument X is the topic token information passed by attention heads. To achieve the above quantitative and qualitative analysis for MLPs, we proposed a novel analytic method aimed at decomposing the outputs of the MLP into components understandable by humans. Additionally, we observed a universal anti-overconfidence mechanism in the final layer of models, which suppresses correct predictions. We mitigate this suppression by leveraging our interpretation to improve factual recall confidence. The above interpretations are evaluated across diverse tasks spanning various domains of factual knowledge, using various language models from the GPT-2 families, 1.3B OPT, up to 7B Llama-2, and in both zero- and few-shot setups.

We propose an algorithm for meta-learning that is model-agnostic, in the sense that it is compatible with any model trained with gradient descent and applicable to a variety of different learning problems, including classification, regression, and reinforcement learning. The goal of meta-learning is to train a model on a variety of learning tasks, such that it can solve new learning tasks using only a small number of training samples. In our approach, the parameters of the model are explicitly trained such that a small number of gradient steps with a small amount of training data from a new task will produce good generalization performance on that task. In effect, our method trains the model to be easy to fine-tune. We demonstrate that this approach leads to state-of-the-art performance on two few-shot image classification benchmarks, produces good results on few-shot regression, and accelerates fine-tuning for policy gradient reinforcement learning with neural network policies.

Symbolic regression (SR) is an area of interpretable machine learning that aims to identify mathematical expressions, often composed of simple functions, that best fit in a given set of covariates X and response y. In recent years, deep symbolic regression (DSR) has emerged as a popular method in the field by leveraging deep reinforcement learning to solve the complicated combinatorial search problem. In this work, we propose an alternative framework (GFN-SR) to approach SR with deep learning. We model the construction of an expression tree as traversing through a directed acyclic graph (DAG) so that GFlowNet can learn a stochastic policy to generate such trees sequentially. Enhanced with an adaptive reward baseline, our method is capable of generating a diverse set of best-fitting expressions. Notably, we observe that GFN-SR outperforms other SR algorithms in noisy data regimes, owing to its ability to learn a distribution of rewards over a space of candidate solutions.

An algorithm to estimate the evolution of learning curves on the whole of a training data base, based on the results obtained from a portion and using a functional strategy, is introduced. We approximate iteratively the sought value at the desired time, independently of the learning technique used and once a point in the process, called prediction level, has been passed. The proposal proves to be formally correct with respect to our working hypotheses and includes a reliable proximity condition. This allows the user to fix a convergence threshold with respect to the accuracy finally achievable, which extends the concept of stopping criterion and seems to be effective even in the presence of distorting observations. Our aim is to evaluate the training effort, supporting decision making in order to reduce the need for both human and computational resources during the learning process. The proposal is of interest in at least three operational procedures. The first is the anticipation of accuracy gain, with the purpose of measuring how much work is needed to achieve a certain degree of performance. The second relates the comparison of efficiency between systems at training time, with the objective of completing this task only for the one that best suits our requirements. The prediction of accuracy is also a valuable item of information for customizing systems, since we can estimate in advance the impact of settings on both the performance and the development costs. Using the generation of part-of-speech taggers as an example application, the experimental results are consistent with our expectations.

We study the cost of overfitting in noisy kernel ridge regression (KRR), which we define as the ratio between the test error of the interpolating ridgeless model and the test error of the optimally-tuned model. We take an "agnostic" view in the following sense: we consider the cost as a function of sample size for any target function, even if the sample size is not large enough for consistency or the target is outside the RKHS. We analyze the cost of overfitting under a Gaussian universality ansatz using recently derived (non-rigorous) risk estimates in terms of the task eigenstructure. Our analysis provides a more refined characterization of benign, tempered and catastrophic overfitting (cf. Mallinar et al. 2022).

State-of-the-art results on image recognition tasks are achieved using over-parameterized learning algorithms that (nearly) perfectly fit the training set and are known to fit well even random labels. This tendency to memorize the labels of the training data is not explained by existing theoretical analyses. Memorization of the training data also presents significant privacy risks when the training data contains sensitive personal information and thus it is important to understand whether such memorization is necessary for accurate learning. We provide the first conceptual explanation and a theoretical model for this phenomenon. Specifically, we demonstrate that for natural data distributions memorization of labels is necessary for achieving close-to-optimal generalization error. Crucially, even labels of outliers and noisy labels need to be memorized. The model is motivated and supported by the results of several recent empirical works. In our model, data is sampled from a mixture of subpopulations and our results show that memorization is necessary whenever the distribution of subpopulation frequencies is long-tailed. Image and text data is known to be long-tailed and therefore our results establish a formal link between these empirical phenomena. Our results allow to quantify the cost of limiting memorization in learning and explain the disparate effects that privacy and model compression have on different subgroups.

Recent advancements in meta-learning have enabled the automatic discovery of novel reinforcement learning algorithms parameterized by surrogate objective functions. To improve upon manually designed algorithms, the parameterization of this learned objective function must be expressive enough to represent novel principles of learning (instead of merely recovering already established ones) while still generalizing to a wide range of settings outside of its meta-training distribution. However, existing methods focus on discovering objective functions that, like many widely used objective functions in reinforcement learning, do not take into account the total number of steps allowed for training, or "training horizon". In contrast, humans use a plethora of different learning objectives across the course of acquiring a new ability. For instance, students may alter their studying techniques based on the proximity to exam deadlines and their self-assessed capabilities. This paper contends that ignoring the optimization time horizon significantly restricts the expressive potential of discovered learning algorithms. We propose a simple augmentation to two existing objective discovery approaches that allows the discovered algorithm to dynamically update its objective function throughout the agent's training procedure, resulting in expressive schedules and increased generalization across different training horizons. In the process, we find that commonly used meta-gradient approaches fail to discover such adaptive objective functions while evolution strategies discover highly dynamic learning rules. We demonstrate the effectiveness of our approach on a wide range of tasks and analyze the resulting learned algorithms, which we find effectively balance exploration and exploitation by modifying the structure of their learning rules throughout the agent's lifetime.

Teaching to improve student models (e.g., knowledge distillation) is an extensively studied methodology in LLMs. However, for humans, teaching not only improves students but also improves teachers. We ask: Can LLMs also learn by teaching (LbT)? If yes, we can potentially unlock the possibility of continuously advancing the models without solely relying on human-produced data or stronger models. In this paper, we provide a preliminary exploration of this ambitious agenda. We show that LbT ideas can be incorporated into existing LLM training/prompting pipelines and provide noticeable improvements. Specifically, we design three methods, each mimicking one of the three levels of LbT in humans: observing students' feedback, learning from the feedback, and learning iteratively, with the goals of improving answer accuracy without training and improving models' inherent capability with fine-tuning. The findings are encouraging. For example, similar to LbT in human, we see that: (1) LbT can induce weak-to-strong generalization: strong models can improve themselves by teaching other weak models; (2) Diversity in students might help: teaching multiple students could be better than teaching one student or the teacher itself. We hope that this early promise can inspire future research on LbT and more broadly adopting the advanced techniques in education to improve LLMs. The code is available at https://github.com/imagination-research/lbt.

We study the problem of teaching multiple learners simultaneously in the nonparametric iterative teaching setting, where the teacher iteratively provides examples to the learner for accelerating the acquisition of a target concept. This problem is motivated by the gap between current single-learner teaching setting and the real-world scenario of human instruction where a teacher typically imparts knowledge to multiple students. Under the new problem formulation, we introduce a novel framework -- Multi-learner Nonparametric Teaching (MINT). In MINT, the teacher aims to instruct multiple learners, with each learner focusing on learning a scalar-valued target model. To achieve this, we frame the problem as teaching a vector-valued target model and extend the target model space from a scalar-valued reproducing kernel Hilbert space used in single-learner scenarios to a vector-valued space. Furthermore, we demonstrate that MINT offers significant teaching speed-up over repeated single-learner teaching, particularly when the multiple learners can communicate with each other. Lastly, we conduct extensive experiments to validate the practicality and efficiency of MINT.

Large Language Models (LLMs) are smart but forgetful. Recent studies, (e.g., (Bubeck et al., 2023)) on modern LLMs have shown that they are capable of performing amazing tasks typically necessitating human-level intelligence. However, unlike humans, frozen LLMs do not improve over time; they neither acquire new knowledge nor learn from their successes or failures. Some approaches to improving the intelligence of LLMs include fine-tuning models based on problem-solving performance (Zelikman et al., 2022), and building bigger and more sophisticated models (Bubeck et al., 2023). However, these methods have the drawback of requiring substantial data and computational resources to retrain existing models. In this paper, we explore the use of Retrieval Augmented Generation, also known as RAG (Lewis et al., 2021) to improve problem-solving performance. We propose ARM-RAG (Auxiliary Rationale Memory for Retrieval Augmented Generation), a system that learns from its successes without incurring high training costs. We demonstrate that the storage and subsequent retrieval of reasoning chains have a positive influence on performance in grade-school math problems.

A key property of deep neural networks (DNNs) is their ability to learn new features during training. This intriguing aspect of deep learning stands out most clearly in recently reported Grokking phenomena. While mainly reflected as a sudden increase in test accuracy, Grokking is also believed to be a beyond lazy-learning/Gaussian Process (GP) phenomenon involving feature learning. Here we apply a recent development in the theory of feature learning, the adaptive kernel approach, to two teacher-student models with cubic-polynomial and modular addition teachers. We provide analytical predictions on feature learning and Grokking properties of these models and demonstrate a mapping between Grokking and the theory of phase transitions. We show that after Grokking, the state of the DNN is analogous to the mixed phase following a first-order phase transition. In this mixed phase, the DNN generates useful internal representations of the teacher that are sharply distinct from those before the transition.

Recent advances in reinforcement learning (RL)-based post-training have led to notable improvements in large language models (LLMs), particularly in enhancing their reasoning capabilities to handle complex tasks. However, most existing methods treat the training data as a unified whole, overlooking the fact that modern LLM training often involves a mixture of data from diverse distributions-varying in both source and difficulty. This heterogeneity introduces a key challenge: how to adaptively schedule training across distributions to optimize learning efficiency. In this paper, we present a principled curriculum learning framework grounded in the notion of distribution-level learnability. Our core insight is that the magnitude of policy advantages reflects how much a model can still benefit from further training on a given distribution. Based on this, we propose a distribution-level curriculum learning framework for RL-based LLM post-training, which leverages the Upper Confidence Bound (UCB) principle to dynamically adjust sampling probabilities for different distrubutions. This approach prioritizes distributions with either high average advantage (exploitation) or low sample count (exploration), yielding an adaptive and theoretically grounded training schedule. We instantiate our curriculum learning framework with GRPO as the underlying RL algorithm and demonstrate its effectiveness on logic reasoning datasets with multiple difficulties and sources. Our experiments show that our framework significantly improves convergence speed and final performance, highlighting the value of distribution-aware curriculum strategies in LLM post-training. Code: https://github.com/ZhentingWang/DUMP.

Much research in machine learning involves finding appropriate inductive biases (e.g. convolutional neural networks, momentum-based optimizers, transformers) to promote generalization on tasks. However, quantification of the amount of inductive bias associated with these architectures and hyperparameters has been limited. We propose a novel method for efficiently computing the inductive bias required for generalization on a task with a fixed training data budget; formally, this corresponds to the amount of information required to specify well-generalizing models within a specific hypothesis space of models. Our approach involves modeling the loss distribution of random hypotheses drawn from a hypothesis space to estimate the required inductive bias for a task relative to these hypotheses. Unlike prior work, our method provides a direct estimate of inductive bias without using bounds and is applicable to diverse hypothesis spaces. Moreover, we derive approximation error bounds for our estimation approach in terms of the number of sampled hypotheses. Consistent with prior results, our empirical results demonstrate that higher dimensional tasks require greater inductive bias. We show that relative to other expressive model classes, neural networks as a model class encode large amounts of inductive bias. Furthermore, our measure quantifies the relative difference in inductive bias between different neural network architectures. Our proposed inductive bias metric provides an information-theoretic interpretation of the benefits of specific model architectures for certain tasks and provides a quantitative guide to developing tasks requiring greater inductive bias, thereby encouraging the development of more powerful inductive biases.

Labeling neural network submodules with human-legible descriptions is useful for many downstream tasks: such descriptions can surface failures, guide interventions, and perhaps even explain important model behaviors. To date, most mechanistic descriptions of trained networks have involved small models, narrowly delimited phenomena, and large amounts of human labor. Labeling all human-interpretable sub-computations in models of increasing size and complexity will almost certainly require tools that can generate and validate descriptions automatically. Recently, techniques that use learned models in-the-loop for labeling have begun to gain traction, but methods for evaluating their efficacy are limited and ad-hoc. How should we validate and compare open-ended labeling tools? This paper introduces FIND (Function INterpretation and Description), a benchmark suite for evaluating the building blocks of automated interpretability methods. FIND contains functions that resemble components of trained neural networks, and accompanying descriptions of the kind we seek to generate. The functions are procedurally constructed across textual and numeric domains, and involve a range of real-world complexities, including noise, composition, approximation, and bias. We evaluate new and existing methods that use language models (LMs) to produce code-based and language descriptions of function behavior. We find that an off-the-shelf LM augmented with only black-box access to functions can sometimes infer their structure, acting as a scientist by forming hypotheses, proposing experiments, and updating descriptions in light of new data. However, LM-based descriptions tend to capture global function behavior and miss local corruptions. These results show that FIND will be useful for characterizing the performance of more sophisticated interpretability methods before they are applied to real-world models.

In this paper, we study the predict-then-optimize problem where the output of a machine learning prediction task is used as the input of some downstream optimization problem, say, the objective coefficient vector of a linear program. The problem is also known as predictive analytics or contextual linear programming. The existing approaches largely suffer from either (i) optimization intractability (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii) requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach to the problem called maximum optimality margin which designs the machine learning loss function by the optimality condition of the downstream optimization. The max-margin formulation enjoys both computational efficiency and good theoretical properties for the learning procedure. More importantly, our new approach only needs the observations of the optimal solution in the training data rather than the objective function, which makes it a new and natural approach to the inverse linear programming problem under both contextual and context-free settings; we also analyze the proposed method under both offline and online settings, and demonstrate its performance using numerical experiments.

Large Language Models (LLMs) exhibit strong potential in mathematical reasoning, yet their effectiveness is often limited by a shortage of high-quality queries. This limitation necessitates scaling up computational responses through self-generated data, yet current methods struggle due to spurious correlated data caused by ineffective exploration across all reasoning stages. To address such challenge, we introduce MARGE: Improving Math Reasoning with Guided Exploration, a novel method to address this issue and enhance mathematical reasoning through hit-guided exploration. MARGE systematically explores intermediate reasoning states derived from self-generated solutions, enabling adequate exploration and improved credit assignment throughout the reasoning process. Through extensive experiments across multiple backbone models and benchmarks, we demonstrate that MARGE significantly improves reasoning capabilities without requiring external annotations or training additional value models. Notably, MARGE improves both single-shot accuracy and exploration diversity, mitigating a common trade-off in alignment methods. These results demonstrate MARGE's effectiveness in enhancing mathematical reasoning capabilities and unlocking the potential of scaling self-generated training data. Our code and models are available at https://github.com/georgao35/MARGE{this link}.

This paper introduces a cost-efficient active learning (AL) framework for classification, featuring a novel query design called candidate set query. Unlike traditional AL queries requiring the oracle to examine all possible classes, our method narrows down the set of candidate classes likely to include the ground-truth class, significantly reducing the search space and labeling cost. Moreover, we leverage conformal prediction to dynamically generate small yet reliable candidate sets, adapting to model enhancement over successive AL rounds. To this end, we introduce an acquisition function designed to prioritize data points that offer high information gain at lower cost. Empirical evaluations on CIFAR-10, CIFAR-100, and ImageNet64x64 demonstrate the effectiveness and scalability of our framework. Notably, it reduces labeling cost by 42% on ImageNet64x64.

Autoregressive Large Language Models have transformed the landscape of Natural Language Processing. Pre-train and prompt paradigm has replaced the conventional approach of pre-training and fine-tuning for many downstream NLP tasks. This shift has been possible largely due to LLMs and innovative prompting techniques. LLMs have shown great promise for a variety of downstream tasks owing to their vast parameters and huge datasets that they are pre-trained on. However, in order to fully realize their potential, their outputs must be guided towards the desired outcomes. Prompting, in which a specific input or instruction is provided to guide the LLMs toward the intended output, has become a tool for achieving this goal. In this paper, we discuss the various prompting techniques that have been applied to fully harness the power of LLMs. We present a taxonomy of existing literature on prompting techniques and provide a concise survey based on this taxonomy. Further, we identify some open problems in the realm of prompting in autoregressive LLMs which could serve as a direction for future research.

Large Language Models (LLMs) have a natural role in answering complex queries about data streams, but the high computational cost of LLM inference makes them infeasible in many such tasks. We propose online cascade learning, the first approach to address this challenge. The objective here is to learn a "cascade" of models, starting with lower-capacity models (such as logistic regression) and ending with a powerful LLM, along with a deferral policy that determines the model to be used on a given input. We formulate the task of learning cascades online as an imitation-learning problem, where smaller models are updated over time imitating the collected LLM demonstrations, and give a no-regret algorithm for the problem. Experimental results across four benchmarks show that our method parallels LLMs in accuracy while cutting down inference costs by as much as 90% with strong robustness against input distribution shifts, underscoring its efficacy and adaptability in stream processing.

Approximating nonlinear differential equations using a neural network provides a robust and efficient tool for various scientific computing tasks, including real-time predictions, inverse problems, optimal controls, and surrogate modeling. Previous works have focused on embedding dynamical systems into networks through two approaches: learning a single solution operator (i.e., the mapping from input parametrized functions to solutions) or learning the governing system of equations (i.e., the constitutive model relative to the state variables). Both of these approaches yield different representations for the same underlying data or function. Additionally, observing that families of differential equations often share key characteristics, we seek one network representation across a wide range of equations. Our method, called Predicting Operators and Symbolic Expressions (PROSE), learns maps from multimodal inputs to multimodal outputs, capable of generating both numerical predictions and mathematical equations. By using a transformer structure and a feature fusion approach, our network can simultaneously embed sets of solution operators for various parametric differential equations using a single trained network. Detailed experiments demonstrate that the network benefits from its multimodal nature, resulting in improved prediction accuracy and better generalization. The network is shown to be able to handle noise in the data and errors in the symbolic representation, including noisy numerical values, model misspecification, and erroneous addition or deletion of terms. PROSE provides a new neural network framework for differential equations which allows for more flexibility and generality in learning operators and governing equations from data.

As AI systems become more capable, we would like to enlist their help to supervise other AIs. We experiment with methods for training a harmless AI assistant through self-improvement, without any human labels identifying harmful outputs. The only human oversight is provided through a list of rules or principles, and so we refer to the method as 'Constitutional AI'. The process involves both a supervised learning and a reinforcement learning phase. In the supervised phase we sample from an initial model, then generate self-critiques and revisions, and then finetune the original model on revised responses. In the RL phase, we sample from the finetuned model, use a model to evaluate which of the two samples is better, and then train a preference model from this dataset of AI preferences. We then train with RL using the preference model as the reward signal, i.e. we use 'RL from AI Feedback' (RLAIF). As a result we are able to train a harmless but non-evasive AI assistant that engages with harmful queries by explaining its objections to them. Both the SL and RL methods can leverage chain-of-thought style reasoning to improve the human-judged performance and transparency of AI decision making. These methods make it possible to control AI behavior more precisely and with far fewer human labels.

The success of the machine learning field has reliably depended on training on large datasets. While effective, this trend comes at an extraordinary cost. This is due to two deeply intertwined factors: the size of models and the size of datasets. While promising research efforts focus on reducing the size of models, the other half of the equation remains fairly mysterious. Indeed, it is surprising that the standard approach to training remains to iterate over and over, uniformly sampling the training dataset. In this paper we explore a series of alternative training paradigms that leverage insights from hard-data-mining and dropout, simple enough to implement and use that can become the new training standard. The proposed Progressive Data Dropout reduces the number of effective epochs to as little as 12.4% of the baseline. This savings actually do not come at any cost for accuracy. Surprisingly, the proposed method improves accuracy by up to 4.82%. Our approach requires no changes to model architecture or optimizer, and can be applied across standard training pipelines, thus posing an excellent opportunity for wide adoption. Code can be found here: https://github.com/bazyagami/LearningWithRevision

Most positive and unlabeled data is subject to selection biases. The labeled examples can, for example, be selected from the positive set because they are easier to obtain or more obviously positive. This paper investigates how learning can be ena BHbled in this setting. We propose and theoretically analyze an empirical-risk-based method for incorporating the labeling mechanism. Additionally, we investigate under which assumptions learning is possible when the labeling mechanism is not fully understood and propose a practical method to enable this. Our empirical analysis supports the theoretical results and shows that taking into account the possibility of a selection bias, even when the labeling mechanism is unknown, improves the trained classifiers.

Language model post-training is applied to refine behaviors and unlock new skills across a wide range of recent language models, but open recipes for applying these techniques lag behind proprietary ones. The underlying training data and recipes for post-training are simultaneously the most important pieces of the puzzle and the portion with the least transparency. To bridge this gap, we introduce T\"ULU 3, a family of fully-open state-of-the-art post-trained models, alongside its data, code, and training recipes, serving as a comprehensive guide for modern post-training techniques. T\"ULU 3, which builds on Llama 3.1 base models, achieves results surpassing the instruct versions of Llama 3.1, Qwen 2.5, Mistral, and even closed models such as GPT-4o-mini and Claude 3.5-Haiku. The training algorithms for our models include supervised finetuning (SFT), Direct Preference Optimization (DPO), and a novel method we call Reinforcement Learning with Verifiable Rewards (RLVR). With T\"ULU 3, we introduce a multi-task evaluation scheme for post-training recipes with development and unseen evaluations, standard benchmark implementations, and substantial decontamination of existing open datasets on said benchmarks. We conclude with analysis and discussion of training methods that did not reliably improve performance. In addition to the T\"ULU 3 model weights and demo, we release the complete recipe -- including datasets for diverse core skills, a robust toolkit for data curation and evaluation, the training code and infrastructure, and, most importantly, a detailed report for reproducing and further adapting the T\"ULU 3 approach to more domains.

When selecting data for training large-scale models, standard practice is to filter for examples that match human notions of data quality. Such filtering yields qualitatively clean datapoints that intuitively should improve model behavior. However, in practice the opposite can often happen: we find that selecting according to similarity with "high quality" data sources may not increase (and can even hurt) performance compared to randomly selecting data. To develop better methods for selecting data, we start by framing dataset selection as an optimization problem that we can directly solve for: given target tasks, a learning algorithm, and candidate data, select the subset that maximizes model performance. This framework thus avoids handpicked notions of data quality, and instead models explicitly how the learning process uses train datapoints to predict on the target tasks. Our resulting method greatly improves language model (LM) performance on both pre-specified tasks and previously unseen tasks. Specifically, choosing target tasks representative of standard LM problems and evaluating on diverse held-out benchmarks, our selected datasets provide a 2x compute multiplier over baseline methods.

Robot manipulation is a common task in fields like industrial manufacturing. Detecting when objects slip from a robot's grasp is crucial for safe and reliable operation. Event cameras, which register pixel-level brightness changes at high temporal resolution (called ``events''), offer an elegant feature when mounted on a robot's end effector: since they only detect motion relative to their viewpoint, a properly grasped object produces no events, while a slipping object immediately triggers them. To research this feature, representative datasets are essential, both for analytic approaches and for training machine learning models. The majority of current research on slip detection with event-based data is done on real-world scenarios and manual data collection, as well as additional setups for data labeling. This can result in a significant increase in the time required for data collection, a lack of flexibility in scene setups, and a high level of complexity in the repetition of experiments. This paper presents a simulation pipeline for generating slip data using the described camera-gripper configuration in a robot arm, and demonstrates its effectiveness through initial data-driven experiments. The use of a simulator, once it is set up, has the potential to reduce the time spent on data collection, provide the ability to alter the setup at any time, simplify the process of repetition and the generation of arbitrarily large data sets. Two distinct datasets were created and validated through visual inspection and artificial neural networks (ANNs). Visual inspection confirmed photorealistic frame generation and accurate slip modeling, while three ANNs trained on this data achieved high validation accuracy and demonstrated good generalization capabilities on a separate test set, along with initial applicability to real-world data. Project page: https://github.com/tub-rip/event_slip

We present an empirical evaluation of various outputs generated by nine of the most widely-available large language models (LLMs). Our analysis is done with off-the-shelf, readily-available tools. We find a correlation between percentage of memorized text, percentage of unique text, and overall output quality, when measured with respect to output pathologies such as counterfactual and logically-flawed statements, and general failures like not staying on topic. Overall, 80.0% of the outputs evaluated contained memorized data, but outputs containing the most memorized content were also more likely to be considered of high quality. We discuss and evaluate mitigation strategies, showing that, in the models evaluated, the rate of memorized text being output is reduced. We conclude with a discussion on potential implications around what it means to learn, to memorize, and to evaluate quality text.

Lectures are a learning experience for both students and teachers. Students learn from teachers about the subject material, while teachers learn from students about how to refine their instruction. However, online student feedback is unstructured and abundant, making it challenging for teachers to learn and improve. We take a step towards tackling this challenge. First, we contribute a dataset for studying this problem: SIGHT is a large dataset of 288 math lecture transcripts and 15,784 comments collected from the Massachusetts Institute of Technology OpenCourseWare (MIT OCW) YouTube channel. Second, we develop a rubric for categorizing feedback types using qualitative analysis. Qualitative analysis methods are powerful in uncovering domain-specific insights, however they are costly to apply to large data sources. To overcome this challenge, we propose a set of best practices for using large language models (LLMs) to cheaply classify the comments at scale. We observe a striking correlation between the model's and humans' annotation: Categories with consistent human annotations (>0.9 inter-rater reliability, IRR) also display higher human-model agreement (>0.7), while categories with less consistent human annotations (0.7-0.8 IRR) correspondingly demonstrate lower human-model agreement (0.3-0.5). These techniques uncover useful student feedback from thousands of comments, costing around 0.002$ per comment. We conclude by discussing exciting future directions on using online student feedback and improving automated annotation techniques for qualitative research.

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the optimization dynamics for the inner objective. Depending on the specific setting, the outer variables take either the meaning of hyperparameters in a supervised learning problem or parameters of a meta-learner. We provide sufficient conditions under which solutions of the approximate problem converge to those of the exact problem. We instantiate our approach for meta-learning in the case of deep learning where representation layers are treated as hyperparameters shared across a set of training episodes. In experiments, we confirm our theoretical findings, present encouraging results for few-shot learning and contrast the bilevel approach against classical approaches for learning-to-learn.

While the history of machine learning so far largely encompasses a series of problems posed by researchers and algorithms that learn their solutions, an important question is whether the problems themselves can be generated by the algorithm at the same time as they are being solved. Such a process would in effect build its own diverse and expanding curricula, and the solutions to problems at various stages would become stepping stones towards solving even more challenging problems later in the process. The Paired Open-Ended Trailblazer (POET) algorithm introduced in this paper does just that: it pairs the generation of environmental challenges and the optimization of agents to solve those challenges. It simultaneously explores many different paths through the space of possible problems and solutions and, critically, allows these stepping-stone solutions to transfer between problems if better, catalyzing innovation. The term open-ended signifies the intriguing potential for algorithms like POET to continue to create novel and increasingly complex capabilities without bound. Our results show that POET produces a diverse range of sophisticated behaviors that solve a wide range of environmental challenges, many of which cannot be solved by direct optimization alone, or even through a direct-path curriculum-building control algorithm introduced to highlight the critical role of open-endedness in solving ambitious challenges. The ability to transfer solutions from one environment to another proves essential to unlocking the full potential of the system as a whole, demonstrating the unpredictable nature of fortuitous stepping stones. We hope that POET will inspire a new push towards open-ended discovery across many domains, where algorithms like POET can blaze a trail through their interesting possible manifestations and solutions.

Large language models (LLMs) undergo a three-phase training process: unsupervised pre-training, supervised fine-tuning (SFT), and learning from human feedback (RLHF/DPO). Notably, it is during the final phase that these models are exposed to negative examples -- incorrect, rejected, or suboptimal responses to queries. This paper delves into the role of negative examples in the training of LLMs, using a likelihood-ratio (Likra) model on multiple-choice question answering benchmarks to precisely manage the influence and the volume of negative examples. Our findings reveal three key insights: (1) During a critical phase in training, Likra with negative examples demonstrates a significantly larger improvement per training example compared to SFT using only positive examples. This leads to a sharp jump in the learning curve for Likra unlike the smooth and gradual improvement of SFT; (2) negative examples that are plausible but incorrect (near-misses) exert a greater influence; and (3) while training with positive examples fails to significantly decrease the likelihood of plausible but incorrect answers, training with negative examples more accurately identifies them. These results indicate a potentially significant role for negative examples in improving accuracy and reducing hallucinations for LLMs.

In this paper, we conduct a comprehensive study of In-Context Learning (ICL) by addressing several open questions: (a) What type of ICL estimator is learned by large language models? (b) What is a proper performance metric for ICL and what is the error rate? (c) How does the transformer architecture enable ICL? To answer these questions, we adopt a Bayesian view and formulate ICL as a problem of predicting the response corresponding to the current covariate, given a number of examples drawn from a latent variable model. To answer (a), we show that, without updating the neural network parameters, ICL implicitly implements the Bayesian model averaging algorithm, which is proven to be approximately parameterized by the attention mechanism. For (b), we analyze the ICL performance from an online learning perspective and establish a O(1/T) regret bound for perfectly pretrained ICL, where T is the number of examples in the prompt. To answer (c), we show that, in addition to encoding Bayesian model averaging via attention, the transformer architecture also enables a fine-grained statistical analysis of pretraining under realistic assumptions. In particular, we prove that the error of pretrained model is bounded by a sum of an approximation error and a generalization error, where the former decays to zero exponentially as the depth grows, and the latter decays to zero sublinearly with the number of tokens in the pretraining dataset. Our results provide a unified understanding of the transformer and its ICL ability with bounds on ICL regret, approximation, and generalization, which deepens our knowledge of these essential aspects of modern language models.

Recent empirical works have successfully used unlabeled data to learn feature representations that are broadly useful in downstream classification tasks. Several of these methods are reminiscent of the well-known word2vec embedding algorithm: leveraging availability of pairs of semantically "similar" data points and "negative samples," the learner forces the inner product of representations of similar pairs with each other to be higher on average than with negative samples. The current paper uses the term contrastive learning for such algorithms and presents a theoretical framework for analyzing them by introducing latent classes and hypothesizing that semantically similar points are sampled from the same latent class. This framework allows us to show provable guarantees on the performance of the learned representations on the average classification task that is comprised of a subset of the same set of latent classes. Our generalization bound also shows that learned representations can reduce (labeled) sample complexity on downstream tasks. We conduct controlled experiments in both the text and image domains to support the theory.

The goal of machine learning is generalization. While the No Free Lunch Theorem states that we cannot obtain theoretical guarantees for generalization without further assumptions, in practice we observe that simple models which explain the training data generalize best: a principle called Occam's razor. Despite the need for simple models, most current approaches in machine learning only minimize the training error, and at best indirectly promote simplicity through regularization or architecture design. Here, we draw a connection between Occam's razor and in-context learning: an emergent ability of certain sequence models like Transformers to learn at inference time from past observations in a sequence. In particular, we show that the next-token prediction loss used to train in-context learners is directly equivalent to a data compression technique called prequential coding, and that minimizing this loss amounts to jointly minimizing both the training error and the complexity of the model that was implicitly learned from context. Our theory and the empirical experiments we use to support it not only provide a normative account of in-context learning, but also elucidate the shortcomings of current in-context learning methods, suggesting ways in which they can be improved. We make our code available at https://github.com/3rdCore/PrequentialCode.

This paper surveys research in the rapidly growing field of enhancing large language models (LLMs) with reinforcement learning (RL), a technique that enables LLMs to improve their performance by receiving feedback in the form of rewards based on the quality of their outputs, allowing them to generate more accurate, coherent, and contextually appropriate responses. In this work, we make a systematic review of the most up-to-date state of knowledge on RL-enhanced LLMs, attempting to consolidate and analyze the rapidly growing research in this field, helping researchers understand the current challenges and advancements. Specifically, we (1) detail the basics of RL; (2) introduce popular RL-enhanced LLMs; (3) review researches on two widely-used reward model-based RL techniques: Reinforcement Learning from Human Feedback (RLHF) and Reinforcement Learning from AI Feedback (RLAIF); and (4) explore Direct Preference Optimization (DPO), a set of methods that bypass the reward model to directly use human preference data for aligning LLM outputs with human expectations. We will also point out current challenges and deficiencies of existing methods and suggest some avenues for further improvements. Project page of this work can be found at: https://github.com/ShuheWang1998/Reinforcement-Learning-Enhanced-LLMs-A-Survey.

We propose a new learning paradigm called Deep Memory. It has the potential to completely revolutionize the Machine Learning field. Surprisingly, this paradigm has not been reinvented yet, unlike Deep Learning. At the core of this approach is the Learning By Heart principle, well studied in primary schools all over the world. Inspired by poem recitation, or by pi decimal memorization, we propose a concrete algorithm that mimics human behavior. We implement this paradigm on the task of generative modeling, and apply to images, natural language and even the pi decimals as long as one can print them as text. The proposed algorithm even generated this paper, in a one-shot learning setting. In carefully designed experiments, we show that the generated samples are indistinguishable from the training examples, as measured by any statistical tests or metrics.

When users can benefit from certain predictive outcomes, they may be prone to act to achieve those outcome, e.g., by strategically modifying their features. The goal in strategic classification is therefore to train predictive models that are robust to such behavior. However, the conventional framework assumes that changing features does not change actual outcomes, which depicts users as "gaming" the system. Here we remove this assumption, and study learning in a causal strategic setting where true outcomes do change. Focusing on accuracy as our primary objective, we show how strategic behavior and causal effects underlie two complementing forms of distribution shift. We characterize these shifts, and propose a learning algorithm that balances between these two forces and over time, and permits end-to-end training. Experiments on synthetic and semi-synthetic data demonstrate the utility of our approach.

This paper presents our winning submission to the AI Mathematical Olympiad - Progress Prize 2 (AIMO-2) competition. Our recipe for building state-of-the-art mathematical reasoning models relies on three key pillars. First, we create a large-scale dataset comprising 540K unique high-quality math problems, including olympiad-level problems, and their 3.2M long-reasoning solutions. Second, we develop a novel method to integrate code execution with long reasoning models through iterative training, generation, and quality filtering, resulting in 1.7M high-quality Tool-Integrated Reasoning solutions. Third, we create a pipeline to train models to select the most promising solution from many candidates. We show that such generative solution selection (GenSelect) can significantly improve upon majority voting baseline. Combining these ideas, we train a series of models that achieve state-of-the-art results on mathematical reasoning benchmarks. To facilitate further research, we release our code, models, and the complete OpenMathReasoning dataset under a commercially permissive license.

We introduce transductive program synthesis, a new formulation of the program synthesis task that explicitly leverages test inputs during synthesis. While prior approaches to program synthesis--whether based on natural language descriptions or input-output examples--typically aim to generalize from training examples, they often struggle with robustness, especially in real-world settings where training examples are limited and test inputs involve various edge cases. To address this, we propose a novel framework that improves robustness by treating synthesis as an active learning over a finite hypothesis class defined by programs' outputs. We use an LLM to predict outputs for selected test inputs and eliminate inconsistent hypotheses, where the inputs are chosen via a greedy maximin algorithm to minimize the number of LLM queries required. We evaluate our approach on four benchmarks: Playgol, MBPP+, 1D-ARC, and programmatic world modeling on MiniGrid. We demonstrate that our method significantly improves program synthesis in both accuracy and efficiency. We release our code at https://github.com/klee972/SYNTRA.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form h = g circ p where p : R^d rightarrow R is a degree k polynomial and g: R rightarrow R is a degree q polynomial. This function class generalizes the single-index model, which corresponds to k=1, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree k polynomials p, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target h up to vanishing test error in mathcal{O}(d^k) samples and polynomial time. This is a strict improvement over kernel methods, which require widetilde Theta(d^{kq}) samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of p being a quadratic. When p is indeed a quadratic, we achieve the information-theoretically optimal sample complexity mathcal{O}(d^2), which is an improvement over prior work~nichani2023provable requiring a sample size of widetildeTheta(d^4). Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature p with mathcal{O}(d^k) samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

We investigate the mathematical capabilities of ChatGPT by testing it on publicly available datasets, as well as hand-crafted ones, and measuring its performance against other models trained on a mathematical corpus, such as Minerva. We also test whether ChatGPT can be a useful assistant to professional mathematicians by emulating various use cases that come up in the daily professional activities of mathematicians (question answering, theorem searching). In contrast to formal mathematics, where large databases of formal proofs are available (e.g., the Lean Mathematical Library), current datasets of natural-language mathematics, used to benchmark language models, only cover elementary mathematics. We address this issue by introducing a new dataset: GHOSTS. It is the first natural-language dataset made and curated by working researchers in mathematics that (1) aims to cover graduate-level mathematics and (2) provides a holistic overview of the mathematical capabilities of language models. We benchmark ChatGPT on GHOSTS and evaluate performance against fine-grained criteria. We make this new dataset publicly available to assist a community-driven comparison of ChatGPT with (future) large language models in terms of advanced mathematical comprehension. We conclude that contrary to many positive reports in the media (a potential case of selection bias), ChatGPT's mathematical abilities are significantly below those of an average mathematics graduate student. Our results show that ChatGPT often understands the question but fails to provide correct solutions. Hence, if your goal is to use it to pass a university exam, you would be better off copying from your average peer!

Large language models (LLMs) have seen considerable advancements in natural language understanding tasks, yet there remains a gap to bridge before attaining true artificial general intelligence, especially concerning shortcomings in mathematical reasoning capabilities. We postulate that the inherent nature of LLM training, which focuses on predicting probabilities of next token, presents challenges in effectively modeling mathematical reasoning that demands exact calculations, both from data-driven and theoretical standpoints. In this paper, we address this challenge by enriching the data landscape and introducing a novel math dataset, enhanced with a capability to utilize a Python code interpreter. This dataset is derived from GSM8K and MATH and has been further refined through a combination of GPT-4 annotations, human review, and self-training processes, where the errors in the original GSM8K training set have been fixed. Additionally, we propose a tentative, easily replicable protocol for the fine-tuning of math-specific LLMs, which has led to a significant improvement in the performance of a 7B-parameter LLM on the GSM8K and MATH datasets. We are committed to advancing the field of mathematical reasoning in LLMs and, to that end, we have made the model checkpoints and will make the dataset publicly available. We hope this will facilitate further research and development within the community.

Recent large language models (LLMs) have shown indications of mathematical reasoning ability. However it has not been clear how they would fare on more challenging competition-level problems. And while self-generated verbalizations of intermediate reasoning steps (i.e., chain-of-thought prompting) have been shown to be helpful, whether LLMs can make use of helpful side information such as problem-specific hints has not been investigated before. In this paper, we propose a challenging benchmark dataset for enabling such analyses. The Concept and Hint-Annotated Math Problems (CHAMP) consists of high school math competition problems, annotated with concepts, or general math facts, and hints, or problem-specific tricks. These annotations allow us to explore the effects of additional information, such as relevant hints, misleading concepts, or related problems. This benchmark is difficult, with the best model only scoring 58.1% in standard settings. With concepts and hints, performance sometimes improves, indicating that some models can make use of such side information. We further annotate model-generated solutions for their correctness. Using this corpus, we find that models often arrive at the correct final answer through wrong reasoning steps. In addition, we test whether models are able to verify these solutions, and find that most models struggle. The dataset and code are available on the project website.

Recent advancements in language models have demonstrated remarkable improvements in various natural language processing (NLP) tasks such as web navigation. Supervised learning (SL) approaches have achieved impressive performance while utilizing significantly less training data compared to previous methods. However, these SL-based models fall short when compared to reinforcement learning (RL) approaches, which have shown superior results. In this paper, we propose a novel approach that combines SL and RL techniques over the MiniWoB benchmark to leverage the strengths of both methods. We also address a critical limitation in previous models' understanding of HTML content, revealing a tendency to memorize target elements rather than comprehend the underlying structure. To rectify this, we propose methods to enhance true understanding and present a new baseline of results. Our experiments demonstrate that our approach outperforms previous SL methods on certain tasks using less data and narrows the performance gap with RL models, achieving 43.58\% average accuracy in SL and 36.69\% when combined with a multimodal RL approach. This study sets a new direction for future web navigation and offers insights into the limitations and potential of language modeling for computer tasks.

In this paper, we introduce the novel concept of pedagogically aligned Large Language Models (LLMs) that signifies a transformative shift in the application of LLMs within educational contexts. Rather than providing direct responses to user queries, pedagogically-aligned LLMs function as scaffolding tools, breaking complex problems into manageable subproblems and guiding students towards the final answer through constructive feedback and hints. The objective is to equip learners with problem-solving strategies that deepen their understanding and internalization of the subject matter. Previous research in this field has primarily applied the supervised finetuning approach without framing the objective as an alignment problem, hence not employing reinforcement learning through human feedback (RLHF) methods. This study reinterprets the narrative by viewing the task through the lens of alignment and demonstrates how RLHF methods emerge naturally as a superior alternative for aligning LLM behaviour. Building on this perspective, we propose a novel approach for constructing a reward dataset specifically designed for the pedagogical alignment of LLMs. We apply three state-of-the-art RLHF algorithms and find that they outperform SFT significantly. Our qualitative analyses across model differences and hyperparameter sensitivity further validate the superiority of RLHF over SFT. Also, our study sheds light on the potential of online feedback for enhancing the performance of pedagogically-aligned LLMs, thus providing valuable insights for the advancement of these models in educational settings.

Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems. We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.

Scaling law principles indicate a power-law correlation between loss and variables such as model size, dataset size, and computational resources utilized during training. These principles play a vital role in optimizing various aspects of model pre-training, ultimately contributing to the success of large language models such as GPT-4, Llama and Gemini. However, the original scaling law paper by OpenAI did not disclose the complete details necessary to derive the precise scaling law formulas, and their conclusions are only based on models containing up to 1.5 billion parameters. Though some subsequent works attempt to unveil these details and scale to larger models, they often neglect the training dependency of important factors such as the learning rate, context length and batch size, leading to their failure to establish a reliable formula for predicting the test loss trajectory. In this technical report, we confirm that the scaling law formulations proposed in the original OpenAI paper remain valid when scaling the model size up to 33 billion, but the constant coefficients in these formulas vary significantly with the experiment setup. We meticulously identify influential factors and provide transparent, step-by-step instructions to estimate all constant terms in scaling-law formulas by training on models with only 1M~60M parameters. Using these estimated formulas, we showcase the capability to accurately predict various attributes for models with up to 33B parameters before their training, including (1) the minimum possible test loss; (2) the minimum required training steps and processed tokens to achieve a specific loss; (3) the critical batch size with an optimal time/computation trade-off at any loss value; and (4) the complete test loss trajectory with arbitrary batch size.

This study focuses on improving the performance of lightweight Large Language Models (LLMs) in mathematical reasoning tasks. We introduce a novel method for measuring mathematical logic similarity and design an automatic screening mechanism to construct a set of reference problems that integrate both semantic and logical similarity. By employing carefully crafted positive and negative example prompts, we guide the model towards adopting sound reasoning logic. To the best of our knowledge, this is the first attempt to utilize retrieval-enhanced generation for mathematical problem-solving. Experimental results demonstrate that our method achieves a 15.8% improvement over the Chain of Thought approach on the SVAMP dataset and a 21.5 % improvement on the GSM8K dataset. Further application of this method to a large-scale model with 175 billion parameters yields performance comparable to the best results on both aforementioned datasets. Finally, we conduct an analysis of errors during the reasoning process, providing valuable insights and directions for future research on reasoning tasks using large language models.

In self-supervised learning, a system is tasked with achieving a surrogate objective by defining alternative targets on a set of unlabeled data. The aim is to build useful representations that can be used in downstream tasks, without costly manual annotation. In this work, we propose a novel self-supervised formulation of relational reasoning that allows a learner to bootstrap a signal from information implicit in unlabeled data. Training a relation head to discriminate how entities relate to themselves (intra-reasoning) and other entities (inter-reasoning), results in rich and descriptive representations in the underlying neural network backbone, which can be used in downstream tasks such as classification and image retrieval. We evaluate the proposed method following a rigorous experimental procedure, using standard datasets, protocols, and backbones. Self-supervised relational reasoning outperforms the best competitor in all conditions by an average 14% in accuracy, and the most recent state-of-the-art model by 3%. We link the effectiveness of the method to the maximization of a Bernoulli log-likelihood, which can be considered as a proxy for maximizing the mutual information, resulting in a more efficient objective with respect to the commonly used contrastive losses.

Deep learning has recently seen rapid development and received significant attention due to its state-of-the-art performance on previously-thought hard problems. However, because of the internal complexity and nonlinear structure of deep neural networks, the underlying decision making processes for why these models are achieving such performance are challenging and sometimes mystifying to interpret. As deep learning spreads across domains, it is of paramount importance that we equip users of deep learning with tools for understanding when a model works correctly, when it fails, and ultimately how to improve its performance. Standardized toolkits for building neural networks have helped democratize deep learning; visual analytics systems have now been developed to support model explanation, interpretation, debugging, and improvement. We present a survey of the role of visual analytics in deep learning research, which highlights its short yet impactful history and thoroughly summarizes the state-of-the-art using a human-centered interrogative framework, focusing on the Five W's and How (Why, Who, What, How, When, and Where). We conclude by highlighting research directions and open research problems. This survey helps researchers and practitioners in both visual analytics and deep learning to quickly learn key aspects of this young and rapidly growing body of research, whose impact spans a diverse range of domains.

Learning from preference labels plays a crucial role in fine-tuning large language models. There are several distinct approaches for preference fine-tuning, including supervised learning, on-policy reinforcement learning (RL), and contrastive learning. Different methods come with different implementation tradeoffs and performance differences, and existing empirical findings present different conclusions, for instance, some results show that online RL is quite important to attain good fine-tuning results, while others find (offline) contrastive or even purely supervised methods sufficient. This raises a natural question: what kind of approaches are important for fine-tuning with preference data and why? In this paper, we answer this question by performing a rigorous analysis of a number of fine-tuning techniques on didactic and full-scale LLM problems. Our main finding is that, in general, approaches that use on-policy sampling or attempt to push down the likelihood on certain responses (i.e., employ a "negative gradient") outperform offline and maximum likelihood objectives. We conceptualize our insights and unify methods that use on-policy sampling or negative gradient under a notion of mode-seeking objectives for categorical distributions. Mode-seeking objectives are able to alter probability mass on specific bins of a categorical distribution at a fast rate compared to maximum likelihood, allowing them to relocate masses across bins more effectively. Our analysis prescribes actionable insights for preference fine-tuning of LLMs and informs how data should be collected for maximal improvement.

We begin the study of list-decodable linear regression using batches. In this setting only an alpha in (0,1] fraction of the batches are genuine. Each genuine batch contains ge n i.i.d. samples from a common unknown distribution and the remaining batches may contain arbitrary or even adversarial samples. We derive a polynomial time algorithm that for any nge tilde Omega(1/alpha) returns a list of size mathcal O(1/alpha^2) such that one of the items in the list is close to the true regression parameter. The algorithm requires only mathcal{O}(d/alpha^2) genuine batches and works under fairly general assumptions on the distribution. The results demonstrate the utility of batch structure, which allows for the first polynomial time algorithm for list-decodable regression, which may be impossible for the non-batch setting, as suggested by a recent SQ lower bound diakonikolas2021statistical for the non-batch setting.

Large Language Models (LLMs) are foundational to AI advancements, facilitating applications like predictive text generation. Nonetheless, they pose risks by potentially memorizing and disseminating sensitive, biased, or copyrighted information from their vast datasets. Machine unlearning emerges as a cutting-edge solution to mitigate these concerns, offering techniques for LLMs to selectively discard certain data. This paper reviews the latest in machine unlearning for LLMs, introducing methods for the targeted forgetting of information to address privacy, ethical, and legal challenges without necessitating full model retraining. It divides existing research into unlearning from unstructured/textual data and structured/classification data, showcasing the effectiveness of these approaches in removing specific data while maintaining model efficacy. Highlighting the practicality of machine unlearning, this analysis also points out the hurdles in preserving model integrity, avoiding excessive or insufficient data removal, and ensuring consistent outputs, underlining the role of machine unlearning in advancing responsible, ethical AI.

We introduce Confucius3-Math, an open-source large language model with 14B parameters that (1) runs efficiently on a single consumer-grade GPU; (2) achieves SOTA performances on a range of mathematical reasoning tasks, outperforming many models with significantly larger sizes. In particular, as part of our mission to enhancing education and knowledge dissemination with AI, Confucius3-Math is specifically committed to mathematics learning for Chinese K-12 students and educators. Built via post-training with large-scale reinforcement learning (RL), Confucius3-Math aligns with national curriculum and excels at solving main-stream Chinese K-12 mathematical problems with low cost. In this report we share our development recipe, the challenges we encounter and the techniques we develop to overcome them. In particular, we introduce three technical innovations: Targeted Entropy Regularization, Recent Sample Recovery and Policy-Specific Hardness Weighting. These innovations encompass a new entropy regularization, a novel data scheduling policy, and an improved group-relative advantage estimator. Collectively, they significantly stabilize the RL training, improve data efficiency, and boost performance. Our work demonstrates the feasibility of building strong reasoning models in a particular domain at low cost. We open-source our model and code at https://github.com/netease-youdao/Confucius3-Math.

Reinforcement Learning (RL) has played a central role in the recent surge of LLMs' math abilities by enabling self-improvement through binary verifier signals. In contrast, Supervised Learning (SL) is rarely considered for such verification-driven training, largely due to its heavy reliance on reference answers and inability to reflect on mistakes. In this work, we challenge the prevailing notion that self-improvement is exclusive to RL and propose Negative-aware Fine-Tuning (NFT) -- a supervised approach that enables LLMs to reflect on their failures and improve autonomously with no external teachers. In online training, instead of throwing away self-generated negative answers, NFT constructs an implicit negative policy to model them. This implicit policy is parameterized with the same positive LLM we target to optimize on positive data, enabling direct policy optimization on all LLMs' generations. We conduct experiments on 7B and 32B models in math reasoning tasks. Results consistently show that through the additional leverage of negative feedback, NFT significantly improves over SL baselines like Rejection sampling Fine-Tuning, matching or even surpassing leading RL algorithms like GRPO and DAPO. Furthermore, we demonstrate that NFT and GRPO are actually equivalent in strict-on-policy training, even though they originate from entirely different theoretical foundations. Our experiments and theoretical findings bridge the gap between SL and RL methods in binary-feedback learning systems.

We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations whose training dynamics and dependence on training set size can be predicted by our effective theory in a toy setting. We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a "Goldilocks zone" (including comprehension and grokking) between memorization and confusion. We find on transformers the grokking phase stays closer to the memorization phase (compared to the comprehension phase), leading to delayed generalization. The Goldilocks phase is reminiscent of "intelligence from starvation" in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning.

Self-Supervised Learning (SSL) has emerged as the solution of choice to learn transferable representations from unlabeled data. However, SSL requires to build samples that are known to be semantically akin, i.e. positive views. Requiring such knowledge is the main limitation of SSL and is often tackled by ad-hoc strategies e.g. applying known data-augmentations to the same input. In this work, we generalize and formalize this principle through Positive Active Learning (PAL) where an oracle queries semantic relationships between samples. PAL achieves three main objectives. First, it unveils a theoretically grounded learning framework beyond SSL, that can be extended to tackle supervised and semi-supervised learning depending on the employed oracle. Second, it provides a consistent algorithm to embed a priori knowledge, e.g. some observed labels, into any SSL losses without any change in the training pipeline. Third, it provides a proper active learning framework yielding low-cost solutions to annotate datasets, arguably bringing the gap between theory and practice of active learning that is based on simple-to-answer-by-non-experts queries of semantic relationships between inputs.

In symbolic regression, the goal is to find an analytical expression that accurately fits experimental data with the minimal use of mathematical symbols such as operators, variables, and constants. However, the combinatorial space of possible expressions can make it challenging for traditional evolutionary algorithms to find the correct expression in a reasonable amount of time. To address this issue, Neural Symbolic Regression (NSR) algorithms have been developed that can quickly identify patterns in the data and generate analytical expressions. However, these methods, in their current form, lack the capability to incorporate user-defined prior knowledge, which is often required in natural sciences and engineering fields. To overcome this limitation, we propose a novel neural symbolic regression method, named Neural Symbolic Regression with Hypothesis (NSRwH) that enables the explicit incorporation of assumptions about the expected structure of the ground-truth expression into the prediction process. Our experiments demonstrate that the proposed conditioned deep learning model outperforms its unconditioned counterparts in terms of accuracy while also providing control over the predicted expression structure.

We study the data selection problem, whose aim is to select a small representative subset of data that can be used to efficiently train a machine learning model. We present a new data selection approach based on k-means clustering and sensitivity sampling. Assuming access to an embedding representation of the data with respect to which the model loss is H\"older continuous, our approach provably allows selecting a set of ``typical'' k + 1/varepsilon^2 elements whose average loss corresponds to the average loss of the whole dataset, up to a multiplicative (1pmvarepsilon) factor and an additive varepsilon lambda Phi_k, where Phi_k represents the k-means cost for the input embeddings and lambda is the H\"older constant. We furthermore demonstrate the performance and scalability of our approach on fine-tuning foundation models and show that it outperforms state-of-the-art methods. We also show how it can be applied on linear regression, leading to a new sampling strategy that surprisingly matches the performances of leverage score sampling, while being conceptually simpler and more scalable.

Language model fine-tuning is essential for modern natural language processing, but is computationally expensive and time-consuming. Further, the effectiveness of fine-tuning is limited by the inclusion of training examples that negatively affect performance. Here we present a general fine-tuning method that we call information gain filtration for improving the overall training efficiency and final performance of language model fine-tuning. We define the information gain of an example as the improvement on a test metric after training on that example. A secondary learner is then trained to approximate this quantity. During fine-tuning, this learner selects informative examples and skips uninformative ones. We show that our method has consistent improvement across datasets, fine-tuning tasks, and language model architectures. For example, we achieve a median perplexity of 54.0 on a books dataset compared to 57.3 for standard fine-tuning. We present statistical evidence that offers insight into the improvements of our method over standard fine-tuning. The generality of our method leads us to propose a new paradigm for language model fine-tuning -- we encourage researchers to release pretrained secondary learners on common corpora to promote efficient and effective fine-tuning, thereby improving the performance and reducing the overall energy footprint of language model fine-tuning.

We present a novel framework for learning on CW-complex structured data points. Recent advances have discussed CW-complexes as ideal learning representations for problems in cheminformatics. However, there is a lack of available machine learning methods suitable for learning on CW-complexes. In this paper we develop notions of convolution and attention that are well defined for CW-complexes. These notions enable us to create the first Hodge informed neural network that can receive a CW-complex as input. We illustrate and interpret this framework in the context of supervised prediction.

Knowledge probing assesses to which degree a language model (LM) has successfully learned relational knowledge during pre-training. Probing is an inexpensive way to compare LMs of different sizes and training configurations. However, previous approaches rely on the objective function used in pre-training LMs and are thus applicable only to masked or causal LMs. As a result, comparing different types of LMs becomes impossible. To address this, we propose an approach that uses an LM's inherent ability to estimate the log-likelihood of any given textual statement. We carefully design an evaluation dataset of 7,731 instances (40,916 in a larger variant) from which we produce alternative statements for each relational fact, one of which is correct. We then evaluate whether an LM correctly assigns the highest log-likelihood to the correct statement. Our experimental evaluation of 22 common LMs shows that our proposed framework, BEAR, can effectively probe for knowledge across different LM types. We release the BEAR datasets and an open-source framework that implements the probing approach to the research community to facilitate the evaluation and development of LMs.

Solving algebraic word problems requires executing a series of arithmetic operations---a program---to obtain a final answer. However, since programs can be arbitrarily complicated, inducing them directly from question-answer pairs is a formidable challenge. To make this task more feasible, we solve these problems by generating answer rationales, sequences of natural language and human-readable mathematical expressions that derive the final answer through a series of small steps. Although rationales do not explicitly specify programs, they provide a scaffolding for their structure via intermediate milestones. To evaluate our approach, we have created a new 100,000-sample dataset of questions, answers and rationales. Experimental results show that indirect supervision of program learning via answer rationales is a promising strategy for inducing arithmetic programs.

Machine learning has been highly successful in data-intensive applications but is often hampered when the data set is small. Recently, Few-Shot Learning (FSL) is proposed to tackle this problem. Using prior knowledge, FSL can rapidly generalize to new tasks containing only a few samples with supervised information. In this paper, we conduct a thorough survey to fully understand FSL. Starting from a formal definition of FSL, we distinguish FSL from several relevant machine learning problems. We then point out that the core issue in FSL is that the empirical risk minimized is unreliable. Based on how prior knowledge can be used to handle this core issue, we categorize FSL methods from three perspectives: (i) data, which uses prior knowledge to augment the supervised experience; (ii) model, which uses prior knowledge to reduce the size of the hypothesis space; and (iii) algorithm, which uses prior knowledge to alter the search for the best hypothesis in the given hypothesis space. With this taxonomy, we review and discuss the pros and cons of each category. Promising directions, in the aspects of the FSL problem setups, techniques, applications and theories, are also proposed to provide insights for future research.

Large language models have transformed natural language processing, yet supervised fine-tuning (SFT) remains computationally intensive. This paper formally proves that capabilities acquired through SFT can be approximated by a base transformer model using inference-time techniques, specifically in-context learning (ICL), without altering model parameters, under idealized assumptions including unbounded computational resources and access to the fine-tuning dataset. We extend these results to practical scenarios with finite context lengths and partial dataset access. For text generation tasks with fixed output length l, datasets of size Oleft( m V{varepsilon^2} log m{delta} right) or, with bounded context, Oleft( l log V{varepsilon^2} log 1{delta} right) suffice to approximate fine-tuned behavior across m contexts within error varepsilon, where V is the vocabulary size and delta is the failure probability. For linear classification, datasets of size Oleft( d{varepsilon} right) or, with fixed context, Oleft( 1{varepsilon^2} log 1{delta} right) are sufficient, where d is the input dimension. Grounded in the Turing completeness of transformers, these results provide a theoretical foundation for resource-efficient deployment of large language models, with practical techniques like retrieval-augmented generation bridging theory to real-world applications.

Intelligent tutoring systems optimize the selection and timing of learning materials to enhance understanding and long-term retention. This requires estimates of both the learner's progress (''knowledge tracing''; KT), and the prerequisite structure of the learning domain (''knowledge mapping''). While recent deep learning models achieve high KT accuracy, they do so at the expense of the interpretability of psychologically-inspired models. In this work, we present a solution to this trade-off. PSI-KT is a hierarchical generative approach that explicitly models how both individual cognitive traits and the prerequisite structure of knowledge influence learning dynamics, thus achieving interpretability by design. Moreover, by using scalable Bayesian inference, PSI-KT targets the real-world need for efficient personalization even with a growing body of learners and learning histories. Evaluated on three datasets from online learning platforms, PSI-KT achieves superior multi-step predictive accuracy and scalable inference in continual-learning settings, all while providing interpretable representations of learner-specific traits and the prerequisite structure of knowledge that causally supports learning. In sum, predictive, scalable and interpretable knowledge tracing with solid knowledge mapping lays a key foundation for effective personalized learning to make education accessible to a broad, global audience.

Large language models have demonstrated impressive capabilities across various natural language processing tasks, especially in solving mathematical problems. However, large language models are not good at math theorem proving using formal languages like Lean. A significant challenge in this area is the scarcity of training data available in these formal languages. To address this issue, we propose a novel pipeline that iteratively generates and filters synthetic data to translate natural language mathematical problems into Lean 4 statements, and vice versa. Our results indicate that the synthetic data pipeline can provide useful training data and improve the performance of LLMs in translating and understanding complex mathematical problems and proofs. Our final dataset contains about 57K formal-informal question pairs along with searched proof from the math contest forum and 21 new IMO questions. We open-source our code at https://github.com/InternLM/InternLM-Math and our data at https://huggingface.co/datasets/InternLM/Lean-Workbook.

Rote learning is a memorization technique based on repetition. It is commonly believed to hinder generalization by encouraging verbatim memorization rather than deeper understanding. This insight holds for even learning factual knowledge that inevitably requires a certain degree of memorization. In this work, we demonstrate that LLMs can be trained to generalize from rote memorized data. We introduce a two-phase memorize-then-generalize framework, where the model first rote memorizes factual subject-object associations using a semantically meaningless token and then learns to generalize by fine-tuning on a small set of semantically meaningful prompts. Extensive experiments over 8 LLMs show that the models can reinterpret rote memorized data through the semantically meaningful prompts, as evidenced by the emergence of structured, semantically aligned latent representations between the two. This surprising finding opens the door to both effective and efficient knowledge injection and possible risks of repurposing the memorized data for malicious usage.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

We show that learning-rate schedules for large model training behave surprisingly similar to a performance bound from non-smooth convex optimization theory. We provide a bound for the constant schedule with linear cooldown; in particular, the practical benefit of cooldown is reflected in the bound due to the absence of logarithmic terms. Further, we show that this surprisingly close match between optimization theory and practice can be exploited for learning-rate tuning: we achieve noticeable improvements for training 124M and 210M Llama-type models by (i) extending the schedule for continued training with optimal learning-rate, and (ii) transferring the optimal learning-rate across schedules.

We consider the problem of conformal prediction under covariate shift. Given labeled data from a source domain and unlabeled data from a covariate shifted target domain, we seek to construct prediction sets with valid marginal coverage in the target domain. Most existing methods require estimating the unknown likelihood ratio function, which can be prohibitive for high-dimensional data such as images. To address this challenge, we introduce the likelihood ratio regularized quantile regression (LR-QR) algorithm, which combines the pinball loss with a novel choice of regularization in order to construct a threshold function without directly estimating the unknown likelihood ratio. We show that the LR-QR method has coverage at the desired level in the target domain, up to a small error term that we can control. Our proofs draw on a novel analysis of coverage via stability bounds from learning theory. Our experiments demonstrate that the LR-QR algorithm outperforms existing methods on high-dimensional prediction tasks, including a regression task for the Communities and Crime dataset, an image classification task from the WILDS repository, and an LLM question-answering task on the MMLU benchmark.

Language models have achieved remarkable performance on a wide range of tasks that require natural language understanding. Nevertheless, state-of-the-art models have generally struggled with tasks that require quantitative reasoning, such as solving mathematics, science, and engineering problems at the college level. To help close this gap, we introduce Minerva, a large language model pretrained on general natural language data and further trained on technical content. The model achieves state-of-the-art performance on technical benchmarks without the use of external tools. We also evaluate our model on over two hundred undergraduate-level problems in physics, biology, chemistry, economics, and other sciences that require quantitative reasoning, and find that the model can correctly answer nearly a third of them.

Optimization modeling enables critical decisions across industries but remains difficult to automate: informal language must be mapped to precise mathematical formulations and executable solver code. Prior LLM approaches either rely on brittle prompting or costly retraining with limited generalization. We present AlphaOPT, a self-improving experience library that enables an LLM to learn from limited demonstrations (even answers alone, without gold-standard programs) and solver feedback - without annotated reasoning traces or parameter updates. AlphaOPT operates in a continual two-phase cycle: (i) a Library Learning phase that reflects on failed attempts, extracting solver-verified, structured insights as {taxonomy, condition, explanation, example}; and (ii) a Library Evolution phase that diagnoses retrieval misalignments and refines the applicability conditions of stored insights, improving transfer across tasks. This design (1) learns efficiently from limited demonstrations without curated rationales, (2) expands continually without costly retraining by updating the library rather than model weights, and (3) makes knowledge explicit and interpretable for human inspection and intervention. Experiments show that AlphaOPT steadily improves with more data (65% to 72% from 100 to 300 training items) and surpasses the strongest baseline by 7.7% on the out-of-distribution OptiBench dataset when trained only on answers. Code and data are available at: https://github.com/Minw913/AlphaOPT.

Large Language Models (LLMs) have emerged as powerful tools in mathematical theorem proving, particularly when utilizing formal languages such as LEAN. The major learning paradigm is expert iteration, which necessitates a pre-defined dataset comprising numerous mathematical problems. In this process, LLMs attempt to prove problems within the dataset and iteratively refine their capabilities through self-training on the proofs they discover. We propose to use large scale LEAN problem datasets Lean-workbook for expert iteration with more than 20,000 CPU days. During expert iteration, we found log-linear trends between solved problem amount with proof length and CPU usage. We train a critic model to select relatively easy problems for policy models to make trials and guide the model to search for deeper proofs. InternLM2.5-StepProver achieves open-source state-of-the-art on MiniF2F, Lean-Workbook-Plus, ProofNet, and Putnam benchmarks. Specifically, it achieves a pass of 65.9% on the MiniF2F-test and proves (or disproves) 17.0% of problems in Lean-Workbook-Plus which shows a significant improvement compared to only 9.5% of problems proved when Lean-Workbook-Plus was released. We open-source our models and searched proofs at https://github.com/InternLM/InternLM-Math and https://huggingface.co/datasets/internlm/Lean-Workbook.

Proof assistants like Lean have revolutionized mathematical proof verification, ensuring high accuracy and reliability. Although large language models (LLMs) show promise in mathematical reasoning, their advancement in formal theorem proving is hindered by a lack of training data. To address this issue, we introduce an approach to generate extensive Lean 4 proof data derived from high-school and undergraduate-level mathematical competition problems. This approach involves translating natural language problems into formal statements, filtering out low-quality statements, and generating proofs to create synthetic data. After fine-tuning the DeepSeekMath 7B model on this synthetic dataset, which comprises 8 million formal statements with proofs, our model achieved whole-proof generation accuracies of 46.3% with 64 samples and 52% cumulatively on the Lean 4 miniF2F test, surpassing the baseline GPT-4 at 23.0% with 64 samples and a tree search reinforcement learning method at 41.0%. Additionally, our model successfully proved 5 out of 148 problems in the Lean 4 Formalized International Mathematical Olympiad (FIMO) benchmark, while GPT-4 failed to prove any. These results demonstrate the potential of leveraging large-scale synthetic data to enhance theorem-proving capabilities in LLMs. Both the synthetic dataset and the model will be made available to facilitate further research in this promising field.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

Although Large Language Models (LLMs) achieve remarkable performance across various tasks, they often struggle with complex reasoning tasks, such as answering mathematical questions. Recent efforts to address this issue have primarily focused on leveraging mathematical datasets through supervised fine-tuning or self-improvement techniques. However, these methods often depend on high-quality datasets that are difficult to prepare, or they require substantial computational resources for fine-tuning. Inspired by findings that LLMs know how to produce the right answer but struggle to select the correct reasoning path, we propose a purely inference-based searching method -- MindStar (M*). This method formulates reasoning tasks as searching problems and proposes two search ideas to identify the optimal reasoning paths. We evaluate the M* framework on both the GSM8K and MATH datasets, comparing its performance with existing open and closed-source LLMs. Our results demonstrate that M* significantly enhances the reasoning abilities of open-source models, such as Llama-2-13B and Mistral-7B, and achieves comparable performance to GPT-3.5 and Grok-1, but with substantially reduced model size and computational costs.

A major problem with Active Learning (AL) is high training costs since models are typically retrained from scratch after every query round. We start by demonstrating that standard AL on neural networks with warm starting fails, both to accelerate training and to avoid catastrophic forgetting when using fine-tuning over AL query rounds. We then develop a new class of techniques, circumventing this problem, by biasing further training towards previously labeled sets. We accomplish this by employing existing, and developing novel, replay-based Continual Learning (CL) algorithms that are effective at quickly learning the new without forgetting the old, especially when data comes from an evolving distribution. We call this paradigm Continual Active Learning (CAL). We show CAL achieves significant speedups using a plethora of replay schemes that use model distillation and that select diverse, uncertain points from the history. We conduct experiments across many data domains, including natural language, vision, medical imaging, and computational biology, each with different neural architectures and dataset sizes. CAL consistently provides a 3x reduction in training time, while retaining performance.

Preference learning algorithms (e.g., RLHF and DPO) are frequently used to steer LLMs to produce generations that are more preferred by humans, but our understanding of their inner workings is still limited. In this work, we study the conventional wisdom that preference learning trains models to assign higher likelihoods to more preferred outputs than less preferred outputs, measured via ranking accuracy. Surprisingly, we find that most state-of-the-art preference-tuned models achieve a ranking accuracy of less than 60% on common preference datasets. We furthermore derive the idealized ranking accuracy that a preference-tuned LLM would achieve if it optimized the DPO or RLHF objective perfectly. We demonstrate that existing models exhibit a significant alignment gap -- i.e., a gap between the observed and idealized ranking accuracies. We attribute this discrepancy to the DPO objective, which is empirically and theoretically ill-suited to fix even mild ranking errors in the reference model, and derive a simple and efficient formula for quantifying the difficulty of learning a given preference datapoint. Finally, we demonstrate that ranking accuracy strongly correlates with the empirically popular win rate metric when the model is close to the reference model used in the objective, shedding further light on the differences between on-policy (e.g., RLHF) and off-policy (e.g., DPO) preference learning algorithms.

Pre-training and fine-tuning have achieved significant advances in the information retrieval (IR). A typical approach is to fine-tune all the parameters of large-scale pre-trained models (PTMs) on downstream tasks. As the model size and the number of tasks increase greatly, such approach becomes less feasible and prohibitively expensive. Recently, a variety of parameter-efficient tuning methods have been proposed in natural language processing (NLP) that only fine-tune a small number of parameters while still attaining strong performance. Yet there has been little effort to explore parameter-efficient tuning for IR. In this work, we first conduct a comprehensive study of existing parameter-efficient tuning methods at both the retrieval and re-ranking stages. Unlike the promising results in NLP, we find that these methods cannot achieve comparable performance to full fine-tuning at both stages when updating less than 1\% of the original model parameters. More importantly, we find that the existing methods are just parameter-efficient, but not learning-efficient as they suffer from unstable training and slow convergence. To analyze the underlying reason, we conduct a theoretical analysis and show that the separation of the inserted trainable modules makes the optimization difficult. To alleviate this issue, we propose to inject additional modules alongside the PTM to make the original scattered modules connected. In this way, all the trainable modules can form a pathway to smooth the loss surface and thus help stabilize the training process. Experiments at both retrieval and re-ranking stages show that our method outperforms existing parameter-efficient methods significantly, and achieves comparable or even better performance over full fine-tuning.

Symbolic regression (SR) is a powerful technique for discovering the underlying mathematical expressions from observed data. Inspired by the success of deep learning, recent efforts have focused on two categories for SR methods. One is using a neural network or genetic programming to search the expression tree directly. Although this has shown promising results, the large search space poses difficulties in learning constant factors and processing high-dimensional problems. Another approach is leveraging a transformer-based model training on synthetic data and offers advantages in inference speed. However, this method is limited to fixed small numbers of dimensions and may encounter inference problems when given data is out-of-distribution compared to the synthetic data. In this work, we propose DySymNet, a novel neural-guided Dynamic Symbolic Network for SR. Instead of searching for expressions within a large search space, we explore DySymNet with various structures and optimize them to identify expressions that better-fitting the data. With a topology structure like neural networks, DySymNet not only tackles the challenge of high-dimensional problems but also proves effective in optimizing constants. Based on extensive numerical experiments using low-dimensional public standard benchmarks and the well-known SRBench with more variables, our method achieves state-of-the-art performance in terms of fitting accuracy and robustness to noise.

We study the problem of learning a single neuron with respect to the L_2^2-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal L_2^2-error within a constant factor. Our algorithm applies under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.

AI researchers and practitioners increasingly apply large language models (LLMs) to what we call reasoning-intensive regression (RiR), i.e. deducing subtle numerical properties from text. Unlike standard language regression tasks, e.g. for sentiment or similarity, RiR often appears instead in ad-hoc problems like rubric-based scoring or domain-specific retrieval, where much deeper analysis of text is required while only limited task-specific training data and computation are available. We cast three realistic problems as RiR tasks to establish an initial benchmark, and use that to test our hypothesis that prompting frozen LLMs and finetuning Transformer encoders via gradient descent will both often struggle in RiR. We then propose MENTAT, a simple and lightweight method that combines batch-reflective prompt optimization with neural ensemble learning. MENTAT achieves up to 65% improvement over both baselines, though substantial room remains for future advances in RiR.

We carry out an information-theoretical analysis of a two-layer neural network trained from input-output pairs generated by a teacher network with matching architecture, in overparametrized regimes. Our results come in the form of bounds relating i) the mutual information between training data and network weights, or ii) the Bayes-optimal generalization error, to the same quantities but for a simpler (generalized) linear model for which explicit expressions are rigorously known. Our bounds, which are expressed in terms of the number of training samples, input dimension and number of hidden units, thus yield fundamental performance limits for any neural network (and actually any learning procedure) trained from limited data generated according to our two-layer teacher neural network model. The proof relies on rigorous tools from spin glasses and is guided by ``Gaussian equivalence principles'' lying at the core of numerous recent analyses of neural networks. With respect to the existing literature, which is either non-rigorous or restricted to the case of the learning of the readout weights only, our results are information-theoretic (i.e. are not specific to any learning algorithm) and, importantly, cover a setting where all the network parameters are trained.