Efficient Contextual Bandits in Non-stationary Worlds

Most contextual bandit algorithms minimize regret against the best fixed policy, a questionable benchmark for non-stationary environments that are ubiquitous in applications. In this work, we develop several efficient contextual bandit algorithms for non-stationary environments by equipping existing methods for i.i.d. problems with sophisticated statistical tests so as to dynamically adapt to a change in distribution. We analyze various standard notions of regret suited to non-stationary environments for these algorithms, including interval regret, switching regret, and dynamic regret. When competing with the best policy at each time, one of our algorithms achieves regret $\mathcal{O}(\sqrt{ST})$ if there are $T$ rounds with $S$ stationary periods, or more generally $\mathcal{O}(\Delta^{{1/3}T{2/3})$} where $\Delta$ is some non-stationarity measure. These results almost match the optimal guarantees achieved by an inefficient baseline that is a variant of the classic Exp4 algorithm. The dynamic regret result is also the first one for efficient and fully adversarial contextual bandit. Furthermore, while the results above require tuning a parameter based on the unknown quantity $S$ or $\Delta$, we also develop a parameter free algorithm achieving regret $\min{S^{1/4}T{3/4}, \Delta^{{1/5}T{4/5}}$.} This improves and generalizes the best existing result $\Delta^{{0.18}T{0.82}$} by Karnin and Anava (2016) which only holds for the two-armed bandit problem.