Policy-based Primal-Dual Methods for Concave CMDP with Variance Reduction
This addresses optimization challenges in reinforcement learning for constrained decision-making problems, representing an incremental improvement with specific algorithmic enhancements.
The paper tackles the problem of solving Concave Constrained Markov Decision Processes (Concave CMDPs) with non-additive concave objectives and constraints, proposing the VR-PDPG algorithm that achieves an O(T^{-1/3}) convergence rate in exact settings and an O(ε^{-4}) sample complexity for ε-global optimality in sample-based settings, with zero constraint violation possible via a pessimistic term.
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 $\widetilde{O}(ε^{-4})$ sample complexity for $ε$-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.