Polynomial-Time Approximability of Constrained Reinforcement Learning

We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time (0, ϵ)-additive bicriteria approximation algorithm for finding optimal constrained policies across a broad class of recursively computable constraints, including almostsure, chance, expectation, and their anytime variants. Matching lower bounds imply our approximation guarantees are optimal so long as P ̸ = N P . The generality of our approach results in answers to several long-standing open complexity questions in the constrained reinforcement learning literature. Specifically, we are the first to prove polynomial-time approximability for the following settings: policies under chance constraints, deterministic policies under multiple expectation constraints, policies under non-homogeneous constraints (i.e., constraints of different types), and policies under constraints for continuous-state processes.