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 $\boldsymbol{u}_1,\ldots,\boldsymbol{u}_T$ 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 $S_T = \sum_t\mathbb{I}\\{\boldsymbol{u}\_t \neq \boldsymbol{u}\_{t-1}\\}$ 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.