Convergence Analysis for Entropy-Regularized Control Problems: A Probabilistic Approach

TL;DR

This paper proves the convergence of the Policy Iteration Algorithm for entropy-regularized stochastic control problems using a probabilistic approach, achieving super-exponential rates in certain models and extending to diffusion control cases.

Contribution

It introduces a simple probabilistic proof for PIA convergence in continuous-time entropy-regularized control, avoiding complex PDE estimates and extending results to diffusion control in one dimension.

Findings

Proves PIA convergence with a simple probabilistic method.

Achieves super-exponential convergence rates in finite and infinite horizon models.

Extends convergence results to one-dimensional diffusion control cases.

Abstract

In this paper we investigate the convergence of the Policy Iteration Algorithm (PIA) for a class of general continuous-time entropy-regularized stochastic control problems. In particular, instead of employing sophisticated PDE estimates for the iterative PDEs involved in the algorithm (see, e.g., Huang-Wang-Zhou(2025)), we shall provide a simple proof from scratch for the convergence of the PIA. Our approach builds on probabilistic representation formulae for solutions of PDEs and their derivatives. Moreover, in the finite horizon model and in the infinite horizon model with large discount factor, the similar arguments lead to a super-exponential rate of convergence without tear. Finally, with some extra efforts we show that our approach can be extended to the diffusion control case in the one dimensional setting, also with a super-exponential rate of convergence.