An Exploration-by-Optimization Approach to Best of Both Worlds in Linear Bandits

In this paper, we consider how to construct best-of-both-worlds linear bandit algorithms that achieve nearly optimal performance for both stochastic and adversarial environments. For this purpose, we show that a natural approach referred to as exploration by optimization [Lattimore and Szepesvári, 2020] works well. Specifically, an algorithm constructed using this approach achieves $O(d \sqrt{ T \log{T}})$-regret in adversarial environments and $O(\frac{d^2 \log T}{\Delta_{\min}} )$-regret in stochastic environments. Symbols $d$, $T$ and $\Delta_{\min}$ here represent the dimensionality of the action set, the time horizon, and the minimum sub-optimality gap, respectively. We also show that this algorithm has even better theoretical guarantees for important special cases including the multi-armed bandit problem and multitask bandits.