Policy Evaluation in Distributional LQR

• 成果类型: Article
• 署名作者: Wang, Zifan; Gao, Yulong; Wang, Siyi; Zavlanos, Michael M.; Abate, Alessandro; Johansson, Karl H.
• 署名单位: Royal Institute of Technology; Imperial College London; Technical University of Munich; Duke University; University of Oxford
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2025.3575649
• 发表日期: 2025
• 关键词: LINEAR-QUADRATIC CONTROL; reinforcement

摘要

Distributional reinforcement learning (DRL) enhances the understanding of the effects of the randomness in the environment by letting agents learn the distribution of a random return, rather than its expected value as in standard reinforcement learning. Meanwhile, a challenge in DRL is that the policy evaluation typically relies on the representation of the return distribution, which needs to be carefully designed. In this article, we address this challenge for the special class of DRL problems that rely on a discounted linear quadratic regulator (LQR), which we call distributional LQR. Specifically, we provide a closed-form expression for the distribution of the random return, which is applicable for all types of exogenous disturbance as long as it is independent and identically distributed. We show that the variance of the random return is bounded if the fourth moment of the exogenous disturbance is bounded. Furthermore, we investigate the sensitivity of the return distribution to model perturbations. While the proposed exact return distribution consists of infinitely many random variables, we show that this distribution can be well approximated by a finite number of random variables. The associated approximation error can be analytically bounded under mild assumptions. When the model is unknown, we propose a model-free approach for estimating the return distribution, supported by sample complexity guarantees. Finally, we extend our approach to partially observable linear systems. Numerical experiments are provided to illustrate the theoretical results.