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.

Distributional Scaling Laws for Emergent Capabilities

In this paper, we explore the nature of sudden breakthroughs in language model performance at scale, which stands in contrast to smooth improvements governed by scaling laws. While advocates of "emergence" view abrupt performance gains as capabilities unlocking at specific scales, others have suggested that they are produced by thresholding effects and alleviated by continuous metrics. We propose that breakthroughs are instead driven by continuous changes in the probability distribution of training outcomes, particularly when performance is bimodally distributed across random seeds. In synthetic length generalization tasks, we show that different random seeds can produce either highly linear or emergent scaling trends. We reveal that sharp breakthroughs in metrics are produced by underlying continuous changes in their distribution across seeds. Furthermore, we provide a case study of inverse scaling and show that even as the probability of a successful run declines, the average performance of a successful run continues to increase monotonically. We validate our distributional scaling framework on realistic settings by measuring MMLU performance in LLM populations. These insights emphasize the role of random variation in the effect of scale on LLM capabilities.

Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions

In recent years, masked diffusion models (MDMs) have emerged as a promising alternative approach for generative modeling over discrete domains. Compared to autoregressive models (ARMs), MDMs trade off complexity at training time with flexibility at inference time. At training time, they must learn to solve an exponentially large number of infilling problems, but at inference time, they can decode tokens in essentially arbitrary order. In this work, we closely examine these two competing effects. On the training front, we theoretically and empirically demonstrate that MDMs indeed train on computationally intractable subproblems compared to their autoregressive counterparts. On the inference front, we show that a suitable strategy for adaptively choosing the token decoding order significantly enhances the capabilities of MDMs, allowing them to sidestep hard subproblems. On logic puzzles like Sudoku, we show that adaptive inference can boost solving accuracy in pretrained MDMs from $<7$% to $\approx 90$%, even outperforming ARMs with $7\times$ as many parameters and that were explicitly trained via teacher forcing to learn the right order of decoding.

Connections between Schedule-Free Optimizers, AdEMAMix, and Accelerated SGD Variants

Recent advancements in deep learning optimization have introduced new algorithms, such as Schedule-Free optimizers, AdEMAMix, MARS and Lion which modify traditional momentum mechanisms. In a separate line of work, theoretical acceleration of stochastic gradient descent (SGD) in noise-dominated regime has been achieved by decoupling the momentum coefficient from the current gradient's weight. In this paper, we establish explicit connections between these two lines of work. We substantiate our theoretical findings with preliminary experiments on a 150m language modeling task. We find that AdEMAMix, which most closely resembles accelerated versions of stochastic gradient descent, exhibits superior performance. Building on these insights, we introduce a modification to AdEMAMix, termed Simplified-AdEMAMix, which maintains the same performance as AdEMAMix across both large and small batch-size settings while eliminating the need for two different momentum terms. The code for Simplified-AdEMAMix is available on the repository: https://github.com/DepenM/Simplified-AdEMAMix/.

Soup to go: mitigating forgetting during continual learning with model averaging

In continual learning, where task data arrives in a sequence, fine-tuning on later tasks will often lead to performance degradation on earlier tasks. This is especially pronounced when these tasks come from diverse domains. In this setting, how can we mitigate catastrophic forgetting of earlier tasks and retain what the model has learned with minimal computational expenses? Inspired by other merging methods, and L2-regression, we propose Sequential Fine-tuning with Averaging (SFA), a method that merges currently training models with earlier checkpoints during the course of training. SOTA approaches typically maintain a data buffer of past tasks or impose a penalty at each gradient step. In contrast, our method achieves comparable results without the need to store past data, or multiple copies of parameters for each gradient step. Furthermore, our method outperforms common merging techniques such as Task Arithmetic, TIES Merging, and WiSE-FT, as well as other penalty methods like L2 and Elastic Weight Consolidation. In turn, our method offers insight into the benefits of merging partially-trained models during training across both image and language domains.

From an Image to a Scene: Learning to Imagine the World from a Million 360 Videos

Three-dimensional (3D) understanding of objects and scenes play a key role in humans' ability to interact with the world and has been an active area of research in computer vision, graphics, and robotics. Large scale synthetic and object-centric 3D datasets have shown to be effective in training models that have 3D understanding of objects. However, applying a similar approach to real-world objects and scenes is difficult due to a lack of large-scale data. Videos are a potential source for real-world 3D data, but finding diverse yet corresponding views of the same content has shown to be difficult at scale. Furthermore, standard videos come with fixed viewpoints, determined at the time of capture. This restricts the ability to access scenes from a variety of more diverse and potentially useful perspectives. We argue that large scale 360 videos can address these limitations to provide: scalable corresponding frames from diverse views. In this paper, we introduce 360-1M, a 360 video dataset, and a process for efficiently finding corresponding frames from diverse viewpoints at scale. We train our diffusion-based model, Odin, on 360-1M. Empowered by the largest real-world, multi-view dataset to date, Odin is able to freely generate novel views of real-world scenes. Unlike previous methods, Odin can move the camera through the environment, enabling the model to infer the geometry and layout of the scene. Additionally, we show improved performance on standard novel view synthesis and 3D reconstruction benchmarks.