Convergence and sample complexity of natural policy gradient primal-dual methods for constrained MDPs

Abstract

We study sequential decision making problems aimed at maximizing the expected total reward while satisfying a constraint on the expected total utility. We employ the natural policy gradient method to solve the discounted infinite-horizon optimal control problem for Constrained Markov Decision Processes (constrained MDPs). Specifically, we propose a new Natural Policy Gradient Primal-Dual (NPG-PD) method that updates the primal variable via natural policy gradient ascent and the dual variable via projected sub-gradient descent. Although the underlying maximization involves a nonconcave objective function and a nonconvex constraint set, under the softmax policy parametrization we prove that our method achieves global convergence with sublinear rates regarding both the optimality gap and the constraint violation. Such convergence is independent of the size of the state-action space, i.e., it is dimension-free. Furthermore, for log-linear and general smooth policy parametrizations, we establish sublinear convergence rates up to a function approximation error caused by restricted policy parametrization. We also provide convergence and finitesample complexity guarantees for two sample-based NPG-PD algorithms. Finally, we use computational experiments to showcase the merits and the effectiveness of our approach.

Nondominated Policy-Guided Learning in Multi-Objective Reinforcement Learning

“…The primal-dual methods, also called Lagrangian methods, are designed to address Lagrange dual problems. Ding et al (2022) introduced a natural policy gradient-based primal-dual method and demonstrated its convergence to an optimal policy at a specified convergence rate. Another primal-dual method, proposed by Bai et al (2022), ensures that a trained policy results in zero constraint violations during evaluation.…”