We study entropy-regularized constrained Markov decision processes (CMDPs) under the soft-max parameterization, in which an agent aims to maximize the entropy-regularized value function while satisfying constraints on the expected total utility. By leveraging the entropy regularization, our theoretical analysis shows that its Lagrangian dual function is smooth and the Lagrangian duality gap can be decomposed into the primal optimality gap and the constraint violation. Furthermore, we propose an accelerated dual-descent method for entropy-regularized CMDPs. We prove that our method achieves the global convergence rate $\widetilde{\mathcal{O}}(1/T)$ for both the optimality gap and the constraint violation for entropy-regularized CMDPs. A discussion about a linear convergence rate for CMDPs with a single constraint is also provided.

翻译：我们研究在软最大化参数化下的熵正则化的约束马尔可夫决策过程(CMDP)，其中代理人旨在最大化熵正则化的值函数，同时满足对总效用的期望约束。通过利用熵正则化，我们的理论分析表明，它的拉格朗日对偶函数是平滑的，并且拉格朗日对偶间隙可以分解为原始最优性间隙和约束违规。此外，我们提出了一种加速的对偶下降方法，适用于熵正则化的CMDP。我们证明了我们的方法实现了熵正则化的CMDP的原始最优性间隙和约束违规的全局收敛速度$\widetilde{\mathcal{O}}(1/T)$。还提供了关于具有单个约束的CMDP的线性收敛速率的讨论。