Policy-Based Primal-Dual Methods for Convex Constrained Markov Decision Processes

We study convex Constrained Markov Decision Processes (CMDPs) in which the objective is concave and the constraints are convex in the state-action occupancy measure. We propose a policy-based primal-dual algorithm that updates the primal variable via policy gradient ascent and updates the dual variable via projected sub-gradient descent. Despite the loss of additivity structure and the nonconvex nature, we establish the global convergence of the proposed algorithm by leveraging a hidden convexity in the problem, and prove the O (T -1/3 ) convergence rate in terms of both optimality gap and constraint violation. When the objective is strongly concave in the occupancy measure, we prove an improved convergence rate of O (T -1/2

). By introducing a pessimistic term to the constraint, we further show that a zero constraint violation can be achieved while preserving the same convergence rate for the optimality gap. This work is the first one in the literature that establishes non-asymptotic convergence guarantees for policy-based primal-dual methods for solving infinite-horizon discounted convex CMDPs.