Quantum Algorithms for Non-smooth Non-convex Optimization

Chengchang Liu, Chaowen Guan, Jianhao He, John C.S. Lui

Keywords: quantum computing, non-convex non-smooth optimization

TL;DR: We show the quantum speedups for finding the Goldstein stationary point by proposing gradient-free quantum algorithms.

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 novel 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 $\epsilon^{-2/3}$.

Primary Area: Optimization (convex and non-convex, discrete, stochastic, robust)

Submission Number: 8627