Denis Belomestny, Ilya Levin, Alexey Naumov et al.
Policy evaluation is an important instrument for the comparison of different algorithms in Reinforcement Learning (RL). However, even a precise knowledge of the value function $V^π$ corresponding to a policy $π$ does not provide reliable information on how far the policy $π$ is from the optimal one. We present a novel model-free upper value iteration procedure ({\sf UVIP}) that allows us to estimate the suboptimality gap $V^{\star}(x) - V^π(x)$ from above and to construct confidence intervals for \(V^\star\). Our approach relies on upper bounds to the solution of the Bellman optimality equation via the martingale approach. We provide theoretical guarantees for {\sf UVIP} under general assumptions and illustrate its performance on a number of benchmark RL problems.