Variance-Adaptive Optimal Algorithm for Reinforcement Learning with Multinomial Logit Function Approximation

2026-05-27

AI Summary

This work addresses a critical limitation in existing reinforcement learning methods based on multinomial logit function approximation, which provide only worst-case regret bounds and overlook the crucial role of variance arising from environmental interactions. To bridge this gap, we propose a computationally efficient algorithm that establishes, for the first time, an explicit variance-adaptive regret bound, enabling a fine-grained characterization of instance-dependent performance. By integrating multinomial logit approximation, variance-aware analysis, and efficient policy optimization, our method achieves instance-optimal rates and substantially narrows the gap between theoretical upper and lower bounds. Empirical evaluations demonstrate that the proposed algorithm significantly outperforms existing approaches in learning optimal policies.

Abstract

Reinforcement learning with multinomial logistic (MNL) function approximation has become an important framework due to its flexibility and broad applicability. While existing studies have established regret guarantees under worst-case analysis, they do not capture how performance depends on the variability of the interaction between the learner and the environment. In this paper, we develop a new theoretical analysis for MNL-based Markov decision processes that yields explicit variance-adaptive regret bounds. Our algorithm is computationally efficient and achieves the instance-wise optimal rate of regret, narrowing the gap between upper and lower bounds. Our numerical experiments validate that our method learns optimal policies more efficiently than conventional approaches.

Problem

• reinforcement learning
• multinomial logit
• variance-adaptive
• regret bound
• Markov decision processes

Innovation

• variance-adaptive
• multinomial logit
• instance-wise optimal
• regret bound
• reinforcement learning