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This paper investigates stochastic optimization under heavy-tailed gradient noise, introducing a quantum mean estimator that outperforms classical methods in low-dimensional settings. The authors establish optimality results for their quantum estimators and derive dimension-dependent lower bounds that reveal unavoidable complexity in the low-dimensional regime. They also present two quantum algorithms, QNSGD and QPSGD, which significantly reduce query complexity for finding stationary points in both convex and nonconvex optimization problems compared to classical benchmarks.
Quantum estimators can dramatically reduce query complexity in stochastic optimization, achieving optimal performance even in low-dimensional settings.
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 $\epsilon$-stationary point using $\tilde{\mathcal{O}}\big(\sqrt d\,\epsilon^{-\frac{5p-4}{2p-2}}\big)$ 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 $\epsilon$-optimal solution using $\tilde{\mathcal{O}}\big(\sqrt d\,\epsilon^{-\frac{3p-2}{2p-2}}+\epsilon^{-2}\big)$ queries in expectation. These sharper bounds improve upon the classical lower bounds $\Omega\big(\epsilon^{-\frac{3p-2}{p-1}}\big)$ for nonconvex problems and $\Omega\big(\epsilon^{-\frac{p}{p-1}}\big)$ for convex problems in the low-dimensional regimes $d\lesssim\epsilon^{-\frac{p}{p-1}}$ and $d\lesssim\epsilon^{-\frac{2-p}{p-1}}$, respectively.
