Semi-infinitely Constrained Markov Decision Processes

We propose a novel generalization of constrained Markov decision processes (CMDPs) that we call the semi-infinitely constrained Markov decision process (SICMDP). Particularly, we consider a continuum of constraints instead of a finite number of constraints as in the case of ordinary CMDPs. We also devise two reinforcement learning algorithms for SICMDPs that we refer to as SI-CMBRL and SI-CPO. SI-CMBRL is a model-based reinforcement learning algorithm. Given an estimate of the transition model, we first transform the reinforcement learning problem into a linear semi-infinitely programming (LSIP) problem and then use the dual exchange method in the LSIP literature to solve it. SI-CPO is a policy optimization algorithm. Borrowing ideas from the cooperative stochastic approximation approach, we make alternative updates to the policy parameters to maximize the reward or minimize the cost. To the best of our knowledge, we are the first to apply tools from semi-infinitely programming (SIP) to solve constrained reinforcement learning problems. We present theoretical analysis for SI-CMBRL and SI-CPO, identifying their iteration complexity and sample complexity. We also conduct extensive numerical experiments to illustrate the SICMDP model and demonstrate that our proposed algorithms are able to solve complex control tasks leveraging modern deep reinforcement learning techniques.