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.

翻译：我们考虑在给定随机梯度或acles的量子访问权限下最小化连续函数的问题。针对Lipschitz凸函数这一特例，我们提出了两种新方法。每种方法在维度与精度之间取得的权衡在经典计算中已被证明无法实现，同时我们证明其中一种方法在低维场景中具有渐近最优性。此外，我们为光滑非凸函数的关键点计算提供了量子算法，其收敛速率在经典框架下尚未被认知可实现。为获得这些结果，我们基于Cornelissen等人2022年的量子多变量均值估计结果，并发展了一种具有独立价值的一般性量子方差缩减技术。