Directional Alignment Mitigates Reward Hacking in Reinforcement Learning for Language Models

Reward hacking arises when a model improves a proxy reward by exploiting shortcuts rather than solving the intended task. We study this failure mode through the geometry of reinforcement learning updates in language models and argue that hacking emerges when optimization drifts away from a stable low-dimensional learning trajectory. We analyze this drift through dominant singular directions of parameter updates and show that reward-hacking runs exhibit substantially larger directional change than clean runs. Motivated by this observation, we introduce trusted-direction projection, which constrains gradients to remain within a clean reference subspace. Across reward-hacking experiments on mathematical reasoning, the proposed approach delays shortcut exploitation and better preserves task performance.

The Implicit Bias of Depth: From Neural Collapse to Softmax Codes

Neural collapse (NC) describes the structured geometry that emerges in the features and weights of trained classifiers. Recent theory suggests NC can be suboptimal in deep architectures, attributing this to an explicit low-rank bias from L2 regularization. We study the deep unconstrained feature model (UFM)-equivalent to a deep linear network with orthogonal inputs-trained without regularization, to isolate how gradient descent and depth alone shape NC. We show that depth induces an implicit low-rank bias: low-rank matrices propagate norm more efficiently through successive multiplications, promoting low-rank alternatives to NC. These alternatives, we argue, correspond to softmax codes: max-margin solutions previously found in width-bottlenecked networks. Analyzing training dynamics under spectral initialization, we identify an early-time repulsion among singular values that drives low-rank emergence, and characterize how depth shrinks NC's basin of attraction. Finally, we show that some effects act in the opposite direction: for randomly initialized networks, increasing width biases training toward higher-rank solutions. Our results provide the first asymptotic and dynamic characterization of implicit bias in deep UFMs trained with unregularized multiclass cross-entropy.

High-Dimensional Statistics: Reflections on Progress and Open Problems

Over the past two decades, the field of high-dimensional statistics has experienced substantial progress, driven largely by technological advances that have dramatically reduced the cost and effort for data collection and storage across a broad range of domains, including biology, medicine, astronomy, and the social and environmental sciences. Modern datasets are increasingly complex, often exhibiting rich dependency, heterogeneity, and other features that challenge traditional statistical methods. In response, high-dimensional statistics has evolved to address more sophisticated estimation and inference problems. This evolution has, in turn, fostered deep connections with and contributions to a wide range of research areas, including optimization, concentration of measure, random matrix theory, information theory, and theoretical computer science. Given the rapid pace of recent developments in high-dimensional statistics, our goal is to synthesize representative advances, highlight common themes and open problems, and point to important works that offer entry points into the field.

Understanding Contextual Recall in Transformers: How Finetuning Enables In-Context Reasoning over Pretraining Knowledge

Transformer-based language models excel at in-context learning (ICL), where they can adapt to new tasks based on contextual examples, without parameter updates. In a specific form of ICL, which we refer to as \textit{contextual recall}, models pretrained on open-ended text leverage pairwise examples to recall specific facts in novel prompt formats. We investigate whether contextual recall emerges from pretraining alone, what finetuning is required, and what mechanisms drive the necessary representations. For this, we introduce a controlled synthetic framework where pretraining sequences consist of subject-grammar-attribute tuples, with attribute types tied to grammar statistics. We demonstrate that while such pretraining successfully yields factual knowledge, it is insufficient for contextual recall: models fail to implicitly infer attribute types when the grammar statistics are removed in ICL prompts. However, we show that finetuning on tasks requiring implicit inference, distinct from the ICL evaluation, using a subset of subjects, triggers the emergence of contextual recall across all subjects. This transition is accompanied by the formation of low-dimensional latent encodings of the shared attribute type. For mechanistic insight, we derive a construction for an attention-only transformer that replicates the transition from factual to contextual recall, corroborated by empirical validation.

Why Loss Re-weighting Works If You Stop Early: Training Dynamics of Unconstrained Features

The application of loss reweighting in modern deep learning presents a nuanced picture. While it fails to alter the terminal learning phase in overparameterized deep neural networks (DNNs) trained on high-dimensional datasets, empirical evidence consistently shows it offers significant benefits early in training. To transparently demonstrate and analyze this phenomenon, we introduce a small-scale model (SSM). This model is specifically designed to abstract the inherent complexities of both the DNN architecture and the input data, while maintaining key information about the structure of imbalance within its spectral components. On the one hand, the SSM reveals how vanilla empirical risk minimization preferentially learns to distinguish majority classes over minorities early in training, consequently delaying minority learning. In stark contrast, reweighting restores balanced learning dynamics, enabling the simultaneous learning of features associated with both majorities and minorities.

Short-Context Dominance: How Much Local Context Natural Language Actually Needs?

We investigate the short-context dominance hypothesis: that for most sequences, a small local prefix suffices to predict their next tokens. Using large language models as statistical oracles, we measure the minimum context length (MCL) needed to reproduce accurate full-context predictions across datasets with sequences of varying lengths. For sequences with 1-7k tokens from long-context documents, we consistently find that 75-80% require only the last 96 tokens at most. Given the dominance of short-context tokens, we then ask whether it is possible to detect challenging long-context sequences for which a short local prefix does not suffice for prediction. We introduce a practical proxy to MCL, called Distributionally Aware MCL (DaMCL), that does not require knowledge of the actual next-token and is compatible with sampling strategies beyond greedy decoding. Our experiments validate that simple thresholding of the metric defining DaMCL achieves high performance in detecting long vs. short context sequences. Finally, to counter the bias that short-context dominance induces in LLM output distributions, we develop an intuitive decoding algorithm that leverages our detector to identify and boost tokens that are long-range-relevant. Across Q&A tasks and model architectures, we confirm that mitigating the bias improves performance.

Scaling Generative Verifiers For Natural Language Mathematical Proof Verification And Selection

Large language models have achieved remarkable success on final-answer mathematical problems, largely due to the ease of applying reinforcement learning with verifiable rewards. However, the reasoning underlying these solutions is often flawed. Advancing to rigorous proof-based mathematics requires reliable proof verification capabilities. We begin by analyzing multiple evaluation setups and show that focusing on a single benchmark can lead to brittle or misleading conclusions. To address this, we evaluate both proof-based and final-answer reasoning to obtain a more reliable measure of model performance. We then scale two major generative verification methods (GenSelect and LLM-as-a-Judge) to millions of tokens and identify their combination as the most effective framework for solution verification and selection. We further show that the choice of prompt for LLM-as-a-Judge significantly affects the model's performance, but reinforcement learning can reduce this sensitivity. However, despite improving proof-level metrics, reinforcement learning does not enhance final-answer precision, indicating that current models often reward stylistic or procedural correctness rather than mathematical validity. Our results establish practical guidelines for designing and evaluating scalable proof-verification and selection systems.

Advantage Shaping as Surrogate Reward Maximization: Unifying Pass@K Policy Gradients

This note reconciles two seemingly distinct approaches to policy gradient optimization for the Pass@K objective in reinforcement learning with verifiable rewards: (1) direct REINFORCE-style methods, and (2) advantage-shaping techniques that directly modify GRPO. We show that these are two sides of the same coin. By reverse-engineering existing advantage-shaping algorithms, we reveal that they implicitly optimize surrogate rewards. We specifically interpret practical ``hard-example up-weighting'' modifications to GRPO as reward-level regularization. Conversely, starting from surrogate reward objectives, we provide a simple recipe for deriving both existing and new advantage-shaping methods. This perspective provides a lens for RLVR policy gradient optimization beyond our original motivation of Pass@K.

How Muon's Spectral Design Benefits Generalization: A Study on Imbalanced Data

The growing adoption of spectrum-aware matrix-valued optimizers such as Muon and Shampoo in deep learning motivates a systematic study of their generalization properties and, in particular, when they might outperform competitive algorithms. We approach this question by introducing appropriate simplifying abstractions as follows: First, we use imbalanced data as a testbed. Second, we study the canonical form of such optimizers, which is Spectral Gradient Descent (SpecGD) -- each update step is $UV^T$ where $U\Sigma V^T$ is the truncated SVD of the gradient. Third, within this framework we identify a canonical setting for which we precisely quantify when SpecGD outperforms vanilla Euclidean GD. For a Gaussian mixture data model and both linear and bilinear models, we show that unlike GD, which prioritizes learning dominant principal components of the data first, SpecGD learns all principal components of the data at equal rates. We demonstrate how this translates to a growing gap in balanced accuracy favoring SpecGD early in training and further show that the gap remains consistent even when the GD counterpart uses adaptive step-sizes via normalization. By extending the analysis to deep linear models, we show that depth amplifies these effects. We empirically verify our theoretical findings on a variety of imbalanced datasets. Our experiments compare practical variants of spectral methods, like Muon and Shampoo, against their Euclidean counterparts and Adam. The results validate our findings that these spectral optimizers achieve superior generalization by promoting a more balanced learning of the data's underlying components.

Efficient Forward-Only Data Valuation for Pretrained LLMs and VLMs

Quantifying the influence of individual training samples is essential for enhancing the transparency and accountability of large language models (LLMs) and vision-language models (VLMs). However, existing data valuation methods often rely on Hessian information or model retraining, making them computationally prohibitive for billion-parameter models. In this work, we introduce For-Value, a forward-only data valuation framework that enables scalable and efficient influence estimation for both LLMs and VLMs. By leveraging the rich representations of modern foundation models, For-Value computes influence scores using a simple closed-form expression based solely on a single forward pass, thereby eliminating the need for costly gradient computations. Our theoretical analysis demonstrates that For-Value accurately estimates per-sample influence by capturing alignment in hidden representations and prediction errors between training and validation samples. Extensive experiments show that For-Value matches or outperforms gradient-based baselines in identifying impactful fine-tuning examples and effectively detecting mislabeled data.