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：我们研究了基于soft-max参数化的熵正则化约束马尔可夫决策问题（CMDPs），其中智能体旨在最大化熵正则化价值函数，同时满足期望总效用的约束条件。通过利用熵正则化，我们的理论分析表明其拉格朗日对偶函数具有光滑性，且拉格朗日对偶间隙可分解为原最优性间隙与约束违反量。进一步地，我们提出了一种针对熵正则化CMDPs的加速对偶下降法。我们证明该方法在熵正则化CMDPs的最优性间隙和约束违反量上均实现了全局收敛率 $\widetilde{\mathcal{O}}(1/T)$。此外，我们还讨论了单约束CMDPs的线性收敛速率。