Data-dependent Bounds with T-Optimal Best-of-Both-Worlds Guarantees in Multi-Armed Bandits using Stability-Penalty Matching

Existing data-dependent and best-of-both-worlds regret bounds for multi-armed bandits problems have limited adaptivity as they are either data-dependent but not best-of-both-worlds (BOBW), BOBW but not data-dependent or have sub-optimal O(√TlnT) worst-case guarantee in the adversarial regime. To overcome these limitations, we propose real-time stability-penalty matching (SPM), a new method for obtaining regret bounds that are simultaneously data-dependent, best-of-both-worlds and T-optimal for multi-armed bandits problems. In particular, we show that real-time SPM obtains bounds with worst-case guarantees of order O(√T) in the adversarial regime and O(lnT) in the stochastic regime while simultaneously being adaptive to data-dependent quantities such as sparsity, variations, and small losses. Our results are obtained by extending the SPM technique for tuning the learning rates in the follow-the-regularized-leader (FTRL) framework, which further indicates that the combination of SPM and FTRL is a promising approach for proving new adaptive bounds in online learning problems.