On the stability of Lipschitz continuous control problems and its application to reinforcement learning

TL;DR

This paper investigates the stability of Lipschitz continuous control problems within reinforcement learning, establishing theoretical insights and proposing a new HJB-based algorithm tested on benchmarks.

Contribution

It bridges Lipschitz control problems with classical control theory, introduces a generalized framework, and develops a novel HJB-based RL algorithm with stability analysis.

Findings

The proposed method demonstrates improved stability over existing algorithms.

Theoretical analysis links Lipschitz properties to convergence rates.

Benchmark tests show competitive performance of the new approach.

Abstract

We address the crucial yet underexplored stability properties of the Hamilton--Jacobi--Bellman (HJB) equation in model-free reinforcement learning contexts, specifically for Lipschitz continuous optimal control problems. We bridge the gap between Lipschitz continuous optimal control problems and classical optimal control problems in the viscosity solutions framework, offering new insights into the stability of the value function of Lipschitz continuous optimal control problems. By introducing structural assumptions on the dynamics and reward functions, we further study the rate of convergence of value functions. Moreover, we introduce a generalized framework for Lipschitz continuous control problems that incorporates the original problem and leverage it to propose a new HJB-based reinforcement learning algorithm. The stability properties and performance of the proposed method are tested with well-known benchmark examples in comparison with existing approaches.