Quantum speedups for stochastic optimization
Aaron Sidford, Chenyi Zhang
TL;DR
This work introduces quantum algorithms with oracle access to stochastic gradients that surpass classical rates for dimensionally favorable stochastic optimization. By developing a quantum variance reduction framework that combines quantum mean estimation with multilevel Monte Carlo, the authors obtain unbiased gradient estimates and accelerated convergence for both convex and certain nonconvex objectives. The results include two SCO quantum algorithms with improved ε dependences and a pair of nonconvex optimizers, all backed by lower bounds showing optimality in low dimensions. The approach leverages Gaussian smoothing and quantum auxiliary techniques to achieve quantum speedups while carefully handling bias and variance, with practical considerations discussed for implementing quantum oracles. Overall, the paper provides a concrete path to quantum advantages in stochastic optimization in low dimensional regimes and establishes foundational limits for higher dimensional settings.
Abstract
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.