Minimisation of Polyak-\L{}ojasewicz Functions Using Random Zeroth-Order Oracles

TL;DR

This paper introduces a zeroth-order optimization method using random oracles to minimize Polyak-ojasewicz functions, providing convergence guarantees and complexity bounds, with numerical validation.

Contribution

It presents a novel zeroth-order algorithm leveraging random oracles for PL functions, with proven convergence and complexity analysis.

Findings

Converges to a global minimum for unconstrained PL functions.

Achieves convergence to a neighborhood of the minimum in constrained cases.

Provides complexity bounds and numerical demonstrations.

Abstract

The application of a zeroth-order scheme for minimising Polyak-\L{}ojasewicz (PL) functions is considered. The framework is based on exploiting a random oracle to estimate the function gradient. The convergence of the algorithm to a global minimum in the unconstrained case and to a neighbourhood of the global minimum in the constrained case along with their corresponding complexity bounds are presented. The theoretical results are demonstrated via numerical examples.