Large Language Models (LLMs) exhibit impressive performance on complex reasoning tasks, yet they frequently fail on basic numerical problems, producing incorrect outputs. Inspired by Benford's Law, a statistical pattern in which lower digits occur more frequently as leading digits, we hypothesize that the skewed digit distributions in web-collected corpora may be learned by LLMs during pretraining, leading to biased numerical generation. To investigate the hypothesis, we first examine whether digits frequencies in pretraining corpus (OLMo2) follows Benford's law. We then construct an evaluation benchmark in which the ground-truth digits are uniformly distributed within each of the seven numerical reasoning tasks. Our evaluation results demonstrate that leading open-source LLMs show a consistent pattern of digit bias that resembles Benford's law. Through logit-lens tracing and neuron-level dissection, we identify that this bias arises predominantly from a small subset of highly digit-selective feed-forward network (FFN) neurons in the deeper layers. Finally, we demonstrate that pruning these neurons mitigates imbalanced overgeneration and partially corrects erroneous outputs, providing causal evidence that fine-grained pretraining digit bias can propagate into model behavior. Our findings reveal a fundamental connection between corpus-level statistics and symbolic failure modes in LLMs, offering a new lens for diagnosing and mitigating hallucinations in numerical tasks.

Searching through chemical space is an exceptionally challenging problem because the number of possible molecules grows combinatorially with the number of atoms. Large, autoregressive models trained on databases of chemical compounds have yielded powerful generators, but we still lack robust strategies for generating molecules with desired properties. This molecular search problem closely resembles the "alignment" problem for large language models, though for many chemical tasks we have a specific and easily evaluable reward function. Here, we introduce an algorithm called energy rank alignment (ERA) that leverages an explicit reward function to produce a gradient-based objective that we use to optimize autoregressive policies. We show theoretically that this algorithm is closely related to proximal policy optimization (PPO) and direct preference optimization (DPO), but has a minimizer that converges to an ideal Gibbs-Boltzmann distribution with the reward playing the role of an energy function. Furthermore, this algorithm is highly scalable, does not require reinforcement learning, and performs well relative to DPO when the number of preference observations per pairing is small. We deploy this approach to align molecular transformers and protein language models to generate molecules and protein sequences, respectively, with externally specified properties and find that it does so robustly, searching through diverse parts of chemical space.

Large Language Models (LLMs) are increasingly adopted in high-stakes scenarios, yet their safety mechanisms often remain fragile. Simple jailbreak prompts or even benign fine-tuning can bypass internal safeguards, underscoring the need to understand the failure modes of current safety strategies. Recent findings suggest that vulnerabilities emerge when alignment is confined to only the initial output tokens. To address this, we introduce the notion of safety depth, a designated output position where the model refuses to generate harmful content. While deeper alignment appears promising, identifying the optimal safety depth remains an open and underexplored challenge.

Vision-Language Foundation Models (VLMs), trained on large-scale multimodal datasets, have driven significant advances in Artificial Intelligence (AI) by enabling rich cross-modal reasoning. Despite their success in general domains, applying these models to medical imaging remains challenging due to the limited availability of diverse imaging modalities and multilingual clinical data. Most existing medical VLMs are trained on a subset of imaging modalities and focus primarily on high-resource languages, thus limiting their generalizability and clinical utility. To address these limitations, we introduce a novel Vietnamese-language multimodal medical dataset consisting of 2,757whole-body PET/CT volumes from independent patients and their corresponding full-length clinical reports. This dataset is designed to fill two pressing gaps in medical AI development: (1) the lack of PET/CT imaging data in existing VLMs training corpora, which hinders the development of models capable of handling functional imaging tasks; and (2) the underrepresentation of low-resource languages, particularly the Vietnamese language, in medical vision-language research.

Recent advances in large reasoning models (LRMs) demonstrate that sophisticated behaviors such as multi-step reasoning and self-reflection can emerge via reinforcement learning with verifiable rewards (RLVR). However, existing RLVR approaches are inherently "on-policy", limiting learning to a model's own outputs and failing to acquire reasoning abilities beyond its initial capabilities. To address this issue, we introduce LUFFY (Learning to reason Under oFF-policY guidance), a framework that augments RLVR with off-policy reasoning traces. LUFFY dynamically balances imitation and exploration by combining off-policy demonstrations with on-policy rollouts during training. Specifically, LUFFY combines the MixedPolicy GRPO framework, which has a theoretically guaranteed convergence rate, alongside policy shaping via regularized importance sampling to avoid superficial and rigid imitation during mixed-policy training. Compared with previous RLVR methods, LUFFY achieves an over +6.4 average gain across six math benchmarks and an advantage of over +6.2 points in out-of-distribution tasks. Most significantly, we show that LUFFY successfully trains weak models in scenarios where on-policy RLVR completely fails. These results provide compelling evidence that LUFFY transcends the fundamental limitations of on-policy RLVR and demonstrates the great potential of utilizing off-policy guidance in RLVR.

Knowledge distillation from Large Language Models (LLMs) to locally hosted Small Language Models (SLMs) provides advantages for Data Analysis Code Generation (DACG) such as privacy protection. However, achieving effective distillation without resource-intensive training is challenging. This paper investigates whether LLMs can distill knowledge to SLMs through In-Context Learning (ICL), a training-free method for rapid task adaptation. We present the DARGO: Distillation and Adaptive Reasoning-Guided Orchestration framework, which facilitates automatic knowledge distillation from LLMs to SLMs. DARGO consists of three phases: exploration through an Model Orchestration Interface (MOI), Memory Collection of successful trajectories, and Knoweldge-driven Inference. We evaluate DARGO on three challenging DACG benchmarks (WIKITQ, TABMWP, and BIRD-SQL), each with in-domain training sets that enable detailed analysis of knowledge distillation effectiveness. DARGO demonstrates a substantial relative performance improvement of 27.5% on average for the student SLMs. To further observe generalization capabilities, we evaluate the DARGO across different teacher-student model combinations, knowledge transfer scenarios, and unified memory approaches for more advanced, test-only data analysis tasks. Our findings contribute a novel perspective on distillation methods that enhance performance for SLMs while avoiding intensive fine-tuning.

While large transformer models excel in predictive performance, their lack of interpretability restricts their usefulness in high-stakes domains. To remedy this, we propose the Generalized Induction-Head Model (GIM), an interpretable model for next-token prediction inspired by the observation of "induction heads" in LLMs. GIM is a retrieval-based module that identifies similar sequences in the input context by combining exact n-gram matching and fuzzy matching based on a neural similarity metric. We evaluate GIM in two settings: language modeling and fMRI response prediction. In language modeling, GIM improves next-token prediction by up to 25%p over interpretable baselines, significantly narrowing the gap with black-box LLMs. In an fMRI setting, GIM improves neural response prediction by 20% and offers insight into the language selectivity of the brain. GIM represents a significant step toward uniting interpretability and performance across domains.

Policy-based methods currently dominate reinforcement learning (RL) pipelines for large language model (LLM) reasoning, leaving value-based approaches largely unexplored. We revisit the classical paradigm of Bellman Residual Minimization and introduce Trajectory Bellman Residual Minimization (TBRM), an algorithm that naturally adapts this idea to LLMs, yielding a simple yet effective off-policy algorithm that optimizes a single trajectory-level Bellman objective using the model's own logits as Q-values. TBRM removes the need for critics, importancesampling ratios, or clipping, and can operate with only one rollout per prompt. We prove convergence to the near-optimal KL-regularized policy from arbitrary offpolicy data via an improved change-of-trajectory-measure analysis. Experiments on standard mathematical-reasoning benchmarks show that TBRM matches or surpasses policy-based baselines, like PPO and GRPO, with comparable or lower computational and memory overhead. Our results indicate that value-based RL might be a principled and efficient alternative for enhancing reasoning capabilities in LLMs.

Diffusion models transform random noise into images of remarkable fidelity, yet the structure of this noise-to-image map remains largely unexplored. We investigate this relationship using patch-wise Principal Component Analysis (PCA) and empirically demonstrate that low-frequency components of the initial noise predominantly influence the compositional structure of generated images. Our analyses reveal that noise seeds inherently contain universal compositional cues, evident when identical seeds produce images with similar structural attributes across different datasets and model architectures. Leveraging these insights, we develop and theoretically justify a simple yet effective Patch PCA denoiser that extracts underlying structure from noise using only generic natural image statistics. The robustness of these structural cues is observed to persist across both pixel-space models and latent diffusion models, highlighting their fundamental nature. Finally, we introduce a zero-shot editing method that enables injecting compositional control over generated images, providing an intuitive approach to guided generation without requiring model fine-tuning or additional training.

The right batch size is important when training language models at scale: a large batch size is necessary for fast training, but a batch size that is too large will harm token efficiency. To navigate this tradeoff, McCandlish et al. (2018) suggest that a critical batch size (CBS), below which training will not substantially degrade loss, can be estimated based on the gradient noise scale during training. While their method has been adopted in practice, e.g., when training GPT-3, strong assumptions are required to justify gradient noise as a proxy for the CBS, which makes it unclear whether their approach should be trusted in practice, limiting its applicability. In this paper, we introduce a simple, empirical approach to directly measure the CBS and show how the CBS evolves over training. Applying our approach to the OLMo models, we find that CBS is near 0 at initialization, increases rapidly at first, and then plateaus as training progresses. Furthermore, we find that this trend holds across different model sizes (1B and 7B), suggesting CBS from small training runs can inform larger-scale training runs. Our findings about how the CBS changes over training motivate batch size warmup as a natural way to reliably train language models at large batch size: start the batch size small and increase it as the CBS grows. To validate this claim, we use batch size warmup to train OLMo 1B to slightly better loss than the original training run with 43% fewer gradient steps. This shows how our framework can be applied to reliably train language models at larger batch sizes, increasing data parallelism without compromising performance.