Gradient-free optimization of highly smooth functions: improved analysis and a new algorithm

TL;DR

This paper introduces a new zero-order optimization algorithm for highly smooth functions, providing improved theoretical convergence rates and analysis, especially under noisy conditions, with extensions to non-convex and special classes of functions.

Contribution

It presents a novel algorithm based on $ ext{l}_1$ sphere randomization, with improved analysis and bounds over existing methods, and introduces new proof techniques using Poincaré inequalities.

Findings

Improved convergence rates for highly smooth functions under noise.

The $ ext{l}_1$ sphere algorithm outperforms $ ext{l}_2$ sphere and Gaussian methods in bias and variance.

Minimax lower bounds show near optimality of the proposed bounds.

Abstract

This work studies minimization problems with zero-order noisy oracle information under the assumption that the objective function is highly smooth and possibly satisfies additional properties. We consider two kinds of zero-order projected gradient descent algorithms, which differ in the form of the gradient estimator. The first algorithm uses a gradient estimator based on randomization over the $\ell_2$ sphere due to Bach and Perchet (2016). We present an improved analysis of this algorithm on the class of highly smooth and strongly convex functions studied in the prior work, and we derive rates of convergence for two more general classes of non-convex functions. Namely, we consider highly smooth functions satisfying the Polyak-{\L}ojasiewicz condition and the class of highly smooth functions with no additional property. The second algorithm is based on randomization over the $\ell_1$ sphere, and it extends to the highly smooth setting the algorithm that was recently proposed for Lipschitz convex functions in Akhavan et al. (2022). We show that, in the case of noiseless oracle, this novel algorithm enjoys better bounds on bias and variance than the $\ell_2$ randomization and the commonly used Gaussian randomization algorithms, while in the noisy case both $\ell_1$ and $\ell_2$ algorithms benefit from similar improved theoretical guarantees. The improvements are achieved thanks to a new proof techniques based on Poincar\'e type inequalities for uniform distributions on the $\ell_1$ or $\ell_2$ spheres. The results are established under weak (almost adversarial) assumptions on the noise. Moreover, we provide minimax lower bounds proving optimality or near optimality of the obtained upper bounds in several cases.