Recent works have highlighted optimization difficulties faced by gradient descent in training the first and last layers of transformer-based language models, which are overcome by optimizers such as Adam. These works suggest that the difficulty is linked to the heavy-tailed distribution of words in text data, where the frequency of the kth most frequent word πk is proportional to 1/k, following Zipf's law. To better understand the impact of the data distribution on training performance, we study a linear bigram model for next-token prediction when the tokens follow a power law πk 1/kα parameterized by the exponent α > 0. We derive optimization scaling laws for deterministic gradient descent and sign descent as a proxy for Adam as a function of the exponent α. Existing theoretical investigations in scaling laws assume that the eigenvalues of the data decay as a power law with exponent α > 1. This assumption effectively makes the problem "finite dimensional" as most of the loss comes from a few of the largest eigencomponents. In comparison, we show that the problem is more difficult when the data have heavier tails. The case α = 1 as found in language is "worst-case" for gradient descent, in that the number of iterations required to reach a small relative error scales almost linearly with dimension. While the performance of sign descent also depends on the dimension, for Zipf-distributed data the number of iterations scales only with the square-root of the dimension, leading to a large improvement for large vocabularies.

We present a universal theoretical framework for understanding long-context language modeling based on a bipartite mutual information scaling law that we rigorously verify in natural language. We demonstrate that bipartite mutual information captures multi-token interactions distinct from and scaling independently of conventional two-point mutual information, and show that this provides a more complete characterization of the dependencies needed for accurately modeling long sequences. Leveraging this scaling law, we formulate the Long-context Language Modeling (L2M) condition, which lower bounds the necessary scaling of a model's history state--the latent variables responsible for storing past information--for effective long-context modeling.

The current paradigm of test-time scaling relies on generating long reasoning traces ("thinking" more) before producing a response. In agent problems that require interaction, this can be done by generating thinking traces before acting in the world. However, this process does not allow agents to acquire new information from the environment or adapt their behavior over time. In this work, we propose to scale test-time interaction, an untapped dimension of test-time scaling that increases the agent's interaction horizon to enable running rich behaviors such as exploration, backtracking, and dynamic re-planning within a single rollout. To demonstrate the promise of this scaling dimension, we study the domain of web agents.

Neural scaling laws are observed in a range of domains, to date with no universal understanding of why they occur. Recent theories suggest that loss power laws arise from Zipf's law, a power law observed in domains like natural language. One theory suggests that language scaling laws emerge when Zipf-distributed task quanta are learned in descending order of frequency. In this paper we examine power-law scaling in AlphaZero, a reinforcement learning algorithm, using a model of language-model scaling. We find that game states in training and inference data scale with Zipf's law, which is known to arise from the tree structure of the environment, and examine the correlation between scaling-law and Zipf'slaw exponents. In agreement with the quanta scaling model, we find that agents optimize state loss in descending order of frequency, even though this order scales inversely with modelling complexity. We also find that inverse scaling, the failure of models to improve with size, is correlated with unusual Zipf curves where end-game states are among the most frequent states. We show evidence that larger models shift their focus to these less-important states, sacrificing their understanding of important early-game states.

Recent work on neural scaling laws demonstrates that model performance scales predictably with compute budget, model size, and dataset size. In this work, we develop scaling laws based on problem complexity. We analyze two fundamental complexity measures: solution space size and representation space size. Using the Traveling Salesman Problem (TSP) as a case study, we show that combinatorial optimization promotes smooth cost trends, and therefore meaningful scaling laws can be obtained even in the absence of an interpretable loss. We then show that suboptimality grows predictably for fixed-size models when scaling the number of TSP nodes or spatial dimensions, independent of whether the model was trained with reinforcement learning or supervised fine-tuning on a static dataset. We conclude with an analogy to problem complexity scaling in local search, showing that a much simpler gradient descent of the cost landscape produces similar trends.1

Inference-time scaling trades efficiency for increased reasoning accuracy by generating longer or more parallel sequences. However, in Transformer LLMs, generation cost is bottlenecked by the size of the key-value (KV) cache, rather than the number of generated tokens. Hence, we explore inference-time hyper-scaling: by compressing the KV cache, we can generate more tokens within the same compute budget and further improve the accuracy of scaled inference. The success of this approach, however, hinges on the ability of compression methods to preserve accuracy even at high compression ratios. To make hyper-scaling practical, we introduce Dynamic Memory Sparsification (DMS), a novel method for sparsifying KV caches that only requires 1K training steps to achieve 8 compression, while maintaining better accuracy than training-free sparse attention.

As models grow larger and training them becomes expensive, it becomes increasingly important to scale training recipes not just to larger models and more data, but to do so in a compute-optimal manner that extracts maximal performance per unit of compute. While such scaling has been well studied for language modeling, reinforcement learning (RL) has received less attention in this regard. In this paper, we investigate compute scaling for online, value-based deep RL. These methods present two primary axes for compute allocation: model capacity and the update-to-data (UTD) ratio. Given a fixed compute budget, we ask: how should resources be partitioned across these axes to maximize data efficiency? Our analysis reveals a nuanced interplay between model size, batch size, and UTD. In particular, we identify a phenomenon we call TD-overfitting: increasing the batch quickly harms Q-function accuracy for small models, but this effect is absent in large models, enabling effective use of large batch size at scale. We provide a mental model for understanding this phenomenon and build guidelines for choosing batch size and UTD to optimize compute usage. Our findings provide a grounded starting point for compute-optimal scaling in deep RL, mirroring studies in supervised learning but adapted to TD learning.

Several recently introduced deep learning optimizers utilizing matrix-level preconditioning have shown promising speedups relative to the current dominant optimizer AdamW, particularly in relatively small-scale experiments. However, efforts to validate and replicate their successes have reported mixed results. To better understand the effectiveness of these optimizers at scale, in this work we investigate how to scale preconditioned optimizers via hyperparameter transfer, building on prior works such as $\mu$P. We study how the optimal learning rate and weight decay should scale with model width and depth for a wide range of optimizers, including Shampoo, SOAP, and Muon, accounting for the impact of commonly used techniques such as blocking and grafting. We find that scaling the learning rate according to $\mu$P improves transfer, but can still suffer from significant finite-width deviations that cause drifting optimal learning rates, which we show can be mitigated by blocking and explicit spectral normalization. For compute-optimal scaling, we find scaling independent weight decay as $1/\mathrm{width}$ is nearly optimal across optimizers. Applying these scaling rules, we show Muon, SOAP and Shampoo consistently achieve near $1.4\times$ speedup over AdamW for training Llama-architecture language models of sizes ranging from $190$M to $1.4$B, whereas the speedup vanishes rapidly with scale under incorrect scaling. Based on these results and further ablations, we argue that studying optimal hyperparameter transfer is essential for reliably comparing optimizers at scale given a realistic tuning budget.

Inspired by recent findings on the fractal geometry of language, we introduce Recursive INference Scaling (RINS) as a complementary, plug-in recipe for scaling inference time in language and multimodal systems. RINS is a particular form of recursive depth that significantly outperforms +55 other variants, including the recent repeat-all-over (RAO) strategy in Mobile LLM (Liu et al., 2024) and latent recurrent thinking (Geiping et al., 2025). Unlike prior works, we carry out our comparisons on a compute-matched regime, and demonstrate that for a fixed model size and training compute budget, RINS substantially improves language modeling performance.

Neural scaling laws are observed in a range of domains, to date with no universal understanding of why they occur. Recent theories suggest that loss power laws arise from Zipf's law, a power law observed in domains like natural language. One theory suggests that language scaling laws emerge when Zipf-distributed task quanta are learned in descending order of frequency. In this paper we examine power-law scaling in AlphaZero, a reinforcement learning algorithm, using a model of language-model scaling. We find that game states in training and inference data scale with Zipf's law, which is known to arise from the tree structure of the environment, and examine the correlation between scaling-law and Zipf's-law exponents. In agreement with the quanta scaling model, we find that agents optimize state loss in descending order of frequency, even though this order scales inversely with modelling complexity. We also find that inverse scaling, the failure of models to improve with size, is correlated with unusual Zipf curves where end-game states are among the most frequent states. We show evidence that larger models shift their focus to these less-important states, sacrificing their understanding of important early-game states.