Convergence Analysis for Entropy-Regularized Control Problems: A Probabilistic Approach
This provides a simpler convergence proof for control algorithms, though it appears incremental compared to existing PDE-based approaches.
The paper tackles the convergence analysis of the Policy Iteration Algorithm for continuous-time entropy-regularized stochastic control problems, providing a simple probabilistic proof that achieves super-exponential convergence rates in finite horizon and infinite horizon models with large discount factors.
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.