Query Complexity of Derivative-Free Optimization

TL;DR

This paper establishes fundamental lower bounds on the convergence rates of derivative-free optimization with noisy evaluations, introduces a near-optimal Boolean comparison-based algorithm for strongly convex functions, and shows similar convergence rates for both evaluation types.

Contribution

It provides the first lower bounds on DFO convergence rates with noise, and proposes a novel Boolean comparison-based algorithm that is near optimal for strongly convex functions.

Findings

Lower bounds on DFO convergence rates with noise

A new Boolean comparison-based DFO algorithm

Equivalent convergence rates for noisy evaluations and Boolean comparisons

Abstract

This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same.