Optimization of Convex Functions with Random Pursuit

TL;DR

This paper introduces and analyzes the Random Pursuit algorithm for unconstrained convex optimization, demonstrating its convergence, invariance properties, and practical performance improvements over existing methods.

Contribution

It provides the first convergence analysis of Random Pursuit, introduces an accelerated heuristic version, and compares its effectiveness with Nesterov's methods and other gradient-free algorithms.

Findings

Random Pursuit converges for smooth convex functions.

The accelerated version outperforms standard Random Pursuit.

Random Pursuit is effective on strongly convex functions with moderate condition numbers.

Abstract

We consider unconstrained randomized optimization of convex objective functions. We analyze the Random Pursuit algorithm, which iteratively computes an approximate solution to the optimization problem by repeated optimization over a randomly chosen one-dimensional subspace. This randomized method only uses zeroth-order information about the objective function and does not need any problem-specific parametrization. We prove convergence and give convergence rates for smooth objectives assuming that the one-dimensional optimization can be solved exactly or approximately by an oracle. A convenient property of Random Pursuit is its invariance under strictly monotone transformations of the objective function. It thus enjoys identical convergence behavior on a wider function class. To support the theoretical results we present extensive numerical performance results of Random Pursuit, two gradient-free algorithms recently proposed by Nesterov, and a classical adaptive step-size random search scheme. We also present an accelerated heuristic version of the Random Pursuit algorithm which significantly improves standard Random Pursuit on all numerical benchmark problems. A general comparison of the experimental results reveals that (i) standard Random Pursuit is effective on strongly convex functions with moderate condition number, and (ii) the accelerated scheme is comparable to Nesterov's fast gradient method and outperforms adaptive step-size strategies. The appendix contains additional supporting online material.