Title: Convergence Rate of Primal-Dual Approach to Constrained Reinforcement Learning with Softmax Policy. Abstract: In this paper, we consider primal-dual approach to solve constrained reinforcement learning (RL) problems, where we formulate constrained reinforcement learning  under constrained Markov decision process (CMDP). We propose the primal-dual policy gradient (PD-PG) algorithm with softmax policy. Although the constrained RL involves a non-concave maximization problem over the policy parameter space, we show that for both exact policy gradient and model-free learning, the proposed PD-PG needs iteration complexity of $\mathcal{O}\left(\epsilon^{-2}\right)$ to achieve its optimal policy for both constraint and reward performance. Such an iteration complexity outperforms or matches most constrained RL algorithms. For the learning with exact policy gradient, the main challenge is to show the positivity of deterministic optimal policy (at the optimal action) is independent on both state space and iteration times.
For the model-free learning, since we consider the discounted infinite-horizon setting, and the simulator can not rollout with an infinite-horizon sequence; thus one of the main challenges lies in how to design unbiased value function estimators with finite-horizon trajectories. We consider the unbiased estimators with finite-horizon trajectories that involve geometric distribution horizons, which is the key technique for us to obtain the theoretical results for model-free learning.