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 (O) over tilde (d(3/2) root 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 (O) over tilde(root d) gap has remained between the upper and lower bounds, as the known regret lower bound is Omega(d root T). Our study resolves these two issues by introducing an algorithm that achieves an optimal regret bound of (O) over tilde (d root 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 (O) over tilde (d root T) under the assumption that the optimal solution is an interior of the feasible region. Even without this assumption, our algorithm achieves (O) over tilde (d(3/2) root T)-regret.