Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise

2026-07-28

AI Summary

This work addresses stochastic optimization under heavy-tailed noise by proposing a novel algorithm that integrates quantum mean estimation, generalized multilevel Monte Carlo, and unbiased gradient estimation to construct a quantum normalized and projected stochastic gradient descent method. In low-dimensional settings, the algorithm achieves query complexities of $\widetilde{O}(\sqrt{d}\cdot\varepsilon^{-(5p-4)/(2p-2)})$ for non-convex objectives and $\widetilde{O}(\sqrt{d}\cdot\varepsilon^{-(3p-2)/(2p-2)} + \varepsilon^{-2})$ for convex ones, significantly improving upon classical lower bounds. The study also establishes the first dimension-dependent quantum query complexity lower bound and demonstrates that the proposed estimator is optimal up to logarithmic factors.

Abstract

We study stochastic optimization with heavy-tailed gradient noise. We first propose a novel quantum mean estimator for multivariate heavy-tailed random variables that achieves lower query complexity than optimal classical estimators in the low-dimensional regime. We further develop an unbiased quantum mean estimator by applying a generalized multi-level Monte Carlo technique. We prove quantum lower bounds showing that, when the dimension $d$ of the random vector is small and can be viewed as a constant, our quantum estimators are optimal up to logarithmic factors. We further derive stronger dimension-dependent lower bounds for tail index $p>4/3$, showing that a nontrivial dependence on the dimension is unavoidable in the low-dimensional regime. Based on these estimators, we propose a quantum normalized stochastic gradient descent method ($\texttt{QNSGD}$), which finds an $ε$-stationary point using $\tilde{\mathcal{O}}\big(\sqrt d\,ε^{-\frac{5p-4}{2p-2}}\big)$ queries to the quantum stochastic gradient oracle. For a convex objective function, we propose a quantum projected stochastic gradient descent method ($\texttt{QPSGD}$), which computes a solution with $ε$-optimal solution using $\tilde{\mathcal{O}}\big(\sqrt d\,ε^{-\frac{3p-2}{2p-2}}+ε^{-2}\big)$ queries in expectation. These sharper bounds improve upon the classical lower bounds $Ω\big(ε^{-\frac{3p-2}{p-1}}\big)$ for nonconvex problems and $Ω\big(ε^{-\frac{p}{p-1}}\big)$ for convex problems in the low-dimensional regimes $d\lesssimε^{-\frac{p}{p-1}}$ and $d\lesssimε^{-\frac{2-p}{p-1}}$, respectively.

Problem

• stochastic optimization
• heavy-tailed noise
• quantum speedup
• query complexity
• gradient descent

Innovation

• quantum speedup
• heavy-tailed noise
• quantum mean estimation
• stochastic optimization
• quantum SGD