The Potential of Second-Order Optimization for LLMs: A Study with Full Gauss-Newton

Recent efforts to accelerate LLM pretraining have focused on computationally-efficient approximations that exploit second-order structure. This raises a key question for large-scale training: how much performance is forfeited by these approximations? To probe this question, we establish a practical upper bound on iteration complexity by applying full Gauss-Newton (GN) preconditioning to transformer models of up to 150M parameters. Our experiments show that full GN updates yield substantial gains over existing optimizers, achieving a 5.4x reduction in training iterations compared to strong baselines like SOAP and Muon. Furthermore, we find that a precise layerwise GN preconditioner, which ignores cross-layer information, nearly matches the performance of the full GN method. Collectively, our results suggest: (1) the GN approximation is highly effective for preconditioning, implying higher-order loss terms may not be critical for convergence speed; (2) the layerwise Hessian structure contains sufficient information to achieve most of these potential gains; and (3) a significant performance gap exists between current approximate methods and an idealized layerwise oracle.

Fine-Tuning Masked Diffusion for Provable Self-Correction

A natural desideratum for generative models is self-correction--detecting and revising low-quality tokens at inference. While Masked Diffusion Models (MDMs) have emerged as a promising approach for generative modeling in discrete spaces, their capacity for self-correction remains poorly understood. Prior attempts to incorporate self-correction into MDMs either require overhauling MDM architectures/training or rely on imprecise proxies for token quality, limiting their applicability. Motivated by this, we introduce PRISM--Plug-in Remasking for Inference-time Self-correction of Masked Diffusions--a lightweight, model-agnostic approach that applies to any pretrained MDM. Theoretically, PRISM defines a self-correction loss that provably learns per-token quality scores, without RL or a verifier. These quality scores are computed in the same forward pass with MDM and used to detect low-quality tokens. Empirically, PRISM advances MDM inference across domains and scales: Sudoku; unconditional text (170M); and code with LLaDA (8B).

Selective Underfitting in Diffusion Models

Diffusion models have emerged as the principal paradigm for generative modeling across various domains. During training, they learn the score function, which in turn is used to generate samples at inference. They raise a basic yet unsolved question: which score do they actually learn? In principle, a diffusion model that matches the empirical score in the entire data space would simply reproduce the training data, failing to generate novel samples. Recent work addresses this question by arguing that diffusion models underfit the empirical score due to training-time inductive biases. In this work, we refine this perspective, introducing the notion of selective underfitting: instead of underfitting the score everywhere, better diffusion models more accurately approximate the score in certain regions of input space, while underfitting it in others. We characterize these regions and design empirical interventions to validate our perspective. Our results establish that selective underfitting is essential for understanding diffusion models, yielding new, testable insights into their generalization and generative performance.

Characterization and Mitigation of Training Instabilities in Microscaling Formats

Training large language models is an expensive, compute-bound process that must be repeated as models scale, algorithms improve, and new data is collected. To address this, next-generation hardware accelerators increasingly support lower-precision arithmetic formats, such as the Microscaling (MX) formats introduced in NVIDIA's Blackwell architecture. These formats use a shared scale within blocks of parameters to extend representable range and perform forward/backward GEMM operations in reduced precision for efficiency gains. In this work, we investigate the challenges and viability of block-scaled precision formats during model training. Across nearly one thousand language models trained from scratch -- spanning compute budgets from $2 \times 10^{17}$ to $4.8 \times 10^{19}$ FLOPs and sweeping over a broad range of weight-activation precision combinations -- we consistently observe that training in MX formats exhibits sharp, stochastic instabilities in the loss, particularly at larger compute scales. To explain this phenomenon, we conduct controlled experiments and ablations on a smaller proxy model that exhibits similar behavior as the language model, sweeping across architectural settings, hyperparameters, and precision formats. These experiments motivate a simple model in which multiplicative gradient bias introduced by the quantization of layer-norm affine parameters and a small fraction of activations can trigger runaway divergence. Through \emph{in situ} intervention experiments on our proxy model, we demonstrate that instabilities can be averted or delayed by modifying precision schemes mid-training. Guided by these findings, we evaluate stabilization strategies in the LLM setting and show that certain hybrid configurations recover performance competitive with full-precision training. We release our code at https://github.com/Hither1/systems-scaling.

Inside you are many wolves: Using cognitive models to interpret value trade-offs in LLMs

Navigating everyday social situations often requires juggling conflicting goals, such as conveying a harsh truth, maintaining trust, all while still being mindful of another person's feelings. These value trade-offs are an integral part of human decision-making and language use, however, current tools for interpreting such dynamic and multi-faceted notions of values in LLMs are limited. In cognitive science, so-called "cognitive models" provide formal accounts of these trade-offs in humans, by modeling the weighting of a speaker's competing utility functions in choosing an action or utterance. In this work, we use a leading cognitive model of polite speech to interpret the extent to which LLMs represent human-like trade-offs. We apply this lens to systematically evaluate value trade-offs in two encompassing model settings: degrees of reasoning "effort" in frontier black-box models, and RL post-training dynamics of open-source models. Our results highlight patterns of higher informational utility than social utility in reasoning models, and in open-source models shown to be stronger in mathematical reasoning. Our findings from LLMs' training dynamics suggest large shifts in utility values early on in training with persistent effects of the choice of base model and pretraining data, compared to feedback dataset or alignment method. We show that our method is responsive to diverse aspects of the rapidly evolving LLM landscape, with insights for forming hypotheses about other high-level behaviors, shaping training regimes for reasoning models, and better controlling trade-offs between values during model training.

A Simplified Analysis of SGD for Linear Regression with Weight Averaging

Theoretically understanding stochastic gradient descent (SGD) in overparameterized models has led to the development of several optimization algorithms that are widely used in practice today. Recent work by~\citet{zou2021benign} provides sharp rates for SGD optimization in linear regression using constant learning rate, both with and without tail iterate averaging, based on a bias-variance decomposition of the risk. In our work, we provide a simplified analysis recovering the same bias and variance bounds provided in~\citep{zou2021benign} based on simple linear algebra tools, bypassing the requirement to manipulate operators on positive semi-definite (PSD) matrices. We believe our work makes the analysis of SGD on linear regression very accessible and will be helpful in further analyzing mini-batching and learning rate scheduling, leading to improvements in the training of realistic models.

Discovering Hierarchical Latent Capabilities of Language Models via Causal Representation Learning

Faithful evaluation of language model capabilities is crucial for deriving actionable insights that can inform model development. However, rigorous causal evaluations in this domain face significant methodological challenges, including complex confounding effects and prohibitive computational costs associated with extensive retraining. To tackle these challenges, we propose a causal representation learning framework wherein observed benchmark performance is modeled as a linear transformation of a few latent capability factors. Crucially, these latent factors are identified as causally interrelated after appropriately controlling for the base model as a common confounder. Applying this approach to a comprehensive dataset encompassing over 1500 models evaluated across six benchmarks from the Open LLM Leaderboard, we identify a concise three-node linear causal structure that reliably explains the observed performance variations. Further interpretation of this causal structure provides substantial scientific insights beyond simple numerical rankings: specifically, we reveal a clear causal direction starting from general problem-solving capabilities, advancing through instruction-following proficiency, and culminating in mathematical reasoning ability. Our results underscore the essential role of carefully controlling base model variations during evaluation, a step critical to accurately uncovering the underlying causal relationships among latent model capabilities.

Interpreting the Linear Structure of Vision-language Model Embedding Spaces

Vision-language models encode images and text in a joint space, minimizing the distance between corresponding image and text pairs. How are language and images organized in this joint space, and how do the models encode meaning and modality? To investigate this, we train and release sparse autoencoders (SAEs) on the embedding spaces of four vision-language models (CLIP, SigLIP, SigLIP2, and AIMv2). SAEs approximate model embeddings as sparse linear combinations of learned directions, or "concepts". We find that, compared to other methods of linear feature learning, SAEs are better at reconstructing the real embeddings, while also able to retain the most sparsity. Retraining SAEs with different seeds or different data diet leads to two findings: the rare, specific concepts captured by the SAEs are liable to change drastically, but we also show that the key commonly-activating concepts extracted by SAEs are remarkably stable across runs. Interestingly, while most concepts are strongly unimodal in activation, we find they are not merely encoding modality per se. Many lie close to - but not entirely within - the subspace defining modality, suggesting that they encode cross-modal semantics despite their unimodal usage. To quantify this bridging behavior, we introduce the Bridge Score, a metric that identifies concept pairs which are both co-activated across aligned image-text inputs and geometrically aligned in the shared space. This reveals that even unimodal concepts can collaborate to support cross-modal integration. We release interactive demos of the SAEs for all models, allowing researchers to explore the organization of the concept spaces. Overall, our findings uncover a sparse linear structure within VLM embedding spaces that is shaped by modality, yet stitched together through latent bridges-offering new insight into how multimodal meaning is constructed.

Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining

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

Data-Efficient Multi-Agent Spatial Planning with LLMs

In this project, our goal is to determine how to leverage the world-knowledge of pretrained large language models for efficient and robust learning in multiagent decision making. We examine this in a taxi routing and assignment problem where agents must decide how to best pick up passengers in order to minimize overall waiting time. While this problem is situated on a graphical road network, we show that with the proper prompting zero-shot performance is quite strong on this task. Furthermore, with limited fine-tuning along with the one-at-a-time rollout algorithm for look ahead, LLMs can out-compete existing approaches with 50 times fewer environmental interactions. We also explore the benefits of various linguistic prompting approaches and show that including certain easy-to-compute information in the prompt significantly improves performance. Finally, we highlight the LLM's built-in semantic understanding, showing its ability to adapt to environmental factors through simple prompts.