发表机构
The Chinese University of Hong Kong; Westlake University; Tencent Quantum Laboratory（香港中文大学; 西湖大学; 腾讯量子实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究重尾噪声下随机优化，提出量子均值估计器，基于此构建量子归一化和投影随机梯度下降方法，在低维情况下，相比经典方法，能以更少查询次数找到近似平稳点或\(\epsilon\)-最优解。
AI 中文摘要
我们研究了重尾梯度噪声下的随机优化。首先为多元重尾随机变量提出了一种新颖的量子均值估计器，在低维情况下实现了比最优经典估计器更低的查询复杂度。通过应用广义多级蒙特卡罗技术进一步开发了无偏量子均值估计器。证明了量子下界，表明当随机向量维度\(d\)较小时可视为常数时，我们的量子估计器在对数因子范围内是最优的。基于这些估计器，提出了量子归一化随机梯度下降方法（\(\texttt{QNSGD}\)）和量子投影随机梯度下降方法（\(\texttt{QPSGD}\)），并给出了它们在查询次数上的优势，改进了经典下界。
英文摘要
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.
Comments56 pages
