Daily Papers

Neural networks trained with gradient descent often learn solutions of increasing complexity over time, a phenomenon known as simplicity bias. Despite being widely observed across architectures, existing theoretical treatments lack a unifying framework. We present a theoretical framework that explains a simplicity bias arising from saddle-to-saddle learning dynamics for a general class of neural networks, incorporating fully-connected, convolutional, and attention-based architectures. Here, simple means expressible with few hidden units, i.e., hidden neurons, convolutional kernels, or attention heads. Specifically, we show that linear networks learn solutions of increasing rank, ReLU networks learn solutions with an increasing number of kinks, convolutional networks learn solutions with an increasing number of convolutional kernels, and self-attention models learn solutions with an increasing number of attention heads. By analyzing fixed points, invariant manifolds, and dynamics of gradient descent learning, we show that saddle-to-saddle dynamics operates by iteratively evolving near an invariant manifold, approaching a saddle, and switching to another invariant manifold. Our analysis also disentangles data-induced and initialization-induced saddle-to-saddle dynamics. In particular, the former leads to low-rank weights while the latter to sparse weights. Equipped with the theory, we predict the effects of data distribution and weight initialization on the duration and number of plateaus in learning. Overall, our theory offers a framework for understanding when and why gradient descent progressively learns increasingly complex solutions.

Plateaus, where an agent's performance stagnates at a suboptimal level, are a common problem in deep on-policy RL. Focusing on PPO due to its widespread adoption, we show that plateaus in certain regimes arise not because of known exploration, capacity, or optimization challenges, but because sample-based estimates of the loss eventually become poor proxies for the true objective over the course of training. As a recap, PPO switches between sampling rollouts from several parallel environments online using the current policy (which we call the outer loop) and performing repeated minibatch SGD steps against this offline dataset (the inner loop). In our work we consider only the outer loop, and conceptually model it as stochastic optimization. The step size is then controlled by the regularization strength towards the previous policy and the gradient noise by the number of samples collected between policy update steps. This model predicts that performance will plateau at a suboptimal level if the outer step size is too large relative to the noise. Recasting PPO in this light makes it clear that there are two ways to address this particular type of learning stagnation: either reduce the step size or increase the number of samples collected between updates. We first validate the predictions of our model and investigate how hyperparameter choices influence the step size and update noise, concluding that increasing the number of parallel environments is a simple and robust way to reduce both factors. Next, we propose a recipe for how to co-scale the other hyperparameters when increasing parallelization, and show that incorrectly doing so can lead to severe performance degradation. Finally, we vastly outperform prior baselines in a complex open-ended domain by scaling PPO to more than 1M parallel environments, thereby enabling monotonic performance improvement up to one trillion transitions.

Reinforcement Learning with Verifiable Rewards (RLVR) has propelled Large Language Models in complex reasoning, yet its scalability is often hindered by a training bottleneck where performance plateaus as policy entropy collapses, signaling a loss of exploration. Previous methods typically address this by maintaining high policy entropy, yet the precise mechanisms that govern meaningful exploration have remained underexplored. Our analysis suggests that an unselective focus on entropy risks amplifying irrelevant tokens and destabilizing training. This paper investigates the exploration dynamics within RLVR and identifies a key issue: the gradual elimination of valuable low-probability exploratory tokens, which we term \textit{reasoning sparks}. We find that while abundant in pre-trained models, these sparks are systematically extinguished during RLVR due to over-penalization, leading to a degeneracy in exploration. To address this, we introduce Low-probability Regularization (Lp-Reg). Its core mechanism regularizes the policy towards a heuristic proxy distribution. This proxy is constructed by filtering out presumed noise tokens and re-normalizing the distribution over the remaining candidates. The result is a less-noisy proxy where the probability of reasoning sparks is amplified, which then serves as a soft regularization target to shield these valuable tokens from elimination via KL divergence. Experiments show that Lp-Reg enables stable on-policy training for around 1,000 steps, a regime where baseline entropy-control methods collapse. This sustained exploration leads to state-of-the-art performance, achieving a 60.17% average accuracy on five math benchmarks, an improvement of 2.66% over prior methods. Code is available at https://github.com/CarlanLark/Lp-Reg.

Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics on weights that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that as the depth of a network approaches infinity, learning speed can nevertheless remain finite: for a special class of initial conditions on the weights, very deep networks incur only a finite, depth independent, delay in learning speed relative to shallow networks. We show that, under certain conditions on the training data, unsupervised pretraining can find this special class of initial conditions, while scaled random Gaussian initializations cannot. We further exhibit a new class of random orthogonal initial conditions on weights that, like unsupervised pre-training, enjoys depth independent learning times. We further show that these initial conditions also lead to faithful propagation of gradients even in deep nonlinear networks, as long as they operate in a special regime known as the edge of chaos.

Reward models (RMs) used in reinforcement learning from human feedback (RLHF) are vulnerable to reward hacking: as the policy maximizes a learned proxy reward, true quality plateaus or degrades. We make the assumption that reward hacking is often caused by flipped advantage signs: instead of reducing the likelihood of a bad response, a flipped sign causes the update to increase it. By considering an adversarial perturbation in the RM parameter space, we can derive a certified sign-preservation radius, which is the smallest perturbation that can flip the advantage sign during policy optimization. Based on this formulation, we propose Sign-Certified Policy Optimization (SignCert-PO), down-weighting non-robust completions in the policy gradient update. Unlike prior approaches that require multiple RMs or access to the RM training data, SignCert-PO is lightweight and operates purely at the policy optimization stage using only the RM parameters and on-policy completions. On TL;DR summarization and AlpacaFarm benchmarks, SignCert-PO consistently achieves a better win rate than baselines and reduces reward hacking.

The prediction of material degradation is an important problem to solve in many industries. Environmental conditions, such as humidity and temperature, are important drivers of degradation processes, with corrosion being one of the most prominent ones. Quantum machine learning is a promising research field but suffers from well known deficits such as barren plateaus and measurement overheads. To address this problem, recent research has examined quantum reservoir computing to address time-series prediction tasks. Although a promising idea, developing circuits that are expressive enough while respecting the limited depths available on current devices is challenging. In classical reservoir computing, the onion echo state network model (ESN) [https://doi.org/10.1007/978-3-031-72359-9_9] was introduced to increase the interpretability of the representation structure of the embeddings. This onion ESN model utilizes a concatenation of smaller reservoirs that describe different time scales by covering different regions of the eigenvalue spectrum. Here, we use the same idea in the realm of quantum reservoir computing by simultaneously evolving smaller quantum reservoirs to better capture all the relevant time-scales while keeping the circuit depth small. We do this by modifying the rotation angles which we show alters the eigenvalues of the quantum evolution, but also note that modifying the number of mid-circuit measurements accomplishes the same goals of changing the long-term or short-term memory. This onion QRC outperforms a simple model and a single classical reservoir for predicting the degradation of aluminum alloys in different environmental conditions. By combining the onion QRC with an additional classical reservoir layer, the prediction accuracy is further improved.

Critique-guided reinforcement learning (RL) has emerged as a powerful paradigm for training LLM agents by augmenting sparse outcome rewards with natural-language feedback. However, current methods often rely on static or offline critic models, which fail to adapt as the policy evolves. In on-policy RL, the agent's error patterns shift over time, causing stationary critics to become stale and providing feedback of diminishing utility. To address this, we introduce ECHO (Evolving Critic for Hindsight-Guided Optimization)}, a framework that jointly optimizes the policy and critic through a synchronized co-evolutionary loop. ECHO utilizes a cascaded rollout mechanism where the critic generates multiple diagnoses for an initial trajectory, followed by policy refinement to enable group-structured advantage estimation. We address the challenge of learning plateaus via a saturation-aware gain shaping objective, which rewards the critic for inducing incremental improvements in high-performing trajectories. By employing dual-track GRPO updates, ECHO ensures the critic's feedback stays synchronized with the evolving policy. Experimental results show that ECHO yields more stable training and higher long-horizon task success across open-world environments.

Although RLVR has become an essential component for developing advanced reasoning skills in LLMs, contemporary studies have documented training plateaus that emerge following thousands of optimization steps, demonstrating notable decreases in performance gains despite increased computational investment. This limitation stems from the sparse exploration patterns inherent in current RLVR practices, where models rely on limited rollouts that often miss critical reasoning paths and fail to provide systematic coverage of the solution space. We present DeepSearch, a framework that integrates Monte Carlo Tree Search directly into RLVR training. In contrast to existing methods that rely on tree search only at inference, DeepSearch embeds structured search into the training loop, enabling systematic exploration and fine-grained credit assignment across reasoning steps. Through training-time exploration, DeepSearch addresses the fundamental bottleneck of insufficient exploration, which leads to diminishing performance improvements over prolonged training steps. Our contributions include: (1) a global frontier selection strategy that prioritizes promising nodes across the search tree, (2) selection with entropy-based guidance that identifies confident paths for supervision, and (3) adaptive replay buffer training with solution caching for efficiency. Experiments on mathematical reasoning benchmarks show that DeepSearch achieves 62.95% average accuracy and establishes a new state-of-the-art for 1.5B reasoning models - using 5.7x fewer GPU hours than extended training approaches. These results highlight the importance of strategic exploration over brute-force scaling and demonstrate the promise of algorithmic innovation for advancing RLVR methodologies. DeepSearch establishes a new direction for scaling reasoning capabilities through systematic search rather than prolonged computation.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a key ingredient for unlocking complex reasoning capabilities in large language models. Recent work ProRL has shown promise in scaling RL by increasing the number of training steps. However, performance plateaus after thousands of steps, with clear diminishing returns from allocating more computation to additional training. In this work, we investigate a complementary paradigm for scaling RL, BroR-Lincreasing the number of rollouts per example to hundreds to exhaustively Broaden exploration, which yields continuous performance gains beyond the saturation point observed in ProRL when scaling the number of training steps. Our approach is motivated by a mass balance equation analysis allowing us to characterize the rate of change in probability mass for correct and incorrect tokens during the reinforcement process. We show that under a one-step RL assumption, sampled rollout tokens always contribute to correct-mass expansion, while unsampled tokens outside rollouts may lead to gains or losses depending on their distribution and the net reward balance. Importantly, as the number of rollouts per example N increases, the effect of unsampled terms diminishes, ensuring overall correct-mass expansion. To validate our theoretical analysis, we conduct simulations under more relaxed conditions and find that a sufficiently large rollout size N-corresponding to ample exploration-guarantees an increase in the probability mass of all correct tokens. Empirically, BroRL revives models saturated after 3K ProRL training steps and demonstrates robust, continuous improvement, achieving state-of-the-art results for the 1.5B model across diverse benchmarks.

Chain-of-thought (CoT) offers a potential boon for AI safety as it allows monitoring a model's CoT to try to understand its intentions and reasoning processes. However, the effectiveness of such monitoring hinges on CoTs faithfully representing models' actual reasoning processes. We evaluate CoT faithfulness of state-of-the-art reasoning models across 6 reasoning hints presented in the prompts and find: (1) for most settings and models tested, CoTs reveal their usage of hints in at least 1% of examples where they use the hint, but the reveal rate is often below 20%, (2) outcome-based reinforcement learning initially improves faithfulness but plateaus without saturating, and (3) when reinforcement learning increases how frequently hints are used (reward hacking), the propensity to verbalize them does not increase, even without training against a CoT monitor. These results suggest that CoT monitoring is a promising way of noticing undesired behaviors during training and evaluations, but that it is not sufficient to rule them out. They also suggest that in settings like ours where CoT reasoning is not necessary, test-time monitoring of CoTs is unlikely to reliably catch rare and catastrophic unexpected behaviors.

Training Transformers on algorithmic tasks frequently demonstrates an intriguing abrupt learning phenomenon: an extended performance plateau followed by a sudden, sharp improvement. This work investigates the underlying mechanisms for such dynamics, primarily in shallow Transformers. We reveal that during the plateau, the model often develops an interpretable partial solution while simultaneously exhibiting a strong repetition bias in their outputs. This output degeneracy is accompanied by internal representation collapse, where hidden states across different tokens become nearly parallel. We further identify the slow learning of optimal attention maps as a key bottleneck. Hidden progress in attention configuration during the plateau precedes the eventual rapid convergence, and directly intervening on attention significantly alters plateau duration and the severity of repetition bias and representational collapse. We validate that these identified phenomena-repetition bias and representation collapse-are not artifacts of toy setups but also manifest in the early pre-training stage of large language models like Pythia and OLMo.

While reinforcement learning (RL) has proven highly effective for general reasoning in vision-language models, its application to tasks requiring in-depth understanding of information-rich images and generation of structured outputs remains underexplored. Chart-to-code generation exemplifies this challenge, demanding complex reasoning over visual charts to generate structured code. Supervised fine-tuning (SFT) alone is often insufficient, highlighting the need for effective RL strategies that appropriately reward structured outputs. We systematically investigate the performance plateau in SFT through large-scale experiments and propose Multimodal Structured Reinforcement Learning (MSRL) for chart-to-code generation, which substantially breaks through this plateau. We construct the largest training corpus to date, containing 3 million chart-code pairs from real-world arXiv tables to mitigate simplistic patterns of prior synthetic data. Despite reaching state-of-the-art performance, our experiments show that scaling SFT data eventually hits a plateau where further increases yield negligible improvements. Our MSRL method leverages a multi-granularity structured reward system using multimodal textual and visual feedback. At the textual level, rule-based rewards validate fine-grained code details. At the visual level, model-based rewards assess structural similarity by rendering generated code into images and employing an evaluator model. We implement this within a two-stage curriculum for training stability. Results demonstrate that MSRL significantly breaks the SFT plateau, improving high-level metrics by 6.2% and 9.9% on ChartMimic and ReachQA benchmarks respectively, achieving competitive performance with advanced closed-source models.

Training recurrent neural networks (RNNs) is a high-dimensional process that requires updating numerous parameters. Therefore, it is often difficult to pinpoint the underlying learning mechanisms. To address this challenge, we propose to gain mechanistic insights into the phenomenon of abrupt learning by studying RNNs trained to perform diverse short-term memory tasks. In these tasks, RNN training begins with an initial search phase. Following a long period of plateau in accuracy, the values of the loss function suddenly drop, indicating abrupt learning. Analyzing the neural computation performed by these RNNs reveals geometric restructuring (GR) in their phase spaces prior to the drop. To promote these GR events, we introduce a temporal consistency regularization that accelerates (bioplausible) training, facilitates attractor formation, and enables efficient learning in strongly connected networks. Our findings offer testable predictions for neuroscientists and emphasize the need for goal-agnostic secondary mechanisms to facilitate learning in biological and artificial networks.

Can a model learn to escape its own learning plateau? Reinforcement learning methods for finetuning large reasoning models stall on datasets with low initial success rates, and thus little training signal. We investigate a fundamental question: Can a pretrained LLM leverage latent knowledge to generate an automated curriculum for problems it cannot solve? To explore this, we design SOAR: A self-improvement framework designed to surface these pedagogical signals through meta-RL. A teacher copy of the model proposes synthetic problems for a student copy, and is rewarded with its improvement on a small subset of hard problems. Critically, SOAR grounds the curriculum in measured student progress rather than intrinsic proxy rewards. Our study on the hardest subsets of mathematical benchmarks (0/128 success) reveals three core findings. First, we show that it is possible to realize bi-level meta-RL that unlocks learning under sparse, binary rewards by sharpening a latent capacity of pretrained models to generate useful stepping stones. Second, grounded rewards outperform intrinsic reward schemes used in prior LLM self-play, reliably avoiding the instability and diversity collapse modes they typically exhibit. Third, analyzing the generated questions reveals that structural quality and well-posedness are more critical for learning progress than solution correctness. Our results suggest that the ability to generate useful stepping stones does not require the preexisting ability to actually solve the hard problems, paving a principled path to escape reasoning plateaus without additional curated data.

Time-dependent data-generating distributions have proven to be difficult for gradient-based training of neural networks, as the greedy updates result in catastrophic forgetting of previously learned knowledge. Despite the progress in the field of continual learning to overcome this forgetting, we show that a set of common state-of-the-art methods still suffers from substantial forgetting upon starting to learn new tasks, except that this forgetting is temporary and followed by a phase of performance recovery. We refer to this intriguing but potentially problematic phenomenon as the stability gap. The stability gap had likely remained under the radar due to standard practice in the field of evaluating continual learning models only after each task. Instead, we establish a framework for continual evaluation that uses per-iteration evaluation and we define a new set of metrics to quantify worst-case performance. Empirically we show that experience replay, constraint-based replay, knowledge-distillation, and parameter regularization methods are all prone to the stability gap; and that the stability gap can be observed in class-, task-, and domain-incremental learning benchmarks. Additionally, a controlled experiment shows that the stability gap increases when tasks are more dissimilar. Finally, by disentangling gradients into plasticity and stability components, we propose a conceptual explanation for the stability gap.

Large language models accumulate vast knowledge during pre-training, yet the dynamics governing this acquisition remain poorly understood. This work investigates the learning dynamics of language models on a synthetic factual recall task, uncovering three key findings: First, language models learn in three phases, exhibiting a performance plateau before acquiring precise factual knowledge. Mechanistically, this plateau coincides with the formation of attention-based circuits that support recall. Second, the training data distribution significantly impacts learning dynamics, as imbalanced distributions lead to shorter plateaus. Finally, hallucinations emerge simultaneously with knowledge, and integrating new knowledge into the model through fine-tuning is challenging, as it quickly corrupts its existing parametric memories. Our results emphasize the importance of data distribution in knowledge acquisition and suggest novel data scheduling strategies to accelerate neural network training.

Although supervised learning has been highly successful in improving the state-of-the-art in the domain of image-based computer vision in the past, the margin of improvement has diminished significantly in recent years, indicating that a plateau is in sight. Meanwhile, the use of self-supervised learning (SSL) for the purpose of natural language processing (NLP) has seen tremendous successes during the past couple of years, with this new learning paradigm yielding powerful language models. Inspired by the excellent results obtained in the field of NLP, self-supervised methods that rely on clustering, contrastive learning, distillation, and information-maximization, which all fall under the banner of discriminative SSL, have experienced a swift uptake in the area of computer vision. Shortly afterwards, generative SSL frameworks that are mostly based on masked image modeling, complemented and surpassed the results obtained with discriminative SSL. Consequently, within a span of three years, over 100 unique general-purpose frameworks for generative and discriminative SSL, with a focus on imaging, were proposed. In this survey, we review a plethora of research efforts conducted on image-oriented SSL, providing a historic view and paying attention to best practices as well as useful software packages. While doing so, we discuss pretext tasks for image-based SSL, as well as techniques that are commonly used in image-based SSL. Lastly, to aid researchers who aim at contributing to image-focused SSL, we outline a number of promising research directions.

Recently, reinforcement learning with verifiable rewards (RLVR) has been widely used for enhancing the reasoning abilities of large language models (LLMs). A core challenge in RLVR involves managing the exchange between entropy and performance of policies. Despite the importance of this exchange, a fine-grained understanding of when and how this exchange operates most effectively remains limited. To bridge this gap, we conduct a systematic empirical analysis of the entropy-performance exchange mechanism of RLVR across different levels of granularity. Specifically, we first divide the training process into two distinct stages based on entropy dynamics, i.e., rising stage and plateau stage, and then systematically investigate how this mechanism varies across stage-level, instance-level, and token-level granularitiess. Our analysis reveals that, in the rising stage, entropy reduction in negative samples facilitates the learning of effective reasoning patterns, which in turn drives rapid performance gains. Moreover, in the plateau stage, learning efficiency strongly correlates with high-entropy tokens present in low-perplexity samples and those located at the end of sequences. Motivated by these findings, we propose two methods that dynamically adjust the reward signal using perplexity and positional information to focus RL updates on tokens that exhibit high learning potential, achieving improvements compared to the baseline methods on various LLMs.

Loss of plasticity is a phenomenon where neural networks become more difficult to train during the course of learning. Continual learning algorithms seek to mitigate this effect by sustaining good predictive performance while maintaining network trainability. We develop new techniques for improving continual learning by first reconsidering how initialization can ensure trainability during early phases of learning. From this perspective, we derive new regularization strategies for continual learning that ensure beneficial initialization properties are better maintained throughout training. In particular, we investigate two new regularization techniques for continual learning: (i) Wasserstein regularization toward the initial weight distribution, which is less restrictive than regularizing toward initial weights; and (ii) regularizing weight matrix singular values, which directly ensures gradient diversity is maintained throughout training. We present an experimental analysis that shows these alternative regularizers can improve continual learning performance across a range of supervised learning tasks and model architectures. The alternative regularizers prove to be less sensitive to hyperparameters while demonstrating better training in individual tasks, sustaining trainability as new tasks arrive, and achieving better generalization performance.

Quantized neural network training optimizes a discrete, non-differentiable objective. The straight-through estimator (STE) enables backpropagation through surrogate gradients and is widely used. While previous studies have primarily focused on the properties of surrogate gradients and their convergence, the influence of quantization hyperparameters, such as bit width and quantization range, on learning dynamics remains largely unexplored. We theoretically show that in the high-dimensional limit, STE dynamics converge to a deterministic ordinary differential equation. This reveals that STE training exhibits a plateau followed by a sharp drop in generalization error, with plateau length depending on the quantization range. A fixed-point analysis quantifies the asymptotic deviation from the unquantized linear model. We also extend analytical techniques for stochastic gradient descent to nonlinear transformations of weights and inputs.

Neural networks are very effective when trained on large datasets for a large number of iterations. However, when they are trained on non-stationary streams of data and in an online fashion, their performance is reduced (1) by the online setup, which limits the availability of data, (2) due to catastrophic forgetting because of the non-stationary nature of the data. Furthermore, several recent works (Caccia et al., 2022; Lange et al., 2023) arXiv:2205.13452 showed that replay methods used in continual learning suffer from the stability gap, encountered when evaluating the model continually (rather than only on task boundaries). In this article, we study the effect of model ensembling as a way to improve performance and stability in online continual learning. We notice that naively ensembling models coming from a variety of training tasks increases the performance in online continual learning considerably. Starting from this observation, and drawing inspirations from semi-supervised learning ensembling methods, we use a lightweight temporal ensemble that computes the exponential moving average of the weights (EMA) at test time, and show that it can drastically increase the performance and stability when used in combination with several methods from the literature.

Recent years have seen considerable progress in the continual training of deep neural networks, predominantly thanks to approaches that add replay or regularization terms to the loss function to approximate the joint loss over all tasks so far. However, we show that even with a perfect approximation to the joint loss, these approaches still suffer from temporary but substantial forgetting when starting to train on a new task. Motivated by this 'stability gap', we propose that continual learning strategies should focus not only on the optimization objective, but also on the way this objective is optimized. While there is some continual learning work that alters the optimization trajectory (e.g., using gradient projection techniques), this line of research is positioned as alternative to improving the optimization objective, while we argue it should be complementary. To evaluate the merits of our proposition, we plan to combine replay-approximated joint objectives with gradient projection-based optimization routines to test whether the addition of the latter provides benefits in terms of (1) alleviating the stability gap, (2) increasing the learning efficiency and (3) improving the final learning outcome.

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.

Recent automatic curriculum learning algorithms, and in particular Teacher-Student algorithms, rely on the notion of learning progress, making the assumption that the good next tasks are the ones on which the learner is making the fastest progress or digress. In this work, we first propose a simpler and improved version of these algorithms. We then argue that the notion of learning progress itself has several shortcomings that lead to a low sample efficiency for the learner. We finally propose a new algorithm, based on the notion of mastering rate, that significantly outperforms learning progress-based algorithms.

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.

Reinforcement Learning with Verifiable Rewards (RLVR) has substantially improved the reasoning abilities of large language models (LLMs), yet training often stalls as problems become saturated. We identify the core challenge as the poor accessibility of informative failures: learning signals exist but are rarely encountered during standard rollouts. To address this, we propose failure-prefix conditioning, a simple and effective method for learning from saturated problems. Rather than starting from the original question, our approach reallocates exploration by conditioning training on prefixes derived from rare incorrect reasoning trajectories, thereby exposing the model to failure-prone states. We observe that failure-prefix conditioning yields performance gains matching those of training on medium-difficulty problems, while preserving token efficiency. Furthermore, we analyze the model's robustness, finding that our method reduces performance degradation under misleading failure prefixes, albeit with a mild trade-off in adherence to correct early reasoning. Finally, we demonstrate that an iterative approach, which refreshes failure prefixes during training, unlocks additional gains after performance plateaus. Overall, our results suggest that failure-prefix conditioning offers an effective pathway to extend RLVR training on saturated problems.

Grokking, or delayed generalization, is a phenomenon where generalization in a deep neural network (DNN) occurs long after achieving near zero training error. Previous studies have reported the occurrence of grokking in specific controlled settings, such as DNNs initialized with large-norm parameters or transformers trained on algorithmic datasets. We demonstrate that grokking is actually much more widespread and materializes in a wide range of practical settings, such as training of a convolutional neural network (CNN) on CIFAR10 or a Resnet on Imagenette. We introduce the new concept of delayed robustness, whereby a DNN groks adversarial examples and becomes robust, long after interpolation and/or generalization. We develop an analytical explanation for the emergence of both delayed generalization and delayed robustness based on a new measure of the local complexity of a DNN's input-output mapping. Our local complexity measures the density of the so-called 'linear regions' (aka, spline partition regions) that tile the DNN input space, and serves as a utile progress measure for training. We provide the first evidence that for classification problems, the linear regions undergo a phase transition during training whereafter they migrate away from the training samples (making the DNN mapping smoother there) and towards the decision boundary (making the DNN mapping less smooth there). Grokking occurs post phase transition as a robust partition of the input space emerges thanks to the linearization of the DNN mapping around the training points. Website: https://bit.ly/grok-adversarial

In class incremental learning, neural networks typically suffer from catastrophic forgetting. We show that an MLP featuring a sparse activation function and an adaptive learning rate optimizer can compete with established regularization techniques in the Split-MNIST task. We highlight the effectiveness of the Adaptive SwisH (ASH) activation function in this context and introduce a novel variant, Hard Adaptive SwisH (Hard ASH) to further enhance the learning retention.

Modern deep-learning systems are specialized to problem settings in which training occurs once and then never again, as opposed to continual-learning settings in which training occurs continually. If deep-learning systems are applied in a continual learning setting, then it is well known that they may fail to remember earlier examples. More fundamental, but less well known, is that they may also lose their ability to learn on new examples, a phenomenon called loss of plasticity. We provide direct demonstrations of loss of plasticity using the MNIST and ImageNet datasets repurposed for continual learning as sequences of tasks. In ImageNet, binary classification performance dropped from 89\% accuracy on an early task down to 77\%, about the level of a linear network, on the 2000th task. Loss of plasticity occurred with a wide range of deep network architectures, optimizers, activation functions, batch normalization, dropout, but was substantially eased by L^2-regularization, particularly when combined with weight perturbation. Further, we introduce a new algorithm -- continual backpropagation -- which slightly modifies conventional backpropagation to reinitialize a small fraction of less-used units after each example and appears to maintain plasticity indefinitely.

It has been observed that the input space of deep neural network classifiers can exhibit `fragmentation', where the model function rapidly changes class as the input space is traversed. The severity of this fragmentation tends to follow the double descent curve, achieving a maximum at the interpolation regime. We study this phenomenon in the context of image classification and ask whether fragmentation could be predictive of generalization performance. Using a fragmentation-based complexity measure, we show this to be possible by achieving good performance on the PGDL (Predicting Generalization in Deep Learning) benchmark. In addition, we report on new observations related to fragmentation, namely (i) fragmentation is not limited to the input space but occurs in the hidden representations as well, (ii) fragmentation follows the trends in the validation error throughout training, and (iii) fragmentation is not a direct result of increased weight norms. Together, this indicates that fragmentation is a phenomenon worth investigating further when studying the generalization ability of deep neural networks.

We analyze continual learning on a sequence of separable linear classification tasks with binary labels. We show theoretically that learning with weak regularization reduces to solving a sequential max-margin problem, corresponding to a special case of the Projection Onto Convex Sets (POCS) framework. We then develop upper bounds on the forgetting and other quantities of interest under various settings with recurring tasks, including cyclic and random orderings of tasks. We discuss several practical implications to popular training practices like regularization scheduling and weighting. We point out several theoretical differences between our continual classification setting and a recently studied continual regression setting.

Although reinforcement learning (RL) can effectively enhance the reasoning capabilities of vision-language models (VLMs), current methods remain heavily dependent on labor-intensive datasets that require extensive manual construction and verification, leading to extremely high training costs and consequently constraining the practical deployment of VLMs. To address this challenge, we propose Vision-Zero, a domain-agnostic framework enabling VLM self-improvement through competitive visual games generated from arbitrary image pairs. Specifically, Vision-Zero encompasses three main attributes: (1) Strategic Self-Play Framework: Vision-Zero trains VLMs in "Who Is the Spy"-style games, where the models engage in strategic reasoning and actions across multiple roles. Through interactive gameplay, models autonomously generate their training data without human annotation. (2) Gameplay from Arbitrary Images: Unlike existing gamified frameworks, Vision-Zero can generate games from arbitrary images, thereby enhancing the model's reasoning ability across diverse domains and showing strong generalization to different tasks. We demonstrate this versatility using three distinct types of image datasets: CLEVR-based synthetic scenes, charts, and real-world images. (3) Sustainable Performance Gain: We introduce Iterative Self-Play Policy Optimization (Iterative-SPO), a novel training algorithm that alternates between Self-Play and reinforcement learning with verifiable rewards (RLVR), mitigating the performance plateau often seen in self-play-only training and achieving sustained long-term improvements. Despite using label-free data, Vision-Zero achieves state-of-the-art performance on reasoning, chart question answering, and vision-centric understanding tasks, surpassing other annotation-based methods. Models and code has been released at https://github.com/wangqinsi1/Vision-Zero.

Grokking is the intriguing phenomenon where a model learns to generalize long after it has fit the training data. We show both analytically and numerically that grokking can surprisingly occur in linear networks performing linear tasks in a simple teacher-student setup with Gaussian inputs. In this setting, the full training dynamics is derived in terms of the training and generalization data covariance matrix. We present exact predictions on how the grokking time depends on input and output dimensionality, train sample size, regularization, and network initialization. We demonstrate that the sharp increase in generalization accuracy may not imply a transition from "memorization" to "understanding", but can simply be an artifact of the accuracy measure. We provide empirical verification for our calculations, along with preliminary results indicating that some predictions also hold for deeper networks, with non-linear activations.

As the application space of language models continues to evolve, a natural question to ask is how we can quickly adapt models to new tasks. We approach this classic question from a continual learning perspective, in which we aim to continue fine-tuning models trained on past tasks on new tasks, with the goal of "transferring" relevant knowledge. However, this strategy also runs the risk of doing more harm than good, i.e., negative transfer. In this paper, we construct a new benchmark of task sequences that target different possible transfer scenarios one might face, such as a sequence of tasks with high potential of positive transfer, high potential for negative transfer, no expected effect, or a mixture of each. An ideal learner should be able to maximally exploit information from all tasks that have any potential for positive transfer, while also avoiding the negative effects of any distracting tasks that may confuse it. We then propose a simple, yet effective, learner that satisfies many of our desiderata simply by leveraging a selective strategy for initializing new models from past task checkpoints. Still, limitations remain, and we hope this benchmark can help the community to further build and analyze such learners.

Grokking, i.e., test performance keeps improving long after training loss converged, has been recently witnessed in neural network training, making the mechanism of generalization and other emerging capabilities such as reasoning mysterious. While prior studies usually train small models on a few toy or highly-specific tasks for thousands of epochs, we conduct the first study of grokking on checkpoints during one-pass pretraining of a 7B large language model (LLM), i.e., OLMoE. We compute the training loss and evaluate generalization on diverse benchmark tasks, including math reasoning, code generation, and commonsense/domain-specific knowledge retrieval tasks. Our study, for the first time, verifies that grokking still happens in the pretraining of large-scale foundation models, though different data may enter grokking stages asynchronously. We further demystify grokking's "emergence of generalization" by investigating LLM internal dynamics. Specifically, we find that training samples' pathways (i.e., expert choices across layers) evolve from random, instance-specific to more structured and shareable between samples during grokking. Also, the complexity of a sample's pathway reduces despite the converged loss. These indicate a memorization-to-generalization conversion, providing a mechanistic explanation of delayed generalization. In the study, we develop two novel metrics to quantify pathway distance and the complexity of a single pathway. We show their ability to predict the generalization improvement on diverse downstream tasks. They are efficient, simple to compute and solely dependent on training data. Hence, they have practical value for pretraining, enabling us to monitor the generalization performance without finetuning and test. Theoretically, we show that more structured pathways reduce model complexity and improve the generalization bound.

Recent advancements in large language models (LLMs) reveal a perplexing phenomenon in continual learning: despite extensive training, models experience significant performance declines, raising questions about task alignment and underlying knowledge retention. This study first explores the concept of "spurious forgetting", proposing that such performance drops often reflect a decline in task alignment rather than true knowledge loss. Through controlled experiments with a synthesized dataset, we investigate the dynamics of model performance during the initial training phases of new tasks, discovering that early optimization steps can disrupt previously established task alignments. Our theoretical analysis connects these shifts to orthogonal updates in model weights, providing a robust framework for understanding this behavior. Ultimately, we introduce a Freezing strategy that fix the bottom layers of the model, leading to substantial improvements in four continual learning scenarios. Our findings underscore the critical distinction between task alignment and knowledge retention, paving the way for more effective strategies in continual learning.

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.

One major obstacle towards AI is the poor ability of models to solve new problems quicker, and without forgetting previously acquired knowledge. To better understand this issue, we study the problem of continual learning, where the model observes, once and one by one, examples concerning a sequence of tasks. First, we propose a set of metrics to evaluate models learning over a continuum of data. These metrics characterize models not only by their test accuracy, but also in terms of their ability to transfer knowledge across tasks. Second, we propose a model for continual learning, called Gradient Episodic Memory (GEM) that alleviates forgetting, while allowing beneficial transfer of knowledge to previous tasks. Our experiments on variants of the MNIST and CIFAR-100 datasets demonstrate the strong performance of GEM when compared to the state-of-the-art.

Background: Deep learning models are typically trained using stochastic gradient descent or one of its variants. These methods update the weights using their gradient, estimated from a small fraction of the training data. It has been observed that when using large batch sizes there is a persistent degradation in generalization performance - known as the "generalization gap" phenomena. Identifying the origin of this gap and closing it had remained an open problem. Contributions: We examine the initial high learning rate training phase. We find that the weight distance from its initialization grows logarithmically with the number of weight updates. We therefore propose a "random walk on random landscape" statistical model which is known to exhibit similar "ultra-slow" diffusion behavior. Following this hypothesis we conducted experiments to show empirically that the "generalization gap" stems from the relatively small number of updates rather than the batch size, and can be completely eliminated by adapting the training regime used. We further investigate different techniques to train models in the large-batch regime and present a novel algorithm named "Ghost Batch Normalization" which enables significant decrease in the generalization gap without increasing the number of updates. To validate our findings we conduct several additional experiments on MNIST, CIFAR-10, CIFAR-100 and ImageNet. Finally, we reassess common practices and beliefs concerning training of deep models and suggest they may not be optimal to achieve good generalization.

Plasticity, the ability of a neural network to quickly change its predictions in response to new information, is essential for the adaptability and robustness of deep reinforcement learning systems. Deep neural networks are known to lose plasticity over the course of training even in relatively simple learning problems, but the mechanisms driving this phenomenon are still poorly understood. This paper conducts a systematic empirical analysis into plasticity loss, with the goal of understanding the phenomenon mechanistically in order to guide the future development of targeted solutions. We find that loss of plasticity is deeply connected to changes in the curvature of the loss landscape, but that it typically occurs in the absence of saturated units or divergent gradient norms. Based on this insight, we identify a number of parameterization and optimization design choices which enable networks to better preserve plasticity over the course of training. We validate the utility of these findings in larger-scale learning problems by applying the best-performing intervention, layer normalization, to a deep RL agent trained on the Arcade Learning Environment.

The relationship between memorization and generalization in large language models (LLMs) remains an open area of research, with growing evidence that the two are deeply intertwined. In this work, we investigate this relationship by pre-training a series of capacity-limited Transformer models from scratch on two synthetic character-level tasks designed to separately probe generalization (via arithmetic extrapolation) and memorization (via factual recall). We observe a consistent trade-off: small models extrapolate to unseen arithmetic cases but fail to memorize facts, while larger models memorize but fail to extrapolate. An intermediate-capacity model exhibits a similar shift toward memorization. When trained on both tasks jointly, no model (regardless of size) succeeds at extrapolation. These findings suggest that pre-training may intrinsically favor one learning mode over the other. By isolating these dynamics in a controlled setting, our study offers insight into how model capacity shapes learning behavior and offers broader implications for the design and deployment of small language models.

Recent work by Power et al. (2022) highlighted a surprising "grokking" phenomenon in learning arithmetic tasks: a neural net first "memorizes" the training set, resulting in perfect training accuracy but near-random test accuracy, and after training for sufficiently longer, it suddenly transitions to perfect test accuracy. This paper studies the grokking phenomenon in theoretical setups and shows that it can be induced by a dichotomy of early and late phase implicit biases. Specifically, when training homogeneous neural nets with large initialization and small weight decay on both classification and regression tasks, we prove that the training process gets trapped at a solution corresponding to a kernel predictor for a long time, and then a very sharp transition to min-norm/max-margin predictors occurs, leading to a dramatic change in test accuracy.

Large language models have achieved significant reasoning improvements through reinforcement learning with verifiable rewards (RLVR). Yet as model capabilities grow, constructing high-quality reward signals becomes increasingly difficult, making it essential to understand when RLVR can succeed under weaker forms of supervision. We conduct a systematic empirical study across diverse model families and reasoning domains under three weak supervision settings: scarce data, noisy rewards, and self-supervised proxy rewards. We find that generalization is governed by training reward saturation dynamics: models that generalize exhibit a prolonged pre-saturation phase during which training reward and downstream performance climb together, while models that saturate rapidly memorize rather than learn. We identify reasoning faithfulness, defined as the extent to which intermediate steps logically support the final answer, as the pre-RL property that predicts which regime a model falls into, while output diversity alone is uninformative. Motivated by these findings, we disentangle the contributions of continual pre-training and supervised fine-tuning, finding that SFT on explicit reasoning traces is necessary for generalization under weak supervision, while continual pre-training on domain data amplifies the effect. Applied together to Llama3.2-3B-Base, these interventions enable generalization across all three settings where the base model previously failed.

We identify a new phenomenon in neural network optimization which arises from the interaction of depth and a particular heavy-tailed structure in natural data. Our result offers intuitive explanations for several previously reported observations about network training dynamics. In particular, it implies a conceptually new cause for progressive sharpening and the edge of stability; we also highlight connections to other concepts in optimization and generalization including grokking, simplicity bias, and Sharpness-Aware Minimization. Experimentally, we demonstrate the significant influence of paired groups of outliers in the training data with strong opposing signals: consistent, large magnitude features which dominate the network output throughout training and provide gradients which point in opposite directions. Due to these outliers, early optimization enters a narrow valley which carefully balances the opposing groups; subsequent sharpening causes their loss to rise rapidly, oscillating between high on one group and then the other, until the overall loss spikes. We describe how to identify these groups, explore what sets them apart, and carefully study their effect on the network's optimization and behavior. We complement these experiments with a mechanistic explanation on a toy example of opposing signals and a theoretical analysis of a two-layer linear network on a simple model. Our finding enables new qualitative predictions of training behavior which we confirm experimentally. It also provides a new lens through which to study and improve modern training practices for stochastic optimization, which we highlight via a case study of Adam versus SGD.

Plasticity, the ability of a neural network to evolve with new data, is crucial for high-performance and sample-efficient visual reinforcement learning (VRL). Although methods like resetting and regularization can potentially mitigate plasticity loss, the influences of various components within the VRL framework on the agent's plasticity are still poorly understood. In this work, we conduct a systematic empirical exploration focusing on three primary underexplored facets and derive the following insightful conclusions: (1) data augmentation is essential in maintaining plasticity; (2) the critic's plasticity loss serves as the principal bottleneck impeding efficient training; and (3) without timely intervention to recover critic's plasticity in the early stages, its loss becomes catastrophic. These insights suggest a novel strategy to address the high replay ratio (RR) dilemma, where exacerbated plasticity loss hinders the potential improvements of sample efficiency brought by increased reuse frequency. Rather than setting a static RR for the entire training process, we propose Adaptive RR, which dynamically adjusts the RR based on the critic's plasticity level. Extensive evaluations indicate that Adaptive RR not only avoids catastrophic plasticity loss in the early stages but also benefits from more frequent reuse in later phases, resulting in superior sample efficiency.

Neural networks often suffer from catastrophic interference (CI): performance on previously learned tasks drops off significantly when learning a new task. This contrasts strongly with humans, who can sequentially learn new tasks without appreciably forgetting previous tasks. Prior work has explored various techniques for mitigating CI such as regularization, rehearsal, generative replay, and distillation methods. The current work takes a different approach, one guided by cognitive science research showing that in naturalistic environments, the probability of encountering a task decreases as a power-law of the time since it was last performed. We argue that a realistic evaluation of techniques for the mitigation of CI should be performed in simulated naturalistic learning environments. Thus, we evaluate the extent of mitigation of CI when training simple rehearsal-based methods in power-law environments similar to the ones humans face. Our work explores this novel rehearsal-based approach for a domain-incremental task: learning permutations in the MNIST task. We compare our rehearsal environment with other baselines to show its efficacy in promoting continual learning. Additionally, we investigate whether this environment shows forward facilitation, i.e., faster learning of later tasks. Next, we explore the robustness of our learning environment to the number of tasks, model size, and amount of data rehearsed after each task. Notably, our results show that the performance is comparable or superior to that of models trained using popular regularization methods and also to rehearsals in non-power-law environments. The benefits of this training paradigm include simplicity and the lack of a need for extra neural circuitry. In addition, because our method is orthogonal to other methods, future research can combine training in power-law environments with other continual learning mechanisms.

In many real-world deployments of machine learning systems, data arrive piecemeal. These learning scenarios may be passive, where data arrive incrementally due to structural properties of the problem (e.g., daily financial data) or active, where samples are selected according to a measure of their quality (e.g., experimental design). In both of these cases, we are building a sequence of models that incorporate an increasing amount of data. We would like each of these models in the sequence to be performant and take advantage of all the data that are available to that point. Conventional intuition suggests that when solving a sequence of related optimization problems of this form, it should be possible to initialize using the solution of the previous iterate -- to "warm start" the optimization rather than initialize from scratch -- and see reductions in wall-clock time. However, in practice this warm-starting seems to yield poorer generalization performance than models that have fresh random initializations, even though the final training losses are similar. While it appears that some hyperparameter settings allow a practitioner to close this generalization gap, they seem to only do so in regimes that damage the wall-clock gains of the warm start. Nevertheless, it is highly desirable to be able to warm-start neural network training, as it would dramatically reduce the resource usage associated with the construction of performant deep learning systems. In this work, we take a closer look at this empirical phenomenon and try to understand when and how it occurs. We also provide a surprisingly simple trick that overcomes this pathology in several important situations, and present experiments that elucidate some of its properties.

What makes a difference in the post-training of LLMs? We investigate the training patterns of different layers in large language models (LLMs), through the lens of gradient, when training with different responses and initial models. We are specifically interested in how fast vs. slow thinking affects the layer-wise gradients, given the recent popularity of training LLMs on reasoning paths such as chain-of-thoughts (CoT) and process rewards. In our study, fast thinking without CoT leads to larger gradients and larger differences of gradients across layers than slow thinking (Detailed CoT), indicating the learning stability brought by the latter. Moreover, pre-trained LLMs are less affected by the instability of fast thinking than instruction-tuned LLMs. Additionally, we study whether the gradient patterns can reflect the correctness of responses when training different LLMs using slow vs. fast thinking paths. The results show that the gradients of slow thinking can distinguish correct and irrelevant reasoning paths. As a comparison, we conduct similar gradient analyses on non-reasoning knowledge learning tasks, on which, however, trivially increasing the response length does not lead to similar behaviors of slow thinking. Our study strengthens fundamental understandings of LLM training and sheds novel insights on its efficiency and stability, which pave the way towards building a generalizable System-2 agent. Our code, data, and gradient statistics can be found in: https://github.com/MingLiiii/Layer_Gradient.

Work on scaling laws has found that large language models (LMs) show predictable improvements to overall loss with increased scale (model size, training data, and compute). Here, we present evidence for the claim that LMs may show inverse scaling, or worse task performance with increased scale, e.g., due to flaws in the training objective and data. We present empirical evidence of inverse scaling on 11 datasets collected by running a public contest, the Inverse Scaling Prize, with a substantial prize pool. Through analysis of the datasets, along with other examples found in the literature, we identify four potential causes of inverse scaling: (i) preference to repeat memorized sequences over following in-context instructions, (ii) imitation of undesirable patterns in the training data, (iii) tasks containing an easy distractor task which LMs could focus on, rather than the harder real task, and (iv) correct but misleading few-shot demonstrations of the task. We release the winning datasets at https://inversescaling.com/data to allow for further investigation of inverse scaling. Our tasks have helped drive the discovery of U-shaped and inverted-U scaling trends, where an initial trend reverses, suggesting that scaling trends are less reliable at predicting the behavior of larger-scale models than previously understood. Overall, our results suggest that there are tasks for which increased model scale alone may not lead to progress, and that more careful thought needs to go into the data and objectives for training language models.

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.

When using Large Language Models (LLMs) to support Knowledge Graph Engineering (KGE), one of the first indications when searching for an appropriate model is its size. According to the scaling laws, larger models typically show higher capabilities. However, in practice, resource costs are also an important factor and thus it makes sense to consider the ratio between model performance and costs. The LLM-KG-Bench framework enables the comparison of LLMs in the context of KGE tasks and assesses their capabilities of understanding and producing KGs and KG queries. Based on a dataset created in an LLM-KG-Bench run covering 26 open state-of-the-art LLMs, we explore the model size scaling laws specific to KGE tasks. In our analyses, we assess how benchmark scores evolve between different model size categories. Additionally, we inspect how the general score development of single models and families of models correlates to their size. Our analyses revealed that, with a few exceptions, the model size scaling laws generally also apply to the selected KGE tasks. However, in some cases, plateau or ceiling effects occurred, i.e., the task performance did not change much between a model and the next larger model. In these cases, smaller models could be considered to achieve high cost-effectiveness. Regarding models of the same family, sometimes larger models performed worse than smaller models of the same family. These effects occurred only locally. Hence it is advisable to additionally test the next smallest and largest model of the same family.

"Induction heads" are attention heads that implement a simple algorithm to complete token sequences like [A][B] ... [A] -> [B]. In this work, we present preliminary and indirect evidence for a hypothesis that induction heads might constitute the mechanism for the majority of all "in-context learning" in large transformer models (i.e. decreasing loss at increasing token indices). We find that induction heads develop at precisely the same point as a sudden sharp increase in in-context learning ability, visible as a bump in the training loss. We present six complementary lines of evidence, arguing that induction heads may be the mechanistic source of general in-context learning in transformer models of any size. For small attention-only models, we present strong, causal evidence; for larger models with MLPs, we present correlational evidence.

Critical learning periods are periods early in development where temporary sensory deficits can have a permanent effect on behavior and learned representations. Despite the radical differences between biological and artificial networks, critical learning periods have been empirically observed in both systems. This suggests that critical periods may be fundamental to learning and not an accident of biology. Yet, why exactly critical periods emerge in deep networks is still an open question, and in particular it is unclear whether the critical periods observed in both systems depend on particular architectural or optimization details. To isolate the key underlying factors, we focus on deep linear network models, and show that, surprisingly, such networks also display much of the behavior seen in biology and artificial networks, while being amenable to analytical treatment. We show that critical periods depend on the depth of the model and structure of the data distribution. We also show analytically and in simulations that the learning of features is tied to competition between sources. Finally, we extend our analysis to multi-task learning to show that pre-training on certain tasks can damage the transfer performance on new tasks, and show how this depends on the relationship between tasks and the duration of the pre-training stage. To the best of our knowledge, our work provides the first analytically tractable model that sheds light into why critical learning periods emerge in biological and artificial networks.

We introduce the framework of performative reinforcement learning where the policy chosen by the learner affects the underlying reward and transition dynamics of the environment. Following the recent literature on performative prediction~Perdomo et. al., 2020, we introduce the concept of performatively stable policy. We then consider a regularized version of the reinforcement learning problem and show that repeatedly optimizing this objective converges to a performatively stable policy under reasonable assumptions on the transition dynamics. Our proof utilizes the dual perspective of the reinforcement learning problem and may be of independent interest in analyzing the convergence of other algorithms with decision-dependent environments. We then extend our results for the setting where the learner just performs gradient ascent steps instead of fully optimizing the objective, and for the setting where the learner has access to a finite number of trajectories from the changed environment. For both settings, we leverage the dual formulation of performative reinforcement learning and establish convergence to a stable solution. Finally, through extensive experiments on a grid-world environment, we demonstrate the dependence of convergence on various parameters e.g. regularization, smoothness, and the number of samples.

Policy gradient algorithms have found wide adoption in NLP, but have recently become subject to criticism, doubting their suitability for NMT. Choshen et al. (2020) identify multiple weaknesses and suspect that their success is determined by the shape of output distributions rather than the reward. In this paper, we revisit these claims and study them under a wider range of configurations. Our experiments on in-domain and cross-domain adaptation reveal the importance of exploration and reward scaling, and provide empirical counter-evidence to these claims.

We study linear contextual bandits in the misspecified setting, where the expected reward function can be approximated by a linear function class up to a bounded misspecification level zeta>0. We propose an algorithm based on a novel data selection scheme, which only selects the contextual vectors with large uncertainty for online regression. We show that, when the misspecification level zeta is dominated by tilde O (Delta / d) with Delta being the minimal sub-optimality gap and d being the dimension of the contextual vectors, our algorithm enjoys the same gap-dependent regret bound tilde O (d^2/Delta) as in the well-specified setting up to logarithmic factors. In addition, we show that an existing algorithm SupLinUCB (Chu et al., 2011) can also achieve a gap-dependent constant regret bound without the knowledge of sub-optimality gap Delta. Together with a lower bound adapted from Lattimore et al. (2020), our result suggests an interplay between misspecification level and the sub-optimality gap: (1) the linear contextual bandit model is efficiently learnable when zeta leq tilde O(Delta / d); and (2) it is not efficiently learnable when zeta geq tilde Omega({Delta} / {d}). Experiments on both synthetic and real-world datasets corroborate our theoretical results.

Continual pre-training has increasingly become the predominant approach for adapting Large Language Models (LLMs) to new domains. This process involves updating the pre-trained LLM with a corpus from a new domain, resulting in a shift in the training distribution. To study the behavior of LLMs during this shift, we measured the model's performance throughout the continual pre-training process. we observed a temporary performance drop at the beginning, followed by a recovery phase, a phenomenon known as the "stability gap," previously noted in vision models classifying new classes. To address this issue and enhance LLM performance within a fixed compute budget, we propose three effective strategies: (1) Continually pre-training the LLM on a subset with a proper size for multiple epochs, resulting in faster performance recovery than pre-training the LLM on a large corpus in a single epoch; (2) Pre-training the LLM only on high-quality sub-corpus, which rapidly boosts domain performance; and (3) Using a data mixture similar to the pre-training data to reduce distribution gap. We conduct various experiments on Llama-family models to validate the effectiveness of our strategies in both medical continual pre-training and instruction tuning. For example, our strategies improve the average medical task performance of the OpenLlama-3B model from 36.2% to 40.7% with only 40% of the original training budget and enhance the average general task performance without causing forgetting. Furthermore, we apply our strategies to the Llama-3-8B model. The resulting model, Llama-3-Physician, achieves the best medical performance among current open-source models, and performs comparably to or even better than GPT-4 on several medical benchmarks. We release our models at https://huggingface.co/YiDuo1999/Llama-3-Physician-8B-Instruct.

In many real-world applications, deep neural networks are retrained from scratch as a dataset grows in size. Given the computational expense for retraining networks, it has been argued that continual learning could make updating networks more efficient. An obstacle to achieving this goal is the stability gap, which refers to an observation that when updating on new data, performance on previously learned data degrades before recovering. Addressing this problem would enable learning new data with fewer network updates, resulting in increased computational efficiency. We study how to mitigate the stability gap. We test a variety of hypotheses to understand why the stability gap occurs. This leads us to discover a method that vastly reduces this gap. In large-scale class incremental learning experiments, we are able to significantly reduce the number of network updates needed for continual learning. Our work has the potential to advance the state-of-the-art in continual learning for real-world applications along with reducing the carbon footprint required to maintain updated neural networks.

We theoretically explore the relationship between sample-efficiency and adaptivity in reinforcement learning. An algorithm is sample-efficient if it uses a number of queries n to the environment that is polynomial in the dimension d of the problem. Adaptivity refers to the frequency at which queries are sent and feedback is processed to update the querying strategy. To investigate this interplay, we employ a learning framework that allows sending queries in K batches, with feedback being processed and queries updated after each batch. This model encompasses the whole adaptivity spectrum, ranging from non-adaptive 'offline' (K=1) to fully adaptive (K=n) scenarios, and regimes in between. For the problems of policy evaluation and best-policy identification under d-dimensional linear function approximation, we establish Omega(log log d) lower bounds on the number of batches K required for sample-efficient algorithms with n = O(poly(d)) queries. Our results show that just having adaptivity (K>1) does not necessarily guarantee sample-efficiency. Notably, the adaptivity-boundary for sample-efficiency is not between offline reinforcement learning (K=1), where sample-efficiency was known to not be possible, and adaptive settings. Instead, the boundary lies between different regimes of adaptivity and depends on the problem dimension.

Reinforcement learning from verifiable rewards (RLVR) has improved the reasoning abilities of LLMs, yet a fundamental limitation remains: models cannot learn from problems that are too difficult to solve under their current policy, as these yield no meaningful reward signal. We propose a simple yet effective solution based on task reformulation. We transform challenging open-ended problems into cognitively simpler variants -- such as multiple-choice and cloze formats -- that preserve the original answer while reducing the effective search space and providing denser learning signals. These reformulations span a spectrum from discriminative to generative tasks, which we exploit to bootstrap learning: models first learn from structured, easier formats, and this knowledge transfers back to improve performance on the original open-ended problems. Building on this insight, we introduce Cog-DRIFT, a framework that constructs reformulated variants and organizes them into an adaptive curriculum based on difficulty. Training progresses from easier to harder formats, enabling the model to learn from problems that previously yielded zero signal under standard RL post-training. Cog-DRIFT not only improves on the originally unsolvable hard problems (absolute +10.11% for Qwen and +8.64% for Llama) but also generalizes well to other held-out datasets. Across 2 models and 6 reasoning benchmarks, our method consistently outperforms standard GRPO and strong guided-exploration baselines. On average, Cog-DRIFT shows +4.72% (Qwen) and +3.23% (Llama) improvements over the second-best baseline. We further show that Cog-DRIFT improves pass@k at test time, and the curriculum improves sample efficiency. Overall, our results highlight task reformulation and curriculum learning as an effective paradigm for overcoming the exploration barrier in LLM post-training.

In this paper, we make the case that a scientific theory of deep learning is emerging. By this we mean a theory which characterizes important properties and statistics of the training process, hidden representations, final weights, and performance of neural networks. We pull together major strands of ongoing research in deep learning theory and identify five growing bodies of work that point toward such a theory: (a) solvable idealized settings that provide intuition for learning dynamics in realistic systems; (b) tractable limits that reveal insights into fundamental learning phenomena; (c) simple mathematical laws that capture important macroscopic observables; (d) theories of hyperparameters that disentangle them from the rest of the training process, leaving simpler systems behind; and (e) universal behaviors shared across systems and settings which clarify which phenomena call for explanation. Taken together, these bodies of work share certain broad traits: they are concerned with the dynamics of the training process; they primarily seek to describe coarse aggregate statistics; and they emphasize falsifiable quantitative predictions. We argue that the emerging theory is best thought of as a mechanics of the learning process, and suggest the name learning mechanics. We discuss the relationship between this mechanics perspective and other approaches for building a theory of deep learning, including the statistical and information-theoretic perspectives. In particular, we anticipate a symbiotic relationship between learning mechanics and mechanistic interpretability. We also review and address common arguments that fundamental theory will not be possible or is not important. We conclude with a portrait of important open directions in learning mechanics and advice for beginners. We host further introductory materials, perspectives, and open questions at learningmechanics.pub.

One of the most surprising puzzles in neural network generalisation is grokking: a network with perfect training accuracy but poor generalisation will, upon further training, transition to perfect generalisation. We propose that grokking occurs when the task admits a generalising solution and a memorising solution, where the generalising solution is slower to learn but more efficient, producing larger logits with the same parameter norm. We hypothesise that memorising circuits become more inefficient with larger training datasets while generalising circuits do not, suggesting there is a critical dataset size at which memorisation and generalisation are equally efficient. We make and confirm four novel predictions about grokking, providing significant evidence in favour of our explanation. Most strikingly, we demonstrate two novel and surprising behaviours: ungrokking, in which a network regresses from perfect to low test accuracy, and semi-grokking, in which a network shows delayed generalisation to partial rather than perfect test accuracy.

Teams that have trained large Transformer-based models have reported training instabilities at large scale that did not appear when training with the same hyperparameters at smaller scales. Although the causes of such instabilities are of scientific interest, the amount of resources required to reproduce them has made investigation difficult. In this work, we seek ways to reproduce and study training stability and instability at smaller scales. First, we focus on two sources of training instability described in previous work: the growth of logits in attention layers (Dehghani et al., 2023) and divergence of the output logits from the log probabilities (Chowdhery et al., 2022). By measuring the relationship between learning rate and loss across scales, we show that these instabilities also appear in small models when training at high learning rates, and that mitigations previously employed at large scales are equally effective in this regime. This prompts us to investigate the extent to which other known optimizer and model interventions influence the sensitivity of the final loss to changes in the learning rate. To this end, we study methods such as warm-up, weight decay, and the muParam (Yang et al., 2022), and combine techniques to train small models that achieve similar losses across orders of magnitude of learning rate variation. Finally, to conclude our exploration we study two cases where instabilities can be predicted before they emerge by examining the scaling behavior of model activation and gradient norms.

Many recent works have discussed the propensity, or lack thereof, for emergent languages to exhibit properties of natural languages. A favorite in the literature is learning compositionality. We note that most of those works have focused on communicative bandwidth as being of primary importance. While important, it is not the only contributing factor. In this paper, we investigate the learning biases that affect the efficacy and compositionality of emergent languages. Our foremost contribution is to explore how capacity of a neural network impacts its ability to learn a compositional language. We additionally introduce a set of evaluation metrics with which we analyze the learned languages. Our hypothesis is that there should be a specific range of model capacity and channel bandwidth that induces compositional structure in the resulting language and consequently encourages systematic generalization. While we empirically see evidence for the bottom of this range, we curiously do not find evidence for the top part of the range and believe that this is an open question for the community.

There are two widely known issues with properly training Recurrent Neural Networks, the vanishing and the exploding gradient problems detailed in Bengio et al. (1994). In this paper we attempt to improve the understanding of the underlying issues by exploring these problems from an analytical, a geometric and a dynamical systems perspective. Our analysis is used to justify a simple yet effective solution. We propose a gradient norm clipping strategy to deal with exploding gradients and a soft constraint for the vanishing gradients problem. We validate empirically our hypothesis and proposed solutions in the experimental section.

Improvements in language models are often driven by improving the quality of the data we train them on, which can be limiting when strong supervision is scarce. In this work, we show that paired preference data consisting of individually weak data points can enable gains beyond the strength of each individual data point. We formulate the delta learning hypothesis to explain this phenomenon, positing that the relative quality delta between points suffices to drive learning via preference tuning--even when supervised finetuning on the weak data hurts. We validate our hypothesis in controlled experiments and at scale, where we post-train 8B models on preference data generated by pairing a small 3B model's responses with outputs from an even smaller 1.5B model to create a meaningful delta. Strikingly, on a standard 11-benchmark evaluation suite (MATH, MMLU, etc.), our simple recipe matches the performance of Tulu 3, a state-of-the-art open model tuned from the same base model while relying on much stronger supervisors (e.g., GPT-4o). Thus, delta learning enables simpler and cheaper open recipes for state-of-the-art post-training. To better understand delta learning, we prove in logistic regression that the performance gap between two weak teacher models provides useful signal for improving a stronger student. Overall, our work shows that models can learn surprisingly well from paired data that might typically be considered weak.

In this work, we investigate how a model's tendency to broadly integrate its parametric knowledge evolves throughout pretraining, and how this behavior affects overall performance, particularly in terms of knowledge acquisition and forgetting. We introduce the concept of knowledge entropy, which quantifies the range of memory sources the model engages with; high knowledge entropy indicates that the model utilizes a wide range of memory sources, while low knowledge entropy suggests reliance on specific sources with greater certainty. Our analysis reveals a consistent decline in knowledge entropy as pretraining advances. We also find that the decline is closely associated with a reduction in the model's ability to acquire and retain knowledge, leading us to conclude that diminishing knowledge entropy (smaller number of active memory sources) impairs the model's knowledge acquisition and retention capabilities. We find further support for this by demonstrating that increasing the activity of inactive memory sources enhances the model's capacity for knowledge acquisition and retention.

Inspired by the phenomenon of catastrophic forgetting, we investigate the learning dynamics of neural networks as they train on single classification tasks. Our goal is to understand whether a related phenomenon occurs when data does not undergo a clear distributional shift. We define a `forgetting event' to have occurred when an individual training example transitions from being classified correctly to incorrectly over the course of learning. Across several benchmark data sets, we find that: (i) certain examples are forgotten with high frequency, and some not at all; (ii) a data set's (un)forgettable examples generalize across neural architectures; and (iii) based on forgetting dynamics, a significant fraction of examples can be omitted from the training data set while still maintaining state-of-the-art generalization performance.

Local prediction-error-based curiosity rewards focus on the current transition without considering the world model's cumulative prediction error across all visited transitions. We introduce Curiosity-Critic, which grounds its intrinsic reward in the improvement of this cumulative objective, and show that it reduces to a tractable per-step form: the difference between the current prediction error and the asymptotic error baseline of the current state transition. We estimate this baseline online with a learned critic co-trained alongside the world model; regressing a single scalar, the critic converges well before the world model saturates, redirecting exploration toward learnable transitions without oracle knowledge of the noise floor. The reward is higher for learnable transitions and collapses toward the baseline for stochastic ones, effectively separating epistemic (reducible) from aleatoric (irreducible) prediction error online. Prior prediction-error curiosity formulations, from Schmidhuber (1991) to learned-feature-space variants, emerge as special cases corresponding to specific approximations of this baseline. Experiments on a stochastic grid world show that Curiosity-Critic outperforms prediction-error and visitation-count baselines in convergence speed and final world model accuracy.

How can we teach large multimodal models (LMMs) new skills without erasing prior abilities? We study sequential fine-tuning on five target skills while monitoring general ability on eight held-out benchmarks across three model families. We observe that apparent "forgetting" on held-out tasks after narrow fine-tuning can partly recover at later stages. We trace this behavior to a measurable shift in the output token distribution, manifested through a simple counting-bias probe that co-varies with forgetting. Guided by this picture, we identify two simple, robust tuning recipes that learn strongly while limiting drift: (i) updating only the self-attention projection layers, and (ii) updating only the MLP Gate&Up while freezing the Down projection. Across models and tasks, these choices deliver strong target gains while largely preserving held-out performance. Code is available at https://github.com/jessemelpolio/LMM_CL

We identify and formalize a fundamental gradient descent phenomenon resulting in a learning proclivity in over-parameterized neural networks. Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task, despite the presence of other predictive features that fail to be discovered. This work provides a theoretical explanation for the emergence of such feature imbalance in neural networks. Using tools from Dynamical Systems theory, we identify simple properties of learning dynamics during gradient descent that lead to this imbalance, and prove that such a situation can be expected given certain statistical structure in training data. Based on our proposed formalism, we develop guarantees for a novel regularization method aimed at decoupling feature learning dynamics, improving accuracy and robustness in cases hindered by gradient starvation. We illustrate our findings with simple and real-world out-of-distribution (OOD) generalization experiments.

Learning to solve complex sequences of tasks--while both leveraging transfer and avoiding catastrophic forgetting--remains a key obstacle to achieving human-level intelligence. The progressive networks approach represents a step forward in this direction: they are immune to forgetting and can leverage prior knowledge via lateral connections to previously learned features. We evaluate this architecture extensively on a wide variety of reinforcement learning tasks (Atari and 3D maze games), and show that it outperforms common baselines based on pretraining and finetuning. Using a novel sensitivity measure, we demonstrate that transfer occurs at both low-level sensory and high-level control layers of the learned policy.

Machine Learning (ML) workloads have rapidly grown in importance, but raised concerns about their carbon footprint. Four best practices can reduce ML training energy by up to 100x and CO2 emissions up to 1000x. By following best practices, overall ML energy use (across research, development, and production) held steady at <15% of Google's total energy use for the past three years. If the whole ML field were to adopt best practices, total carbon emissions from training would reduce. Hence, we recommend that ML papers include emissions explicitly to foster competition on more than just model quality. Estimates of emissions in papers that omitted them have been off 100x-100,000x, so publishing emissions has the added benefit of ensuring accurate accounting. Given the importance of climate change, we must get the numbers right to make certain that we work on its biggest challenges.

On a variety of tasks, the performance of neural networks predictably improves with training time, dataset size and model size across many orders of magnitude. This phenomenon is known as a neural scaling law. Of fundamental importance is the compute-optimal scaling law, which reports the performance as a function of units of compute when choosing model sizes optimally. We analyze a random feature model trained with gradient descent as a solvable model of network training and generalization. This reproduces many observations about neural scaling laws. First, our model makes a prediction about why the scaling of performance with training time and with model size have different power law exponents. Consequently, the theory predicts an asymmetric compute-optimal scaling rule where the number of training steps are increased faster than model parameters, consistent with recent empirical observations. Second, it has been observed that early in training, networks converge to their infinite-width dynamics at a rate 1/width but at late time exhibit a rate width^{-c}, where c depends on the structure of the architecture and task. We show that our model exhibits this behavior. Lastly, our theory shows how the gap between training and test loss can gradually build up over time due to repeated reuse of data.

Recent advancements in Large Language Models have yielded significant improvements in complex reasoning tasks such as mathematics and programming. However, these models remain heavily dependent on annotated data and exhibit limited adaptability in unsupervised scenarios. To address these limitations, test-time reinforcement learning (TTRL) has been proposed, which enables self-optimization by leveraging model-generated pseudo-labels. Despite its promise, TTRL faces several key challenges, including high inference costs due to parallel rollouts and early-stage estimation bias that fosters overconfidence, reducing output diversity and causing performance plateaus. To address these challenges, we introduce an entropy-based mechanism to enhance the exploration-exploitation balance in test-time reinforcement learning through two strategies: Entropy-fork Tree Majority Rollout (ETMR) and Entropy-based Advantage Reshaping (EAR). Compared with the baseline, our approach enables Llama3.1-8B to achieve a 68 percent relative improvement in Pass at 1 metric on the AIME 2024 benchmark, while consuming only 60 percent of the rollout tokens budget. This highlights our method's ability to effectively optimize the trade-off between inference efficiency, diversity, and estimation robustness, thereby advancing unsupervised reinforcement learning for open-domain reasoning tasks.

The ability of language models to learn a task from a few examples in context has generated substantial interest. Here, we provide a perspective that situates this type of supervised few-shot learning within a much broader spectrum of meta-learned in-context learning. Indeed, we suggest that any distribution of sequences in which context non-trivially decreases loss on subsequent predictions can be interpreted as eliciting a kind of in-context learning. We suggest that this perspective helps to unify the broad set of in-context abilities that language models exhibit x2014 such as adapting to tasks from instructions or role play, or extrapolating time series. This perspective also sheds light on potential roots of in-context learning in lower-level processing of linguistic dependencies (e.g. coreference or parallel structures). Finally, taking this perspective highlights the importance of generalization, which we suggest can be studied along several dimensions: not only the ability to learn something novel, but also flexibility in learning from different presentations, and in applying what is learned. We discuss broader connections to past literature in meta-learning and goal-conditioned agents, and other perspectives on learning and adaptation. We close by suggesting that research on in-context learning should consider this broader spectrum of in-context capabilities and types of generalization.

Neural networks often exhibit emergent behavior, where qualitatively new capabilities arise from scaling up the amount of parameters, training data, or training steps. One approach to understanding emergence is to find continuous progress measures that underlie the seemingly discontinuous qualitative changes. We argue that progress measures can be found via mechanistic interpretability: reverse-engineering learned behaviors into their individual components. As a case study, we investigate the recently-discovered phenomenon of ``grokking'' exhibited by small transformers trained on modular addition tasks. We fully reverse engineer the algorithm learned by these networks, which uses discrete Fourier transforms and trigonometric identities to convert addition to rotation about a circle. We confirm the algorithm by analyzing the activations and weights and by performing ablations in Fourier space. Based on this understanding, we define progress measures that allow us to study the dynamics of training and split training into three continuous phases: memorization, circuit formation, and cleanup. Our results show that grokking, rather than being a sudden shift, arises from the gradual amplification of structured mechanisms encoded in the weights, followed by the later removal of memorizing components.

In this paper we propose to study generalization of neural networks on small algorithmically generated datasets. In this setting, questions about data efficiency, memorization, generalization, and speed of learning can be studied in great detail. In some situations we show that neural networks learn through a process of "grokking" a pattern in the data, improving generalization performance from random chance level to perfect generalization, and that this improvement in generalization can happen well past the point of overfitting. We also study generalization as a function of dataset size and find that smaller datasets require increasing amounts of optimization for generalization. We argue that these datasets provide a fertile ground for studying a poorly understood aspect of deep learning: generalization of overparametrized neural networks beyond memorization of the finite training dataset.

Continual learning systems are increasingly deployed in environments where retraining or reset is infeasible, yet many approaches emphasize task performance rather than the evolution of internal representations over time. In this work, we study a minimal continual learning agent designed to isolate representational dynamics from architectural complexity and optimization objectives. The agent maintains a persistent state vector across executions and incrementally updates it as new textual data is introduced. We quantify representational change using cosine similarity between successive normalized state vectors and define a stability metric over time intervals. Longitudinal experiments across eight executions reveal a transition from an initial plastic regime to a stable representational regime under consistent input. A deliberately introduced semantic perturbation produces a bounded decrease in similarity, followed by recovery and restabilization under subsequent coherent input. These results demonstrate that meaningful stability plasticity tradeoffs can emerge in a minimal, stateful learning system without explicit regularization, replay, or architectural complexity. The work establishes a transparent empirical baseline for studying representational accumulation and adaptation in continual learning systems.

Large language models have demonstrated impressive performance on challenging mathematical reasoning tasks, which has triggered the discussion of whether the performance is achieved by true reasoning capability or memorization. To investigate this question, prior work has constructed mathematical benchmarks when questions undergo simple perturbations -- modifications that still preserve the underlying reasoning patterns of the solutions. However, no work has explored hard perturbations, which fundamentally change the nature of the problem so that the original solution steps do not apply. To bridge the gap, we construct MATH-P-Simple and MATH-P-Hard via simple perturbation and hard perturbation, respectively. Each consists of 279 perturbed math problems derived from level-5 (hardest) problems in the MATH dataset (Hendrycksmath et. al., 2021). We observe significant performance drops on MATH-P-Hard across various models, including o1-mini (-16.49%) and gemini-2.0-flash-thinking (-12.9%). We also raise concerns about a novel form of memorization where models blindly apply learned problem-solving skills without assessing their applicability to modified contexts. This issue is amplified when using original problems for in-context learning. We call for research efforts to address this challenge, which is critical for developing more robust and reliable reasoning models.

We describe a robotic learning system for autonomous exploration and navigation in diverse, open-world environments. At the core of our method is a learned latent variable model of distances and actions, along with a non-parametric topological memory of images. We use an information bottleneck to regularize the learned policy, giving us (i) a compact visual representation of goals, (ii) improved generalization capabilities, and (iii) a mechanism for sampling feasible goals for exploration. Trained on a large offline dataset of prior experience, the model acquires a representation of visual goals that is robust to task-irrelevant distractors. We demonstrate our method on a mobile ground robot in open-world exploration scenarios. Given an image of a goal that is up to 80 meters away, our method leverages its representation to explore and discover the goal in under 20 minutes, even amidst previously-unseen obstacles and weather conditions. Please check out the project website for videos of our experiments and information about the real-world dataset used at https://sites.google.com/view/recon-robot.

While the field of Quality-Diversity (QD) has grown into a distinct branch of stochastic optimization, a few problems, in particular locomotion and navigation tasks, have become de facto standards. Are such benchmarks sufficient? Are they representative of the key challenges faced by QD algorithms? Do they provide the ability to focus on one particular challenge by properly disentangling it from others? Do they have much predictive power in terms of scalability and generalization? Existing benchmarks are not standardized, and there is currently no MNIST equivalent for QD. Inspired by recent works on Reinforcement Learning benchmarks, we argue that the identification of challenges faced by QD methods and the development of targeted, challenging, scalable but affordable benchmarks is an important step. As an initial effort, we identify three problems that are challenging in sparse reward settings, and propose associated benchmarks: (1) Behavior metric bias, which can result from the use of metrics that do not match the structure of the behavior space. (2) Behavioral Plateaus, with varying characteristics, such that escaping them would require adaptive QD algorithms and (3) Evolvability Traps, where small variations in genotype result in large behavioral changes. The environments that we propose satisfy the properties listed above.

We prove theoretically that generalization improves not only through data scaling but also by compressing internal representations. To operationalize this insight, we introduce the Information Bottleneck Language Modeling (IBLM) objective, which reframes language modeling as a constrained optimization problem: minimizing representation entropy subject to optimal prediction performance. Empirically, we observe an emergent memorization-compression cycle during LLM pretraining, evidenced by oscillation positive/negative gradient alignment between cross-entropy and Matrix-Based Entropy (MBE), a measure of representation entropy. This pattern closely mirrors the predictive-compressive trade-off prescribed by IBLM and also parallels the biological alternation between awake learning and sleep consolidation. Motivated by this observation, we propose Gated Phase Transition (GAPT), a training algorithm that adaptively switches between memorization and compression phases. When applied to GPT-2 pretraining on FineWeb dataset, GAPT reduces MBE by 50% and improves cross-entropy by 4.8%. GAPT improves OOD generalizatino by 35% in a pretraining task on arithmetic multiplication. In a setting designed to simulate catastrophic forgetting, GAPT reduces interference by compressing and separating representations, achieving a 97% improvement in separation - paralleling the functional role of sleep consolidation.

While Large Language Models (LLMs) have emerged with remarkable capabilities in complex tasks through Chain-of-Thought reasoning, practical resource constraints have sparked interest in transferring these abilities to smaller models. However, achieving both domain performance and cross-domain generalization remains challenging. Existing approaches typically restrict students to following a single golden rationale and treat different reasoning paths independently. Due to distinct inductive biases and intrinsic preferences, alongside the student's evolving capacity and reasoning preferences during training, a teacher's "optimal" rationale could act as out-of-distribution noise. This misalignment leads to a degeneration of the student's latent reasoning distribution, causing suboptimal performance. To bridge this gap, we propose MIND, a capability-adaptive framework that transitions distillation from passive mimicry to active cognitive construction. We synthesize diverse teacher perspectives through a novel "Teaching Assistant" network. By employing a Feedback-Driven Inertia Calibration mechanism, this network utilizes inertia-filtered training loss to align supervision with the student's current adaptability, effectively enhancing performance while mitigating catastrophic forgetting. Extensive experiments demonstrate that MIND achieves state-of-the-art performance on both in-distribution and out-of-distribution benchmarks, and our sophisticated latent space analysis further confirms the mechanism of reasoning ability internalization.

The emergence of In-Context Learning (ICL) in LLMs remains a significant phenomenon with little understanding. To explain ICL, recent studies try to theoretically connect it to Gradient Descent (GD). We ask, does this connection hold up in actual pre-trained models? We highlight the limiting assumptions in prior works that make their context considerably different from the practical context in which language models are trained. For example, the theoretical hand-constructed weights used in these studies have properties that don't match those of real LLMs. Furthermore, their experimental verification uses ICL objective (training models explicitly for ICL), which differs from the emergent ICL in the wild. We also look for evidence in real models. We observe that ICL and GD have different sensitivity to the order in which they observe demonstrations. Finally, we probe and compare the ICL vs. GD hypothesis in a natural setting. We conduct comprehensive empirical analyses on language models pre-trained on natural data (LLaMa-7B). Our comparisons of three performance metrics highlight the inconsistent behavior of ICL and GD as a function of various factors such as datasets, models, and the number of demonstrations. We observe that ICL and GD modify the output distribution of language models differently. These results indicate that the equivalence between ICL and GD remains an open hypothesis and calls for further studies.

In many environments only a tiny subset of all states yield high reward. In these cases, few of the interactions with the environment provide a relevant learning signal. Hence, we may want to preferentially train on those high-reward states and the probable trajectories leading to them. To this end, we advocate for the use of a backtracking model that predicts the preceding states that terminate at a given high-reward state. We can train a model which, starting from a high value state (or one that is estimated to have high value), predicts and sample for which the (state, action)-tuples may have led to that high value state. These traces of (state, action) pairs, which we refer to as Recall Traces, sampled from this backtracking model starting from a high value state, are informative as they terminate in good states, and hence we can use these traces to improve a policy. We provide a variational interpretation for this idea and a practical algorithm in which the backtracking model samples from an approximate posterior distribution over trajectories which lead to large rewards. Our method improves the sample efficiency of both on- and off-policy RL algorithms across several environments and tasks.

How can we build agents that keep learning from experience, quickly and efficiently, after their initial training? Here we take inspiration from the main mechanism of learning in biological brains: synaptic plasticity, carefully tuned by evolution to produce efficient lifelong learning. We show that plasticity, just like connection weights, can be optimized by gradient descent in large (millions of parameters) recurrent networks with Hebbian plastic connections. First, recurrent plastic networks with more than two million parameters can be trained to memorize and reconstruct sets of novel, high-dimensional 1000+ pixels natural images not seen during training. Crucially, traditional non-plastic recurrent networks fail to solve this task. Furthermore, trained plastic networks can also solve generic meta-learning tasks such as the Omniglot task, with competitive results and little parameter overhead. Finally, in reinforcement learning settings, plastic networks outperform a non-plastic equivalent in a maze exploration task. We conclude that differentiable plasticity may provide a powerful novel approach to the learning-to-learn problem.

The ability to learn different tasks sequentially is essential to the development of artificial intelligence. In general, neural networks lack this capability, the major obstacle being catastrophic forgetting. It occurs when the incrementally available information from non-stationary data distributions is continually acquired, disrupting what the model has already learned. Our approach remembers old tasks by projecting the representations of new tasks close to that of old tasks while keeping the decision boundaries unchanged. We employ the center loss as a regularization penalty that enforces new tasks' features to have the same class centers as old tasks and makes the features highly discriminative. This, in turn, leads to the least forgetting of already learned information. This method is easy to implement, requires minimal computational and memory overhead, and allows the neural network to maintain high performance across many sequentially encountered tasks. We also demonstrate that using the center loss in conjunction with the memory replay outperforms other replay-based strategies. Along with standard MNIST variants for continual learning, we apply our method to continual domain adaptation scenarios with the Digits and PACS datasets. We demonstrate that our approach is scalable, effective, and gives competitive performance compared to state-of-the-art continual learning methods.

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.

The ongoing debate on whether reinforcement learning with verifiable rewards (RLVR) expands or shrinks the reasoning capabilities of large language models (LLMs) remains unresolved. Some studies contend that RLVR mainly improves sampling efficiency but at the expense of diversity and exploratory capacity, resulting in capability boundary shrinkage. In contrast, others demonstrate that prolonged training can lead to the emergence of novel reasoning strategies, suggesting capability boundary expansion. To reconcile these contradictory findings, we theoretically and empirically show that both perspectives are partially valid-each aligning with a separate phase in an inherent two-stage probability mass dynamic: (1) Exploitation stage: initially, the model primarily samples explored high-reward and low-reward tokens, while rarely selecting the potentially optimal token. Positive advantage estimates increase the probability of high-reward tokens and decrease those of low-reward tokens, yet the optimal token's probability remains largely unchanged during this stage. (2) Exploration stage: as training advances, the growth rate of previously acquired high-reward tokens slows as their probabilities approach saturation. When a potentially optimal token-now receiving positive advantage estimates-is occasionally sampled, its probability increases, while those of the originally high-reward tokens decrease. This dynamic suggests that over-exploitation during the exploitation stage may lead to capability boundary shrinkage, whereas prolonged training into the exploration stage can promote an expansion of the reasoning capability boundary. Building upon our insights, we revisit the potential of only using relative negative gradients for prolonging training, providing a theoretical and empirical foundation for the development of more advanced reasoning capabilities.

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time τ_gen at which models begin to generate high-quality samples, and a later time τ_mem beyond which memorization emerges. Crucially, we find that τ_mem increases linearly with the training set size n, while τ_gen remains constant. This creates a growing window of training times with n where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when n becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

We study subliminal learning, a surprising phenomenon where language models transmit behavioral traits via semantically unrelated data. In our main experiments, a "teacher" model with some trait T (such as liking owls or being misaligned) generates a dataset consisting solely of number sequences. Remarkably, a "student" model trained on this dataset learns T. This occurs even when the data is filtered to remove references to T. We observe the same effect when training on code or reasoning traces generated by the same teacher model. However, we do not observe the effect when the teacher and student have different base models. To help explain our findings, we prove a theoretical result showing that subliminal learning occurs in all neural networks under certain conditions, and demonstrate subliminal learning in a simple MLP classifier. We conclude that subliminal learning is a general phenomenon that presents an unexpected pitfall for AI development. Distillation could propagate unintended traits, even when developers try to prevent this via data filtering.

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 .

When an agent encounters a continual stream of new tasks in the lifelong learning setting, it leverages the knowledge it gained from the earlier tasks to help learn the new tasks better. In such a scenario, identifying an efficient knowledge representation becomes a challenging problem. Most research works propose to either store a subset of examples from the past tasks in a replay buffer, dedicate a separate set of parameters to each task or penalize excessive updates over parameters by introducing a regularization term. While existing methods employ the general task-agnostic stochastic gradient descent update rule, we propose a task-aware optimizer that adapts the learning rate based on the relatedness among tasks. We utilize the directions taken by the parameters during the updates by accumulating the gradients specific to each task. These task-based accumulated gradients act as a knowledge base that is maintained and updated throughout the stream. We empirically show that our proposed adaptive learning rate not only accounts for catastrophic forgetting but also allows positive backward transfer. We also show that our method performs better than several state-of-the-art methods in lifelong learning on complex datasets with a large number of tasks.

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.

Despite the advancement of machine learning techniques in recent years, state-of-the-art systems lack robustness to "real world" events, where the input distributions and tasks encountered by the deployed systems will not be limited to the original training context, and systems will instead need to adapt to novel distributions and tasks while deployed. This critical gap may be addressed through the development of "Lifelong Learning" systems that are capable of 1) Continuous Learning, 2) Transfer and Adaptation, and 3) Scalability. Unfortunately, efforts to improve these capabilities are typically treated as distinct areas of research that are assessed independently, without regard to the impact of each separate capability on other aspects of the system. We instead propose a holistic approach, using a suite of metrics and an evaluation framework to assess Lifelong Learning in a principled way that is agnostic to specific domains or system techniques. Through five case studies, we show that this suite of metrics can inform the development of varied and complex Lifelong Learning systems. We highlight how the proposed suite of metrics quantifies performance trade-offs present during Lifelong Learning system development - both the widely discussed Stability-Plasticity dilemma and the newly proposed relationship between Sample Efficient and Robust Learning. Further, we make recommendations for the formulation and use of metrics to guide the continuing development of Lifelong Learning systems and assess their progress in the future.

We characterize the statistical efficiency of knowledge transfer through n samples from a teacher to a probabilistic student classifier with input space mathcal S over labels mathcal A. We show that privileged information at three progressive levels accelerates the transfer. At the first level, only samples with hard labels are known, via which the maximum likelihood estimator attains the minimax rate {|{mathcal S||{mathcal A}|}/{n}}. The second level has the teacher probabilities of sampled labels available in addition, which turns out to boost the convergence rate lower bound to {{|{mathcal S}||{mathcal A}|}/{n}}. However, under this second data acquisition protocol, minimizing a naive adaptation of the cross-entropy loss results in an asymptotically biased student. We overcome this limitation and achieve the fundamental limit by using a novel empirical variant of the squared error logit loss. The third level further equips the student with the soft labels (complete logits) on {mathcal A} given every sampled input, thereby provably enables the student to enjoy a rate {|{mathcal S}|}/{n} free of |{mathcal A}|. We find any Kullback-Leibler divergence minimizer to be optimal in the last case. Numerical simulations distinguish the four learners and corroborate our theory.

Machine unlearning seeks to selectively remove the "influence" of specific training data on a model's outputs. The ideal goal is Retrain Equivalence--behavior identical to a model trained from scratch on only the retained data. This goal was formulated for models trained on i.i.d. data batches, but modern pipelines often involve multi-stage training, with each stage having a distinct data distribution and objective. Examples include LLM fine-tuning for alignment, reasoning ability, etc. Our study shows via theory and experiments that this shift to multi-stage training introduces a fundamental barrier for machine unlearning. The theory indicates that the outcome of local unlearning--methods that only use gradients computed on the forget set--is path-dependent. That is, a model's behavior during unlearning is influenced by the order of its training stages during learning, making it impossible for path-oblivious algorithms to universally achieve Retrain Equivalence. We empirically demonstrate the same phenomenon in LLM post-training across Llama and Qwen models (1B to 14B) with gradient ascent, NPO, and SimNPO local unlearning algorithms. Models fine-tuned via different orderings of identical training stages diverge in behavior during unlearning, with the degradation in GSM8K accuracy after unlearning varying by over 20% across paths. We also observe that some learning paths consistently produce models that unlearn slowly. During unlearning, whether the probability mass gets squeezed into paraphrasing or alternative concepts is also path-dependent. These results consistently show that Retrain Equivalence is an ill-posed target for local unlearning algorithms, so long as the target models are trained in stages. In situations where access to models' training histories is hard, the current work calls for rethinking the definition and desiderata of machine unlearning.

It is widely believed that given the same labeling budget, active learning (AL) algorithms like margin-based active learning achieve better predictive performance than passive learning (PL), albeit at a higher computational cost. Recent empirical evidence suggests that this added cost might be in vain, as margin-based AL can sometimes perform even worse than PL. While existing works offer different explanations in the low-dimensional regime, this paper shows that the underlying mechanism is entirely different in high dimensions: we prove for logistic regression that PL outperforms margin-based AL even for noiseless data and when using the Bayes optimal decision boundary for sampling. Insights from our proof indicate that this high-dimensional phenomenon is exacerbated when the separation between the classes is small. We corroborate this intuition with experiments on 20 high-dimensional datasets spanning a diverse range of applications, from finance and histology to chemistry and computer vision.

Learning from changing tasks and sequential experience without forgetting the obtained knowledge is a challenging problem for artificial neural networks. In this work, we focus on two challenging problems in the paradigm of Continual Learning (CL) without involving any old data: (i) the accumulation of catastrophic forgetting caused by the gradually fading knowledge space from which the model learns the previous knowledge; (ii) the uncontrolled tug-of-war dynamics to balance the stability and plasticity during the learning of new tasks. In order to tackle these problems, we present Progressive Learning without Forgetting (PLwF) and a credit assignment regime in the optimizer. PLwF densely introduces model functions from previous tasks to construct a knowledge space such that it contains the most reliable knowledge on each task and the distribution information of different tasks, while credit assignment controls the tug-of-war dynamics by removing gradient conflict through projection. Extensive ablative experiments demonstrate the effectiveness of PLwF and credit assignment. In comparison with other CL methods, we report notably better results even without relying on any raw data.

Knowledge-dependent tasks typically use two sources of knowledge: parametric, learned at training time, and contextual, given as a passage at inference time. To understand how models use these sources together, we formalize the problem of knowledge conflicts, where the contextual information contradicts the learned information. Analyzing the behaviour of popular models, we measure their over-reliance on memorized information (the cause of hallucinations), and uncover important factors that exacerbate this behaviour. Lastly, we propose a simple method to mitigate over-reliance on parametric knowledge, which minimizes hallucination, and improves out-of-distribution generalization by 4%-7%. Our findings demonstrate the importance for practitioners to evaluate model tendency to hallucinate rather than read, and show that our mitigation strategy encourages generalization to evolving information (i.e., time-dependent queries). To encourage these practices, we have released our framework for generating knowledge conflicts.

Recent studies show evidence for emergent cognitive abilities in Large Pre-trained Language Models (PLMs). The increasing cognitive alignment of these models has made them candidates for cognitive science theories. Prior research into the emergent cognitive abilities of PLMs has largely been path-independent to model training, i.e., has focused on the final model weights and not the intermediate steps. However, building plausible models of human cognition using PLMs would benefit from considering the developmental alignment of their performance during training to the trajectories of children's thinking. Guided by psychometric tests of human intelligence, we choose four sets of tasks to investigate the alignment of ten popular families of PLMs and evaluate their available intermediate and final training steps. These tasks are Numerical ability, Linguistic abilities, Conceptual understanding, and Fluid reasoning. We find a striking regularity: regardless of model size, the developmental trajectories of PLMs consistently exhibit a window of maximal alignment to human cognitive development. Before that window, training appears to endow "blank slate" models with the requisite structure to be poised to rapidly learn from experience. After that window, training appears to serve the engineering goal of reducing loss but not the scientific goal of increasing alignment with human cognition.

Can Large Language Models understand how students learn? As LLMs are deployed for adaptive testing and personalized tutoring, this question becomes urgent -- yet we cannot answer it with existing resources. Current educational datasets provide only question identifiers and binary correctness labels, rendering them opaque to LLMs that reason in natural language. We address this gap with FoundationalASSIST, the first English educational dataset providing the complete information needed for research on LLMs in education: full question text, actual student responses (not just right/wrong), records of which wrong answers students chose, and alignment to Common Core K-12 standards. These 1.7 million interactions from 5,000 students enable research directions that were previously impossible to pursue, from fine-tuning student models to analyzing misconception patterns. To demonstrate the dataset's utility, we evaluate four frontier models (GPT-OSS-120B, Llama-3.3-70B, Qwen3-Next-80B variants) on two complementary task families: Knowledge Tracing, testing whether LLMs can predict student performance on questions, and the exact answer a student will give; and Pedagogical Grounding, testing whether LLMs understand the properties that make assessment items effective. Our evaluation reveals significant gaps in current LLM capabilities. Every model barely achieves a trivial baseline on knowledge tracing. All models fall below random chance on item discrimination, indicating that LLMs do not understand what makes one problem more diagnostic than another. Models do show competence at judging relative difficulty (up to 68.6%), but this partial success only highlights the gaps elsewhere. These results establish that substantial advances are needed before LLMs can reliably support personalized learning at scale. We release FoundationalASSIST to support progress on these foundational challenges.

Large language models learn and continually learn through the accumulation of gradient-based updates, but how individual pieces of new information affect existing knowledge, leading to both beneficial generalization and problematic hallucination, remains poorly understood. We demonstrate that when learning new information, LLMs exhibit a "priming" effect: learning a new fact can cause the model to inappropriately apply that knowledge in unrelated contexts. To systematically study this phenomenon, we introduce "Outlandish," a carefully curated dataset of 1320 diverse text samples designed to probe how new knowledge permeates through an LLM's existing knowledge base. Using this dataset, we show that the degree of priming after learning new information can be predicted by measuring the token probability of key words before learning. This relationship holds robustly across different model architectures (PALM-2, Gemma, Llama), sizes, and training stages. Finally, we develop two novel techniques to modulate how new knowledge affects existing model behavior: (1) a ``stepping-stone'' text augmentation strategy and (2) an ``ignore-k'' update pruning method. These approaches reduce undesirable priming effects by 50-95\% while preserving the model's ability to learn new information. Our findings provide both empirical insights into how LLMs learn and practical tools for improving the specificity of knowledge insertion in language models. Further materials: https://sunchipsster1.github.io/projects/outlandish/

Recent advances in reinforcement learning (RL) with numerical feedback, such as scalar rewards, have significantly enhanced the complex reasoning capabilities of large language models (LLMs). Despite this success, we identify three key challenges encountered by RL with solely numerical feedback: performance plateaus, limited effectiveness of self-reflection, and persistent failures. We then demonstrate that RL-finetuned models, even after exhibiting performance plateaus, can generate correct refinements on persistently failed problems by leveraging natural language feedback in the form of critiques. Building on this insight, we propose Critique-GRPO, an online RL framework that integrates both natural language and numerical feedback for effective policy optimization. Critique-GRPO enables LLMs to learn from initial responses and critique-guided refinements simultaneously while maintaining exploration. Extensive experiments using Qwen2.5-7B-Base and Qwen3-8B-Base show that Critique-GRPO consistently outperforms supervised learning-based and RL-based fine-tuning approaches across eight challenging mathematical, STEM, and general reasoning tasks, improving average pass@1 scores by approximately 4.5% and 5%, respectively. Notably, Critique-GRPO surpasses a strong baseline that incorporates expert demonstrations within online RL. Further analysis reveals two critical insights about policy exploration: (1) higher entropy does not always guarantee efficient learning from exploration, and (2) longer responses do not necessarily lead to more effective exploration.

Vision-language pretraining has advanced image-text alignment, yet progress in radiology remains constrained by the heterogeneity of clinical reports, including abbreviations, impression-only notes, and stylistic variability. Unlike general-domain settings where more data often leads to better performance, naively scaling to large collections of noisy reports can plateau or even degrade model learning. We ask whether large language model (LLM) encoders can provide robust clinical representations that transfer across diverse styles and better guide image-text alignment. We introduce LLM2VEC4CXR, a domain-adapted LLM encoder for chest X-ray reports, and LLM2CLIP4CXR, a dual-tower framework that couples this encoder with a vision backbone. LLM2VEC4CXR improves clinical text understanding over BERT-based baselines, handles abbreviations and style variation, and achieves strong clinical alignment on report-level metrics. LLM2CLIP4CXR leverages these embeddings to boost retrieval accuracy and clinically oriented scores, with stronger cross-dataset generalization than prior medical CLIP variants. Trained on 1.6M CXR studies from public and private sources with heterogeneous and noisy reports, our models demonstrate that robustness -- not scale alone -- is the key to effective multimodal learning. We release models to support further research in medical image-text representation learning.

Large language models (LMs) are able to in-context learn -- perform a new task via inference alone by conditioning on a few input-label pairs (demonstrations) and making predictions for new inputs. However, there has been little understanding of how the model learns and which aspects of the demonstrations contribute to end task performance. In this paper, we show that ground truth demonstrations are in fact not required -- randomly replacing labels in the demonstrations barely hurts performance on a range of classification and multi-choce tasks, consistently over 12 different models including GPT-3. Instead, we find that other aspects of the demonstrations are the key drivers of end task performance, including the fact that they provide a few examples of (1) the label space, (2) the distribution of the input text, and (3) the overall format of the sequence. Together, our analysis provides a new way of understanding how and why in-context learning works, while opening up new questions about how much can be learned from large language models through inference alone.

Continual learning aims to acquire tasks sequentially without catastrophic forgetting, yet standard strategies face a core tradeoff: regularization-based methods (e.g., EWC) can overconstrain updates when task optima are weakly overlapping, while replay-based methods can retain performance but drift due to imperfect replay. We study a hybrid perspective: trust region continual learning that combines generative replay with a Fisher-metric trust region constraint. We show that, under local approximations, the resulting update admits a MAML-style interpretation with a single implicit inner step: replay supplies an old-task gradient signal (query-like), while the Fisher-weighted penalty provides an efficient offline curvature shaping (support-like). This yields an emergent meta-learning property in continual learning: the model becomes an initialization that rapidly re-converges to prior task optima after each task transition, without explicitly optimizing a bilevel objective. Empirically, on task-incremental diffusion image generation and continual diffusion-policy control, trust region continual learning achieves the best final performance and retention, and consistently recovers early-task performance faster than EWC, replay, and continual meta-learning baselines.

In multi-goal Reinforcement Learning, an agent can share experience between related training tasks, resulting in better generalization for new tasks at test time. However, when the goal space has discontinuities and the reward is sparse, a majority of goals are difficult to reach. In this context, a curriculum over goals helps agents learn by adapting training tasks to their current capabilities. In this work we propose Stein Variational Goal Generation (SVGG), which samples goals of intermediate difficulty for the agent, by leveraging a learned predictive model of its goal reaching capabilities. The distribution of goals is modeled with particles that are attracted in areas of appropriate difficulty using Stein Variational Gradient Descent. We show that SVGG outperforms state-of-the-art multi-goal Reinforcement Learning methods in terms of success coverage in hard exploration problems, and demonstrate that it is endowed with a useful recovery property when the environment changes.

In-context learning (ICL) has shown impressive results in few-shot learning tasks, yet its underlying mechanism is still not fully understood. A recent line of work suggests that ICL performs gradient descent (GD)-based optimization implicitly. While appealing, much of the research focuses on simplified settings, where the parameters of a shallow model are optimized. In this work, we revisit evidence for ICL-GD correspondence on realistic NLP tasks and models. We find gaps in evaluation, both in terms of problematic metrics and insufficient baselines. We show that surprisingly, even untrained models achieve comparable ICL-GD similarity scores despite not exhibiting ICL. Next, we explore a major discrepancy in the flow of information throughout the model between ICL and GD, which we term Layer Causality. We propose a simple GD-based optimization procedure that respects layer causality, and show it improves similarity scores significantly.

Adapting one's thought process based on corrective feedback is an essential ability in human learning, particularly in collaborative settings. In contrast, the current large language model training paradigm relies heavily on modeling vast, static corpora. While effective for knowledge acquisition, it overlooks the interactive feedback loops essential for models to adapt dynamically to their context. In this work, we propose a framework that treats this interactive in-context learning ability not as an emergent property, but as a distinct, trainable skill. We introduce a scalable method that transforms single-turn verifiable tasks into multi-turn didactic interactions driven by information asymmetry. We first show that current flagship models struggle to integrate corrective feedback on hard reasoning tasks. We then demonstrate that models trained with our approach dramatically improve the ability to interactively learn from language feedback. More specifically, the multi-turn performance of a smaller model nearly reaches that of a model an order of magnitude larger. We also observe robust out-of-distribution generalization: interactive training on math problems transfers to diverse domains like coding, puzzles and maze navigation. Our qualitative analysis suggests that this improvement is due to an enhanced in-context plasticity. Finally, we show that this paradigm offers a unified path to self-improvement. By training the model to predict the teacher's critiques, effectively modeling the feedback environment, we convert this external signal into an internal capability, allowing the model to self-correct even without a teacher.

Neural network training is inherently sensitive to initialization and the randomness induced by stochastic gradient descent. However, it is unclear to what extent such effects lead to meaningfully different networks, either in terms of the models' weights or the underlying functions that were learned. In this work, we show that during the initial "chaotic" phase of training, even extremely small perturbations reliably causes otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time. We quantify this divergence through (i) L^2 distance between parameters, (ii) the loss barrier when interpolating between networks, (iii) L^2 and barrier between parameters after permutation alignment, and (iv) representational similarity between intermediate activations; revealing how perturbations across different hyperparameter or fine-tuning settings drive training trajectories toward distinct loss minima. Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.

In this work, we formulate a new multi-task active learning setting in which the learner's goal is to solve multiple matrix completion problems simultaneously. At each round, the learner can choose from which matrix it receives a sample from an entry drawn uniformly at random. Our main practical motivation is market segmentation, where the matrices represent different regions with different preferences of the customers. The challenge in this setting is that each of the matrices can be of a different size and also of a different rank which is unknown. We provide and analyze a new algorithm, MAlocate that is able to adapt to the unknown ranks of the different matrices. We then give a lower-bound showing that our strategy is minimax-optimal and demonstrate its performance with synthetic experiments.

Knowledge Distillation (KD) is a model-agnostic technique to improve model quality while having a fixed capacity budget. It is a commonly used technique for model compression, where a larger capacity teacher model with better quality is used to train a more compact student model with better inference efficiency. Through distillation, one hopes to benefit from student's compactness, without sacrificing too much on model quality. Despite the large success of knowledge distillation, better understanding of how it benefits student model's training dynamics remains under-explored. In this paper, we categorize teacher's knowledge into three hierarchical levels and study its effects on knowledge distillation: (1) knowledge of the `universe', where KD brings a regularization effect through label smoothing; (2) domain knowledge, where teacher injects class relationships prior to student's logit layer geometry; and (3) instance specific knowledge, where teacher rescales student model's per-instance gradients based on its measurement on the event difficulty. Using systematic analyses and extensive empirical studies on both synthetic and real-world datasets, we confirm that the aforementioned three factors play a major role in knowledge distillation. Furthermore, based on our findings, we diagnose some of the failure cases of applying KD from recent studies.

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.

Large language models (LLMs) have achieved great success, but their occasional content fabrication, or hallucination, limits their practical application. Hallucination arises because LLMs struggle to admit ignorance due to inadequate training on knowledge boundaries. We call it a limitation of LLMs that they can not accurately express their knowledge boundary, answering questions they know while admitting ignorance to questions they do not know. In this paper, we aim to teach LLMs to recognize and express their knowledge boundary, so they can reduce hallucinations caused by fabricating when they do not know. We propose CoKE, which first probes LLMs' knowledge boundary via internal confidence given a set of questions, and then leverages the probing results to elicit the expression of the knowledge boundary. Extensive experiments show CoKE helps LLMs express knowledge boundaries, answering known questions while declining unknown ones, significantly improving in-domain and out-of-domain performance.

We provide the first proof of learning rate transfer with width in a linear multi-layer perceptron (MLP) parametrized with muP, a neural network parameterization designed to ``maximize'' feature learning in the infinite-width limit. We show that under mu P, the optimal learning rate converges to a non-zero constant as width goes to infinity, providing a theoretical explanation to learning rate transfer. In contrast, we show that this property fails to hold under alternative parametrizations such as Standard Parametrization (SP) and Neural Tangent Parametrization (NTP). We provide intuitive proofs and support the theoretical findings with extensive empirical results.

In this work, we investigate the implicit regularization induced by teacher-student learning dynamics in self-distillation. To isolate its effect, we describe a simple experiment where we consider teachers at random initialization instead of trained teachers. Surprisingly, when distilling a student into such a random teacher, we observe that the resulting model and its representations already possess very interesting characteristics; (1) we observe a strong improvement of the distilled student over its teacher in terms of probing accuracy. (2) The learned representations are data-dependent and transferable between different tasks but deteriorate strongly if trained on random inputs. (3) The student checkpoint contains sparse subnetworks, so-called lottery tickets, and lies on the border of linear basins in the supervised loss landscape. These observations have interesting consequences for several important areas in machine learning: (1) Self-distillation can work solely based on the implicit regularization present in the gradient dynamics without relying on any dark knowledge, (2) self-supervised learning can learn features even in the absence of data augmentation and (3) training dynamics during the early phase of supervised training do not necessarily require label information. Finally, we shed light on an intriguing local property of the loss landscape: the process of feature learning is strongly amplified if the student is initialized closely to the teacher. These results raise interesting questions about the nature of the landscape that have remained unexplored so far. Code is available at https://github.com/safelix/dinopl.

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.

While pretraining on large-scale image-text data from the Web has facilitated rapid progress on many vision-and-language (V&L) tasks, recent work has demonstrated that pretrained models lack "fine-grained" understanding, such as the ability to recognise relationships, verbs, and numbers in images. This has resulted in an increased interest in the community to either develop new benchmarks or models for such capabilities. To better understand and quantify progress in this direction, we investigate four competitive V&L models on four fine-grained benchmarks. Through our analysis, we find that X-VLM (Zeng et al., 2022) consistently outperforms other baselines, and that modelling innovations can impact performance more than scaling Web data, which even degrades performance sometimes. Through a deeper investigation of X-VLM, we highlight the importance of both novel losses and rich data sources for learning fine-grained skills. Finally, we inspect training dynamics, and discover that for some tasks, performance peaks early in training or significantly fluctuates, never converging.