Exchange Policy Optimization Algorithm for Semi-Infinite Safe Reinforcement Learning

Safe reinforcement learning (RL) aims to optimize long-term performance while adhering to safety requirements. However, many practical applications involve an infinite number of constraints, forming semi-infinite safe RL (SI-safe RL). Such scenarios typically appear when safety conditions must be enforced across an entire continuous parameter space, such as ensuring adequate resource distribution at every spatial location. Existing approaches typically tackle these continuous constraints through naive spatial discretization or stochastic sampling. Such methods inherently suffer from residual violations or provide only probabilistic safety guarantees. Therefore, no current framework can handle infinitely many constraints to provide reliable safety certificates. In this paper, we propose exchange policy optimization (EPO), an algorithmic framework that achieves optimal policy performance with provably bounded safety guarantees. EPO operates by iteratively solving safe RL subproblems restricted to a finite constraint set, adaptively adjusting the active set through constraint expansion and deletion. Specifically, at each iteration, constraints violating a predefined tolerance are added to refine the policy, while those with zero Lagrange multipliers are removed after the policy update. This exchange rule limits the subproblem complexity to ensure computational tractability while driving policy convergence. Theoretically, we establish that, under mild assumptions, EPO achieves finite convergence to a policy that both ensures the optimal reward performance and keeps the global constraint violation within the prescribed tolerance. Furthermore, we derive an upper bound on the required number of iterations and quantify the gap between the obtained policy and the true optimum.