A Simple Method for Convex Optimization in the Oracle Model

TL;DR

This paper introduces a straightforward convex optimization method in the oracle model using the Frank--Wolfe algorithm, providing an efficient alternative to traditional cutting plane methods for various applications.

Contribution

It presents a simple, natural Frank--Wolfe based approach for convex optimization with theoretical guarantees and practical advantages over existing cutting plane techniques.

Findings

Method achieves near-optimal solutions with specified iteration complexity.

Algorithm compares favorably to cutting plane methods in experiments.

Easy to implement and adaptable to different convex optimization problems.

Abstract

We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function $f$ over a convex set $K$ given by a separation oracle. Our method utilizes the Frank--Wolfe algorithm over the cone of valid inequalities of $K$ and subgradients of $f$. Under the assumption that $f$ is $L$-Lipschitz and that $K$ contains a ball of radius $r$ and is contained inside the origin centered ball of radius $R$, using $O(\frac{(RL)^2}{\varepsilon^2} \cdot \frac{R^2}{r^2})$ iterations and calls to the oracle, our main method outputs a point $x \in K$ satisfying $f(x) \leq \varepsilon + \min_{z \in K} f(z)$. Our algorithm is easy to implement, and we believe it can serve as a useful alternative to existing cutting plane methods. As evidence towards this, we show that it compares favorably in terms of iteration counts to the standard LP based cutting plane method and the analytic center cutting plane method, on a testbed of combinatorial, semidefinite and machine learning instances.