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This research demonstrates that quantum algorithms can achieve superior query complexity for stochastic optimization problems involving heavy-tailed noise, particularly in low-dimensional regimes where classical methods face inherent limitations. By introducing novel quantum mean estimators and proving corresponding lower bounds, the study establishes a theoretical advantage over standard gradient descent approaches under specific dimensional constraints. The work advances our understanding of how quantum computing can address complex statistical challenges that are computationally expensive for classical systems.
Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise
arXiv:2607.25492v1 Announce Type: new 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 $\epsilon$-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.
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 $\epsilon$-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.
Machine Learning (cs.LG)
arXiv:2607.25492 [cs.LG]
(or arXiv:2607.25492v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.25492 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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