Parameter-Free Dynamic Regret for Unconstrained Linear Bandits

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 boldsymbolu1,ldots,boldsymboluT in ℝd, 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 ST = sumtmathbbIboldsymbolut neq boldsymbolut-1 of an arbitrary comparator sequence. In particular, we provide the first algorithm for linear bandits achieving the optimal regret guarantee of order Obig(√d(1+ST) Tbig) up to poly-logarithmic terms without prior knowledge of ST, thus resolving a long-standing open problem.