We present AutoformBot, a multi-agent system for building an Autoformalized Textbook Library At Scale (Atlas) in Lean 4. AutoformBot orchestrates thousands of LLM agents, equipped with formal verification tools, dependency-aware task scheduling, and collaborative version control, to translate informal textbook prose into machine-checked definitions and proofs. We apply our methods to a corpus of 26 open-access textbooks spanning analysis, algebra, topology, combinatorics, and probability, producing Atlas: a verified library of over 45,000 Lean 4 declarations and 500 thousand lines of code. We release two artifacts: (i) AutoformBot, the open-source multi-agent framework; and (ii) Atlas, the resulting formal library. Our results suggest that autoformalizing the core content of graduate-level mathematics at scale is now economically and technically feasible. This opens the door to the automated verification of both human- and machine-generated mathematics at a research level.

We present a formalization in Lean of the core interior De Giorgi–Nash–Moser theory for uniformly elliptic divergence-form equations with bounded measurable coefficients. The formalized results include local boundedness of weak subsolutions, the weak Harnack inequality for positive weak supersolutions, Moser's Harnack inequality for positive weak solutions, and interior Hölder regularity. This is, to our knowledge, the first machine-checked formalization of a major theorem in modern PDE theory. The development also required substantial new infrastructure for Sobolev spaces on bounded domains, weak solutions of elliptic equations, and quantitative regularity estimates. More broadly, it suggests that large-scale autoformalization of hard analysis in Lean is now within reach.

Fine-tuning large language models (LLMs) on reasoning benchmarks via reinforcement learning requires a specific reward function, often binary, for each benchmark. This comes with two potential limitations: the need to design the reward, and the potentially sparse nature of binary rewards. Here, we systematically investigate rewards derived from the probability or log-probability of emitting the reference answer (or any other prompt continuation present in the data), which have the advantage of not relying on specific verifiers and being available at scale. Several recent works have advocated for the use of similar rewards (e.g., VeriFree, JEPO, RLPR, NOVER). We systematically compare variants of likelihood-based rewards with standard baselines, testing performance both on standard mathematical reasoning benchmarks, and on long-form answers where no external verifier is available. We find that using the log-probability of the reference answer as the reward for chain-of-thought (CoT) learning is the only option that performs well in all setups. This reward is also consistent with the next-token log-likelihood loss used during pretraining. In verifiable settings, log-probability rewards bring comparable or better success rates than reinforcing with standard binary rewards, and yield much better perplexity. In non-verifiable settings, they perform on par with SFT. On the other hand, methods based on probability, such as VeriFree, flatline on non-verifiable settings due to vanishing probabilities of getting the correct answer. Overall, this establishes log-probability rewards as a viable method for CoT fine-tuning, bridging the short, verifiable and long, non-verifiable answer settings.

The use of continuous instead of discrete tokens during the Chain-of-Thought (CoT) phase of reasoning LLMs has garnered attention recently, based on the intuition that a continuous mixture of discrete tokens could simulate a superposition of several reasoning paths simultaneously. Theoretical results have formally proven that continuous tokens have much greater expressivity and can solve specific problems more efficiently. However, practical use of continuous tokens has been limited by strong training difficulties: previous works either just use continuous tokens at inference time on a pre-trained discrete-token model, or must distill the continuous CoT from ground-truth discrete CoTs and face computational costs that limit the CoT to very few tokens. This is the first work introducing a scalable method to learn continuous CoTs via reinforcement learning (RL), without distilling from reference discrete CoTs. We use "soft" tokens: mixtures of tokens together with noise on the input embedding to provide RL exploration. Computational overhead is minimal, enabling us to learn continuous CoTs with hundreds of tokens. On math reasoning benchmarks with Llama and Qwen models up to 8B, training with continuous CoTs match discrete-token CoTs for pass@1 and surpass them for pass@32, showing greater CoT diversity. In systematic comparisons, the best-performing scenario is to train with continuous CoT tokens then use discrete tokens for inference, meaning the "soft" models can be deployed in a standard way. Finally, we show continuous CoT RL training better preserves the predictions of the base model on out-of-domain tasks, thus providing a softer touch to the base model.

To deploy large language models (LLMs) in high-stakes application domains that require substantively accurate responses to open-ended prompts, we need reliable, computationally inexpensive methods that assess the trustworthiness of long-form responses generated by LLMs. However, existing approaches often rely on claim-by-claim fact-checking, which is computationally expensive and brittle in long-form responses to open-ended prompts. In this work, we introduce semantic isotropy—the degree of uniformity across normalized text embeddings on the unit sphere—and use it to assess the trustworthiness of long-form responses generated by LLMs. To do so, we generate several long-form responses, embed them, and estimate the level of semantic isotropy of these responses as the angular dispersion of the embeddings on the unit sphere. We find that higher semantic isotropy—that is, greater embedding dispersion—reliably signals lower factual consistency across samples. Our approach requires no labeled data, no fine-tuning, and no hyperparameter selection, and can be used with open- or closed-weight embedding models. Across multiple domains, our method consistently outperforms existing approaches in predicting nonfactuality in long-form responses using only a handful of samples—offering a practical, low-cost approach for integrating trust assessment into real-world LLM workflows.

While Experience Replay - the practice of storing rollouts and reusing them multiple times during training - is a foundational technique in general RL, it remains largely unexplored in LLM post-training due to the prevailing belief that fresh, on-policy data is essential for high performance. In this work, we challenge this assumption. We present a systematic study of replay buffers for LLM post-training, formalizing the optimal design as a trade-off between staleness-induced variance, sample diversity and the high computational cost of generation. We show that strict on-policy sampling is suboptimal when generation is expensive. Empirically, we show that a well-designed replay buffer can drastically reduce inference compute without degrading - and in some cases even improving - final model performance, while preserving policy entropy.

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.

This workshop explored how modern machine learning can both accelerate mathematical discovery and preserve rigorous standards. It focused on three angles: using AI techniques to help mathematicians make advances on challenging problems; using mathematics to understand AI predictions; and using deep-learning models for automated theorem proving. Key discussions included using machine learning as a tool for constructing interesting mathematical constructions and navigating in mathematical search spaces, to uncover conjectures and high-quality examples (e.g., sphere packings via DiffuseBoost, combinatorial objects via AlphaEvolve); Integrating Large Language Models (LLMs) with formal systems (e.g., Lean/mathlib) to create scalable, certifiable AI-based automated theorem prover; Collaborative formalization (e.g., the Carleson theorem project), autoformalization for high-quality supervised data, and reinforcement learning/search methods for proof generation and algorithmic reasoning.

Transformers have achieved state-of-the-art performance across diverse language and vision tasks. This success drives the imperative to interpret their internal mechanisms with the dual goals of enhancing performance and improving behavioral control. Attribution methods help advance interpretability by assigning model outputs associated with a target concept to specific model components. Current attribution research primarily studies multi-layer perceptron (MLP) neurons and addresses relatively simple concepts such as factual associations (e.g., Paris is located in France). This focus tends to overlook the impact of the attention mechanism and lacks a unified approach for analyzing more complex concepts. To fill these gaps, we introduce Scalable Attention Module Discovery (SAMD), a concept-agnostic method for mapping arbitrary, complex concepts to specific attention heads of general transformer models. We accomplish this by representing each concept as a vector, calculating its cosine similarity with each attention head, and selecting the TopK-scoring heads to construct the concept-associated attention module. We then propose Scalar Attention Module Intervention (SAMI), a simple strategy to diminish or amplify the effects of a concept by adjusting the attention module using only a single scalar parameter. Empirically, we demonstrate SAMD on concepts of varying complexity, and visualize the locations of their corresponding modules. Our results demonstrate that module locations remain stable before and after LLM post-training, and confirm prior work on the mechanics of LLM multi-lingualism. Through SAMI, we facilitate jailbreaking on HarmBench (+72.7%) by diminishing “safety” and improve performance on the GSM8K benchmark (+1.6%) by amplifying “reasoning”. Lastly, we highlight the domain-agnostic nature of our approach by suppressing the image classification accuracy of vision transformers on ImageNet.

We study internalization processes, by which neural-network-based systems absorb an explicit computational procedure into their own weights, and how they facilitate learning. We investigate how transformers internalize the simulation of semiautomata by internalizing chain-of-thought (CoT) tokens, which classes of semiautomata are harder to internalize, and expose the flip side of internalization, that is, a progressive degradation of out-of-distribution performance. We then provide the first provable analysis of successful internalization: for the task of learning parities, we show that a simplified one-layer transformer provably first learns the target with explicit CoT supervision and then internalizes the autoregressive generation as CoT tokens are progressively removed, learning to directly compute the parity. This task is computationally hard to learn from data without CoT supervision. Finally, we discuss how learning through internalization relates to the Positive Distribution Shift phenomenon recently introduced by .

Large reasoning models (LRMs) spend substantial test-time compute on long chain-of-thought (CoT) traces, but what an effective CoT remains unclear. While prior work reports gains from lengthening CoTs and increasing review via appended tokens, recent studies suggest that shorter thinking can outperform longer traces. We therefore conduct a systematic evaluation across ten LRMs on math and scientific reasoning. Contrary to the “longer-is-better” narrative, we find that both naively using longer CoTs and more review behaviors are associated with accuracy. As CoT unfolds step by step, token-level metrics can conflate verbosity with process quality. We introduce a graph view of CoT to extract structure and identify a single statistic—the (FSF), the fraction of steps in abandoned branches—that consistently outpredicts length and review ratio for correctness across models. To probe causality, we design two interventions. First, we rank candidate CoTs by each metric at test time, where FSF yields the largest pass@1 gains; second, we edit CoTs to remove failed branches, which significantly improves accuracy, indicating that failed branches bias subsequent reasoning. Taken together, these results characterize effective CoTs as those that and support test-time scaling over indiscriminately generating long CoTs.

RL 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? We explore this with SOAR: A self-improvement framework designed to surface these pedagogical signals through meta-RL. A teacher model proposes synthetic problems for a student model, and is rewarded with its improvement on a subset of hard problems, thus grounding the curriculum in real student progress rather than proxy rewards. Our study on the hardest subsets of math benchmarks (0/128 success) reveal three core findings. First, 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 problems. Second, grounded rewards outperform intrinsic rewards used in prior LLM self-play, reliably avoiding the typical instability and diversity collapse modes. Third, the structure and well-posedness of questions 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 solve the hard problems, paving a principled path to escape reasoning plateaus without additional curated data.

Recent advances in reasoning domains with neural networks have primarily been enabled by a training recipe that optimizes Large Language Models, previously trained to predict the next-token in a sequence, with reinforcement learning algorithms. We introduce a framework to study the success of this paradigm, and we theoretically expose the optimization mechanisms by which reinforcement learning improves over next-token prediction in this setting. We study learning from mixture distributions of short and long ``chain-of-thought'' sequences encoding a single task. In particular, when the task consists of predicting the parity of $d$ bits and long sequences are rare, we show how reinforcement learning after next-token prediction enables autoregressive transformers to generalize, whereas mere next-token prediction requires extreme statistical or computational resources to do so. We further explain how reinforcement learning leverages increased test-time computation, manifested in longer responses, to facilitate this learning process. In a simplified setting, we theoretically prove that autoregressive linear models following this training recipe can efficiently learn to predict the parity of $d$ bits as long as the proportion of long demonstrations in the data mix is not exponentially small in the input dimension $d$. Finally, we demonstrate these same phenomena in other settings, including the post-training of Llama-series models on mixture variations of common mathematical reasoning benchmarks.

Reliability is key to realizing the promise of autonomous UI-agents, multimodal agents that directly interact with the apps humans use, as users must be able to trust an agent to complete a given task. Current evaluations rely on fixed environments---often clones of existing apps--- which are limited in that they can only shed light on whether or how often an agent can complete a task within a specific environment. When deployed however, agents are likely to encounter variations in app design and content that can affect an agent’s ability to complete a task. To address this blind spot of measuring agent reliability across app variations, we develop OpenApps, a light-weight open-source ecosystem with six apps (messenger, calendar, maps, etc.) that are configurable in appearance and content. OpenApps requires just a single CPU to run, enabling easy generation and deployment of thousands of versions of each app. Specifically, we run more than 10,000 independent evaluations to study reliability across seven leading multimodal agents. We find that while standard reliability within a fixed app is relatively stable, reliability can vary drastically when measured across app variations. Task success rates for many agents can fluctuate by more than 50\% across app variations. For example, Kimi-VL-3B's average success across all tasks fluctuates from 63\% to just 4\% across app versions. We also find agent behaviors such as looping or hallucinating actions can differ drastically depending on the environment configuration. These initial findings highlight the importance of measuring reliability along this new dimension of app variations.

Large Language Models (LLM) are increasingly trained on data generated by other LLMs, either because generated text and images become part of the pre-training corpus, or because synthetized data is used as a replacement for expensive human-annotation. This raises concerns about *model collapse*, a drop in model performance when their training sets include generated data. Considering that it is easier for both humans and machines to tell between good and bad examples than to generate high-quality samples, we investigate the use of verification on synthesized data to prevent model collapse. We provide a theoretical characterization using Gaussian mixtures, linear classifiers, and linear verifiers to derive conditions with measurable proxies to assess whether the verifier can effectively select synthesized data that leads to optimal performance. We experiment with two practical tasks -- computing matrix eigenvalues with transformers and news summarization with LLMs -- which both exhibit model collapse when trained on generated data, and show that verifiers, even imperfect ones, can indeed be harnessed to prevent model collapse and that our proposed proxy measure strongly correlates with performance.

In the era of exceptionally data-hungry models, careful selection of the training data is essential to mitigate the extensive costs of deep learning. Data pruning offers a solution by removing redundant or uninformative samples from the dataset, which yields faster convergence and improved neural scaling laws. However, little is known about its impact on classification bias of the trained models. We conduct the first systematic study of this effect and reveal that existing data pruning algorithms can produce highly biased classifiers. We present theoretical analysis of the classification risk in a mixture of Gaussians to argue that choosing appropriate class pruning ratios, coupled with random pruning within classes has potential to improve worst-class performance. We thus propose DRoP, a distributionally robust approach to pruning and empirically demonstrate its performance on standard computer vision benchmarks. In sharp contrast to existing algorithms, our proposed method continues improving distributional robustness at a tolerable drop of average performance as we prune more from the datasets.

Reinforcement learning with verifiable rewards (RLVR) has become a standard recipe for improving large language models (LLMs) on reasoning tasks, with Group Relative Policy Optimization (GRPO) widely used in practice. Yet GRPO wastes substantial compute on negative groups: groups in which no sampled response is correct yield zero advantage and thus no gradient. We ask whether negative groups can be leveraged without extra supervision. Starting from a maximum-likelihood (MLE) objective in reward modeling, we show that the MLE gradient is equivalent to a policy gradient for a modified value function. This value function adds a confidence-weighted penalty on incorrect responses, imposing larger penalties on more confident mistakes. We refer to this as Likelihood Estimation with Negative Samples (LENS). LENS modifies GRPO to assign non-zero, confidence-dependent rewards to incorrect generations, making negative groups informative and converting previously wasted samples into useful gradient updates. On the MATH benchmark with Llama-3.1-8B and Qwen-2.5-3B, the proposed variant consistently outperforms GRPO baseline, with significant gains on harder items. These results demonstrate a principled and practical way to "rescue" negative groups, improving efficiency and performance in RLVR.

Reinforcement learning (RL) is increasingly used to align large language models (LLMs). Off-policy methods offer greater implementation simplicity and data efficiency than on-policy techniques, but often result in suboptimal performance. In this work, we study the intermediate range of algorithms between off-policy RL and supervised fine-tuning by analyzing a simple off-policy REINFORCE algorithm, where the advantage is defined as $A=r-V$, with $r$ a reward and $V$ some tunable baseline. Intuitively, lowering $V$ emphasizes high-reward samples, while raising it penalizes low-reward ones more heavily. We first provide a theoretical analysis of this off-policy REINFORCE algorithm, showing that when the baseline $V$ lower-bounds the expected reward, the algorithm enjoys a policy improvement guarantee. Our analysis reveals that while on-policy updates can safely leverage both positive and negative signals, off-policy updates benefit from focusing more on positive rewards than on negative ones. We validate our findings experimentally in a controlled stochastic bandit setting and through fine-tuning state-of-the-art LLMs on reasoning tasks.

We study the implicit bias of the general family of steepest descent algorithms with infinitesimal learning rate in deep homogeneous neural networks. We show that: (a) an algorithm-dependent geometric margin starts increasing once the networks reach perfect training accuracy, and (b) any limit point of the training trajectory corresponds to a KKT point of the corresponding margin-maximization problem. We experimentally zoom into the trajectories of neural networks optimized with various steepest descent algorithms, highlighting connections to the implicit bias of popular adaptive methods (Adam and Shampoo).

Within the scaling laws paradigm, which underpins the training of large neural networks like ChatGPT and Llama, we consider a supervised regression setting and establish a strong form of the model collapse phenomenon, a critical performance degradation due to synthetic data in the training corpus. Our results show that even the smallest fraction of synthetic data (e.g., as little as 1 per 1000) can still lead to model collapse: larger and larger training sets do not enhance performance. We further investigate whether increasing model size, an approach aligned with current trends in training large language models, exacerbates or mitigates model collapse. In a simplified regime where neural networks are approximated via random projections of tunable size, we both theoretically and empirically show that larger models can amplify model collapse. Interestingly, our theory also indicates that, beyond the interpolation threshold (which can be extremely high for very large datasets), larger models may mitigate the collapse, although they do not entirely prevent it. Our theoretical findings are empirically verified through experiments on language models and neural networks for images.