On Adaptivity in Non-stationary Stochastic Optimization With Bandit Feedback

In this paper we study the non-stationary stochastic optimization question with bandit feedback and dynamic regret measures. The seminal work of Besbes et al. (2015) shows that, when aggregated function changes is known a priori, a simple re-starting algorithm attains the optimal dynamic regret. In this work, we designed a stochastic optimization algorithm with fixed step sizes, which combined together with the multi-scale sampling framework of Wei and Luo (2021) achieves the optimal dynamic regret in non-stationary stochastic optimization without requiring prior knowledge of function change budget, thereby closes a question that has been open for a while. We also establish an additional result showing that any algorithm achieving good regret against stationary benchmarks with high probability could be automatically converted to an algorithm that achieves good regret against dynamic benchmarks, which is applicable to a wide class of bandit convex optimization algorithms.