Non-Stationary Bandit Convex Optimization: An Optimal Algorithm with Two-Point Feedback

Abstract: This paper studies bandit convex optimization in non-stationary environments with two-point feedback, using dynamic regret as the performance measure. We propose an algorithm based on bandit mirror descent that extends naturally to non-Euclidean settings. Let $T$ be the total number of iterations and $\mathcal{P}{T,p}$ the path variation with respect to the $\ell_p$-norm. In Euclidean space, our algorithm matches the optimal regret bound $\mathcal{O}(\sqrt{dT(1+\mathcal{P}{T,2})})$, improving upon Zhao et al. (2021) by a factor of $\mathcal{O}(\sqrt{d})$. Beyond Euclidean settings, our algorithm achieves an upper bound of $\mathcal{O}(\sqrt{d\log(d) T(1 + \mathcal{P}{T,1})})$ on the simplex, which is optimal up to a $\log(d)$ factor. For the cross-polytope, the bound reduces to $\mathcal{O}(\sqrt{d\log(d)T(1+\mathcal{P}{T,p})})$ for some $p = 1 + 1/\log(d)$.