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

arXiv: https://arxiv.org/abs/2205.10715 · HTML: https://mlsanity.com/paper/2205.10715/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2022.

## Built on by (most cited first, 5 of 5)

- [Last-Iterate Convergent Policy Gradient Primal-Dual Methods for Constrained MDPs](https://mlsanity.com/paper/2306.11700/) (2023)
- [Scalable Multi-Agent Reinforcement Learning with General Utilities](https://mlsanity.com/paper/2302.07938/) (2023)
- [Offline Constrained Reinforcement Learning under Partial Data Coverage](https://mlsanity.com/paper/2505.17506/) (2025)
- [Global Optimality for Constrained Exploration via Penalty Regularization](https://mlsanity.com/paper/2604.28144/) (2026)
- [Benign Nonconvex Landscape for Policy Optimization: Infinite-Horizon Discounted MDPs with General State and Action Spaces](https://mlsanity.com/paper/2609.21433/) (2026)

## Abstract (arXiv)

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 widetildeO(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.