Efficient Convex Optimization with Membership Oracles

TL;DR

This paper presents a new algorithm for convex optimization that efficiently minimizes convex functions over convex sets using only evaluation and membership oracles, improving the complexity of such problems.

Contribution

It introduces a simple, efficient algorithm with improved oracle call complexity for convex optimization with membership and evaluation oracles.

Findings

Algorithm solves convex optimization with $ ilde{O}(n^2)$ oracle calls.

Reduces complexity of basic convex set and function oracles.

Enhances efficiency of convex optimization methods.

Abstract

We consider the problem of minimizing a convex function over a convex set given access only to an evaluation oracle for the function and a membership oracle for the set. We give a simple algorithm which solves this problem with $\tilde{O}(n^2)$ oracle calls and $\tilde{O}(n^3)$ additional arithmetic operations. Using this result, we obtain more efficient reductions among the five basic oracles for convex sets and functions defined by Gr\"otschel, Lovasz and Schrijver.