Quantum Algorithms for Non-smooth Non-convex Optimization

TL;DR

This paper introduces a quantum algorithm for non-smooth, non-convex optimization that improves query complexity over classical methods by leveraging quantum estimators and variance reduction techniques.

Contribution

It develops a zeroth-order quantum estimator and a novel quantum algorithm with improved query complexity for finding stationary points in non-smooth non-convex optimization.

Findings

Quantum algorithm outperforms classical methods in dependency on .

Introduces a variance reduction technique for quantum optimization.

Achieves lower query complexity for -stationary points.

Abstract

This paper considers the problem for finding the $(\delta,\epsilon)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-3})$ on the stochastic function value oracle, where $d$ is the dimension of the problem. We also enhance the query complexity to $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-7/3})$ by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of $\epsilon$ by a factor of $\epsilon^{-2/3}$.