An Optimal Algorithm for Bandit Convex Optimization with Strongly-Convex and Smooth Loss

We consider non-stochastic bandit convex optimization with strongly-convex and smooth loss functions. For this problem, Hazan and Levy have proposed an algorithm with a regret bound of $\tilde{O}(d^{3/2} \sqrt{T})$ given access to an $O(d)$-self-concordant barrier over the feasible region, where $d$ and $T$ stand for the dimensionality of the feasible region and the number of rounds, respectively. However, there are no known efficient ways for constructing self-concordant barriers for general convex sets, and a $\tilde{O}(\sqrt{d})$ gap has remained between the upper and lower bounds, as the known regret lower bound is $\Omega(d\sqrt{T})$. Our study resolves these two issues by introducing an algorithm that achieves an optimal regret bound of $\tilde{O}(d \sqrt{T})$ under a mild assumption, without self-concordant barriers. More precisely, the algorithm requires only a membership oracle for the feasible region, and it achieves an optimal regret bound of $\tilde{O}(d\sqrt{T})$ under the assumption that the optimal solution is an interior of the feasible region. Even without this assumption, our algorithm achieves $\tilde{O}(d^{3/2}\sqrt{T})$-regret.