Minimisation of Quasar-Convex Functions Using Random Zeroth-Order Oracles

TL;DR

This paper analyzes a Gaussian smoothing zeroth-order method for minimizing quasar-convex functions, providing convergence guarantees and complexity bounds in both unconstrained and constrained settings, with applications to machine learning.

Contribution

It introduces the notion of proximal-quasar-convexity and establishes convergence and complexity bounds for zeroth-order optimization of QC and SQC functions.

Findings

Convergence to a global minimum in unconstrained QC/SQC problems.

Complexity bounds for the zeroth-order algorithm.

Practical applications to machine learning problems.

Abstract

This paper explores the performance of a random Gaussian smoothing zeroth-order (ZO) scheme for minimising quasar-convex (QC) and strongly quasar-convex (SQC) functions in both unconstrained and constrained settings. For the unconstrained problem, we establish the ZO algorithm's convergence to a global minimum along with its complexity when applied to both QC and SQC functions. For the constrained problem, we introduce the new notion of proximal-quasar-convexity and prove analogous results to the unconstrained case. Specifically, we derive complexity bounds and prove convergence of the algorithm to a neighbourhood of a global minimum whose size can be controlled under a variance reduction scheme. Beyond the theoretical guarantees, we demonstrate the practical implications of our results on several machine learning problems where quasar-convexity naturally arises, including linear dynamical system identification and generalised linear models.