A New Algorithm for Non-stationary Contextual Bandits: Efficient, Optimal, and Parameter-free

Abstract: We propose the first contextual bandit algorithm that is parameter-free, efficient, and optimal in terms of dynamic regret. Specifically, our algorithm achieves dynamic regret $$\mathcal{O}(\min{\sqrt{ST}, \Delta<sup>{\frac{1}{3}}T<sup>{\frac{2}{3}}})$$ for a contextual bandit problem with $T$ rounds, $S$ switches and $\Delta$ total variation in data distributions. Importantly, our algorithm is adaptive and does not need to know $S$ or $\Delta$ ahead of time, and can be implemented efficiently assuming access to an ERM oracle. Our results strictly improve the $$\mathcal{O}(\min {S^{{\frac{1}{4}}T^{{\frac{3}{4}},}} \Delta<sup>{\frac{1}{5}}T<sup>{\frac{4}{5}}})$$ bound of (Luo et al., 2018), and greatly generalize and improve the $\mathcal{O}(\sqrt{ST})$ result of (Auer et al, 2018) that holds only for the two-armed bandit problem without contextual information. The key novelty of our algorithm is to introduce replay phases, in which the algorithm acts according to its previous decisions for a certain amount of time in order to detect non-stationarity while maintaining a good balance between exploration and exploitation.