Crush Optimism with Pessimism: Structured Bandits Beyond Asymptotic Optimality

In this paper, we study stochastic structured bandits for minimizing regret. The fact that the popular optimistic algorithms do not achieve the asymptotic instancedependent regret optimality (asymptotic optimality for short) has recently allured researchers. On the other hand, it is known that one can achieve a bounded regret (i.e., does not grow indefinitely with n) in certain instances. Unfortunately, existing asymptotically optimal algorithms rely on forced sampling that introduces an ω(1) term w.r.t. the time horizon n in their regret, failing to adapt to the "easiness" of the instance. In this paper, we focus on the finite hypothesis class and ask if one can achieve the asymptotic optimality while enjoying bounded regret whenever possible. We provide a positive answer by introducing a new algorithm called CRush Optimism with Pessimism (CROP) that eliminates optimistic hypotheses by pulling the informative arms indicated by a pessimistic hypothesis. Our finite-time analysis shows that CROP (i) achieves a constant-factor asymptotic optimality and, thanks to the forced-exploration-free design, (ii) adapts to bounded regret, and (iii) its regret bound scales not with the number of arms K but with an effective number of arms K ψ that we introduce. We also discuss a problem class where CROP can be exponentially better than existing algorithms in nonasymptotic regimes. Finally, we observe that even a clairvoyant oracle who plays according to the asymptotically optimal arm pull scheme may suffer a linear worst-case regret, indicating that it may not be the end of optimism. Minimizing this regret poses a well-known challenge in balancing between exploration and exploitation; we refer to Lattimore and Szepesvári [21] for the backgrounds on bandits. We define our problem precisely in Section 2. v2: Added the lower bound result. This version is identical to the NeurIPS'20 camera-ready version. Preprint. Under review.