Breaking the Capability Ceiling of LLM Post-Training by Reintroducing Markov States

Reinforcement learning (RL) post-training is a dominant approach for improving the reasoning performance of large language models (LLMs), yet growing evidence suggests that its gains arise primarily from distribution sharpening rather than the acquisition of new capabilities. Recent work has shown that sampling from the power distribution of LLMs using Markov chain Monte Carlo (MCMC) can recover performance comparable to RL post-training without relying on external rewards; however, the high computational cost of MCMC makes such approaches impractical for widespread adoption. In this work, we propose a theoretically grounded alternative that eliminates the need for iterative MCMC. We derive a token-level decomposition of the autoregressive power distribution, showing that the global power distribution can be approximated by a token-level scaled low-temperature one, where the scaling factor captures future trajectory quality. Leveraging this insight, we introduce a training-free and verifier-free algorithm that sharpens the base model's generative distribution autoregressively. Empirically, we evaluate our method on math, QA, and code tasks, and show that our method matches or surpasses one-shot GRPO without relying on any external rewards, while reducing inference latency by over $10\times$ compared to MCMC-based sampling.

Reinforcement learning (RL) has become central to improving large language models (LLMs) on complex reasoning tasks, yet RL post-training is largely studied in isolation from the pretraining that precedes it. As a result, two basic questions remain open: (1) how do pretraining choices (model size, data) shape the returns to RL compute, and (2) what does RL actually do to the model? These questions are difficult to study in the standard LLM setting: pretraining corpora are vast and uncontrolled, making it hard to attribute behaviors to pretraining versus RL, and systematic compute sweeps across both stages are prohibitively expensive. To address these challenges, we use chess as a controlled testbed for studying reasoning across the full pretraining-to-post-training pipeline. We follow the standard LLM training pipeline by pretraining language models from 5M to 1B parameters on human chess games, supervised fine-tuning on synthetic reasoning traces, and running RL on chess puzzles with verifiable rewards. Using this framework, we find that the post-RL performance at given RL compute level is well-predicted from the pretraining loss, and slope of the RL reward curves improves approximately linearly with the pretraining tokens. Beyond scaling, we find that RL does not simply sharpen the SFT policy: on easy puzzles it amplifies correct moves the SFT policy already preferred, while on hard puzzles it surfaces correct moves that were nearly absent under SFT. We further test whether our findings transfer beyond chess by training a 1B language model on math-domain text, where the same predictive pattern emerges: longer-pretrained checkpoints reach higher post-RL performance and improve faster under RL. In sum, we provide a quantitative account of the pretraining-to-RL interface and a controlled testbed for studying the science of reasoning across the full pretraining-to-post-training pipeline.

Reinforcement learning (RL) post-training has emerged as a powerful framework for enhancing the capabilities of large language models (LLMs), enabling impressive reasoning, math, and coding capabilities. Yet for many researchers and practitioners, the principles behind classical RL remain a "black box". In this work, we deconstruct the RL post-training algorithm, investigating each step to clarify what is actually happening beneath the surface. By isolating the mechanics of RL with Verifiable Rewards in a controlled and simplified environment, we examine how RL outcomes are shaped by the base model's prior distribution, the granularity of the reward signal, the diversity of the prompt distribution, and model scale. We use the entropy of the policy's output distribution as a lens to compare the distributions learned through pretraining, SFT, and RL post-training, revealing how each stage shapes model certainty. Our investigation sheds light on how these choices interact to affect post-training success. For example, we show that the effect of so-called 'spurious rewards' depends on the prompt distribution used for post-training. We also provide insight into why the success of RL post-training depends on whether the base model already places sufficient probability mass on the desired behavior, linking it to the classical concept of exploration in RL. Ultimately, we provide this primer as a resource to those in the NLP community wishing to incorporate RL as a tool in their toolbox.