Policy-based Primal-Dual Methods for Concave CMDP with Variance Reduction

TL;DR

This paper introduces a variance-reduced primal-dual policy gradient algorithm for concave CMDPs, achieving improved convergence rates and zero constraint violation, with theoretical guarantees and numerical validation.

Contribution

It develops VR-PDPG, a novel algorithm for concave CMDPs, with proven global convergence, improved rates, and constraint violation control, addressing challenges of nonconcavity and variance reduction.

Findings

Achieves $O(T^{-1/3})$ convergence rate in exact setting.

Improves to $O(T^{-1/2})$ under strong concavity.

Attains $ ilde{O}(\e^{-4})$ sample complexity in the stochastic setting.

Abstract

We study Concave Constrained Markov Decision Processes (Concave CMDPs) where both the objective and constraints are defined as concave functions of the state-action occupancy measure. We propose the Variance-Reduced Primal-Dual Policy Gradient Algorithm (VR-PDPG), which updates the primal variable via policy gradient ascent and the dual variable via projected sub-gradient descent. Despite the challenges posed by the loss of additivity structure and the nonconcave nature of the problem, we establish the global convergence of VR-PDPG by exploiting a form of hidden concavity. In the exact setting, we prove an $O(T^{-1/3})$ convergence rate for both the average optimality gap and constraint violation, which further improves to $O(T^{-1/2})$ under strong concavity of the objective in the occupancy measure. In the sample-based setting, we demonstrate that VR-PDPG achieves an $\widetilde{O}(\epsilon^{-4})$ sample complexity for $\epsilon$-global optimality. Moreover, by incorporating a diminishing pessimistic term into the constraint, we show that VR-PDPG can attain a zero constraint violation without compromising the convergence rate of the optimality gap. Finally, we validate the effectiveness of our methods through numerical experiments.