An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback

TL;DR

This paper introduces a simple, optimal algorithm for bandit and zero-order convex optimization with two-point feedback, applicable to Lipschitz functions and extendable to non-Euclidean settings.

Contribution

It presents a new, simpler algorithm with optimal performance for convex Lipschitz functions, improving upon previous methods limited to smooth functions.

Findings

Algorithm is optimal for convex Lipschitz functions.

Simpler analysis and extension to non-Euclidean problems.

Improves upon previous methods limited to smooth functions.

Abstract

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 \cite{dujww13}, 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.