Parameter-Free Dynamic Regret for Unconstrained Linear Bandits

Abstract

We study dynamic regret minimization in unconstrained adversarial linear bandit problems. In this setting, a learner must minimize the cumulative loss relative to an arbitrary sequence of comparators \boldsymbol{u}_1,\ldots,\boldsymbol{u}_T in \mathbb{R}^d, but receives only point-evaluation feedback on each round. We provide a simple approach to combining the guarantees of several bandit algorithms, allowing us to optimally adapt to the number of switches S_T = \sum_t\mathbb{I}\{\boldsymbol{u}_t eq \boldsymbol{u}_{t-1}\} of an arbitrary comparator sequence. In particular, we provide the first algorithm for linear bandits achieving the optimal regret guarantee of order \mathcal{O}\big(\sqrt{d(1+S_T) T}\big) up to poly-logarithmic terms without prior knowledge of S_T, thus resolving a long-standing open problem.