Optimization of Smooth Functions with Noisy Observations: Local Minimax Rates

TL;DR

This paper introduces a local minimax framework for zeroth-order optimization of smooth functions with noisy evaluations, providing refined rates and algorithms that adapt to the local structure of the function.

Contribution

It proposes a local minimax analysis for zeroth-order optimization, revealing faster rates for functions with certain local properties and matching classical rates for strongly convex functions.

Findings

Algorithms can identify near global minimizers with fewer queries for functions with rapid level set growth.

For strongly convex functions, the rates match existing zeroth-order convex optimization results.

A new algorithm attains the derived upper bounds on optimization error.

Abstract

We consider the problem of global optimization of an unknown non-convex smooth function with zeroth-order feedback. In this setup, an algorithm is allowed to adaptively query the underlying function at different locations and receives noisy evaluations of function values at the queried points (i.e. the algorithm has access to zeroth-order information). Optimization performance is evaluated by the expected difference of function values at the estimated optimum and the true optimum. In contrast to the classical optimization setup, first-order information like gradients are not directly accessible to the optimization algorithm. We show that the classical minimax framework of analysis, which roughly characterizes the worst-case query complexity of an optimization algorithm in this setting, leads to excessively pessimistic results. We propose a local minimax framework to study the fundamental difficulty of optimizing smooth functions with adaptive function evaluations, which provides a refined picture of the intrinsic difficulty of zeroth-order optimization. We show that for functions with fast level set growth around the global minimum, carefully designed optimization algorithms can identify a near global minimizer with many fewer queries. For the special case of strongly convex and smooth functions, our implied convergence rates match the ones developed for zeroth-order convex optimization problems. At the other end of the spectrum, for worst-case smooth functions no algorithm can converge faster than the minimax rate of estimating the entire unknown function in the $\ell_\infty$-norm. We provide an intuitive and efficient algorithm that attains the derived upper error bounds.