A Comparative Theoretical Analysis of Entropy Control Methods in Reinforcement Learning

TL;DR

This paper compares two entropy control methods in reinforcement learning, analyzing their theoretical properties and implications for large language model training.

Contribution

It introduces a unified framework for entropy dynamics and reveals the advantages of covariance-based regularization over traditional methods.

Findings

Traditional entropy regularization biases the stationary policy.

Covariance-based methods selectively regularize high-covariance tokens.

Covariance-based regularization achieves asymptotic unbiasedness with annealing.

Abstract

Reinforcement learning (RL) has become a key approach for enhancing reasoning in large language models (LLMs), yet scalable training is often hindered by the rapid collapse of policy entropy, which leads to premature convergence and performance saturation. This paper provides a comparative theoretical analysis of two entropy control strategies: traditional entropy regularization and the recently proposed covariance-based mechanism. We establish a unified framework for entropy dynamics under softmax parameterization, showing that entropy change is governed by the covariance between log-probabilities and logit updates. Our analysis reveals that traditional entropy regularization introduces a dense, persistent bias that modifies the stationary condition, leading to suboptimal policies, while covariance-based methods selectively regularize a sparse subset of high-covariance tokens and achieve asymptotic unbiasedness when the regularization coefficient is annealed. These results provide principled guidelines for entropy control in LLM posttraining, with implications for scaling RL to larger models and more complex reasoning tasks.