Approximate Solutions to Optimal Stopping Problems
We propose and analyze an algorithm that approximates solutions to the problem of optimal stopping in a discounted irreducible aperiodic Markov chain. The scheme involves the use of linear combinations of fixed basis functions to approximate a Q-function. The weights of the linear combination are incrementally updated through an iterative process similar to Q-learning, involving simulation of the underlying Markov chain. Due to space limitations, we only provide an overview of a proof of convergence (with probability 1) and bounds on the approximation error. This is the first theoretical result that establishes the soundness of a Q-learning-like algorithm when combined with arbitrary linear function approximators to solve a sequential decision problem. Though this paper focuses on the case of finite state spaces, the results extend naturally to continuous and unbounded state spaces, which are addressed in a forthcoming full-length paper.