Quantum speedups for stochastic optimization

TL;DR

This paper introduces quantum algorithms that outperform classical methods in stochastic optimization, achieving better dimension-accuracy trade-offs and optimal rates for convex and non-convex functions.

Contribution

It presents two novel quantum methods for Lipschitz convex minimization with provable advantages and extends quantum algorithms to find critical points in non-convex optimization.

Findings

Quantum algorithms outperform classical in dimension-accuracy trade-offs.

One method is asymptotically optimal in low-dimensional settings.

Quantum algorithms achieve rates for non-convex critical point computation not known classically.

Abstract

We consider the problem of minimizing a continuous function given quantum access to a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. 2022 and provide a general quantum-variance reduction technique of independent interest.