Last-Iterate Convergence of General Parameterized Policies in Constrained MDPs

TL;DR

This paper introduces a new primal-dual accelerated natural policy gradient algorithm for constrained MDPs, achieving improved last-iterate convergence guarantees with specific sample complexities depending on policy class completeness.

Contribution

It proposes the PDR-ANPG algorithm with entropy and quadratic regularizers, providing the first last-iterate convergence guarantees for general parameterized policies in CMDPs.

Findings

Achieves last-iterate $ ilde{O}(rac{1}{ ext{epsilon}^4})$ sample complexity for complete policies.

Reduces sample complexity to $ ilde{O}(rac{1}{ ext{epsilon}^2})$ when the policy class is incomplete.

Improves upon existing state-of-the-art guarantees for parameterized CMDPs.

Abstract

This paper focuses on learning a Constrained Markov Decision Process (CMDP) via general parameterized policies. We propose a Primal-Dual based Regularized Accelerated Natural Policy Gradient (PDR-ANPG) algorithm that uses entropy and quadratic regularizers to reach this goal. For parameterized policy classes with a transferred compatibility approximation error, $\epsilon_{\mathrm{bias}}$, PDR-ANPG achieves a last-iterate $\epsilon$ optimality gap and $\epsilon$ constraint violation with a sample complexity of $\tilde{\mathcal{O}}(\epsilon^{-2}\min\{\epsilon^{-2},\epsilon_{\mathrm{bias}}^{-\frac{1}{3}}\})$. If the class is incomplete ($\epsilon_{\mathrm{bias}}>0$), then the sample complexity reduces to $\tilde{\mathcal{O}}(\epsilon^{-2})$ for $\epsilon<(\epsilon_{\mathrm{bias}})^{\frac{1}{6}}$. Moreover, for complete policies with $\epsilon_{\mathrm{bias}}=0$, our algorithm achieves a last-iterate $\epsilon$ optimality gap and $\epsilon$ constraint violation with $\tilde{\mathcal{O}}(\epsilon^{-4})$ sample complexity. It is a significant improvement over the state-of-the-art last-iterate guarantees of general parameterized CMDPs.