Uniform Last-Iterate Guarantee for Bandits and Reinforcement Learning

Existing metrics for reinforcement learning (RL) such as regret, PAC bounds, or uniform-PAC (Dann et al., 2017), typically evaluate the cumulative performance, while allowing the play of an arbitrarily bad policy at any finite time t. Such a behavior can be highly detrimental in high-stakes applications. This paper introduces a stronger metric, uniform last-iterate (ULI) guarantee, capturing both cumulative and instantaneous performance of RL algorithms. Specifically, ULI characterizes the instantaneous performance since it ensures that the per-round suboptimality of the played policy is bounded by a function, monotonically decreasing w.r.t. (large) round t, preventing revisits to bad policies when sufficient samples are available. We demonstrate that a near-optimal ULI guarantee directly implies near-optimal cumulative performance across aforementioned metrics, but not the other way around. To examine the achievability of ULI, we first provide two positive results for bandit problems with finite arms, showing that some elimination-based algorithms and high-probability adversarial algorithms with stronger analysis or additional designs, can attain near-optimal ULI guarantees. We also provide a negative result, indicating that optimistic algorithms cannot achieve a near-optimal ULI guarantee. Furthermore, we propose an efficient algorithm for linear bandits with infinitely many arms, which achieves the ULI guarantee, given access to an optimization oracle. Finally, we propose an algorithm that achieves a near-optimal ULI guarantee for the online reinforcement learning setting.