Rate-Optimal Algorithm for Adversarial Linear CMDPs

Kihyun Yu, Honghao Wei, Dabeen Lee

We study episodic adversarial linear constrained Markov decision processes (CMDPs) with unknown transitions, where both the loss and constraint functions may vary adversarially across episodes. The best previous algorithm achieves $\widetilde{\mathcal{O}}(K^{3/4})$ regret and cumulative constraint violation, leaving a gap to the optimal $\widetilde{\mathcal{O}}(\sqrt{K})$ dependence on the number of episodes $K$. We close this gap by proposing a new primal dual algorithm that achieves $\widetilde{\mathcal{O}}(\sqrt{K})$ regret and cumulative constraint violation without assuming Slater's condition. We further extend the algorithm to achieve the same $\widetilde{\mathcal{O}}(\sqrt{K})$ guarantees for regret and hard constraint violation, which does not allow constraint violations to cancel across episodes. The main challenge is that learning linear CMDPs requires uniform concentration over a value function class with a controlled covering number, whereas standard techniques in constrained online learning, such as policy mixing, can make this class more complex. Our algorithm combines adaptive Follow the Regularized Leader (FTRL), contracted value estimation, and an exponential…