An optimal algorithm for bandit and zero-order convex optimization with two-point feedback

We consider the closely related problems of bandit convex optimization with two-point feedback, and zero-order stochastic convex optimization with two function evaluations per round. We provide a simple algorithm and analysis which is optimal for convex Lipschitz functions. This improves on Duchi et al. (2015), which only provides an optimal result for smooth functions; Moreover, the algorithm and analysis are simpler, and readily extend to non-Euclidean problems. The algorithm is based on a small but surprisingly powerful modification of the gradient estimator.