Isotropic Noise in Stochastic and Quantum Convex Optimization

TL;DR

This paper studies stochastic convex optimization with isotropic noise, proposing an improved algorithm and quantum method that enhance convergence rates and establish new theoretical bounds in high-dimensional settings.

Contribution

It introduces an algorithm tailored for isotropic noise in stochastic convex optimization and develops a quantum isotropifier to improve quantum sampling efficiency.

Findings

Improved convergence rates for isotropic noise in stochastic optimization.

New quantum algorithm for unbiased isotropic error estimation.

Matching lower bounds up to polylogarithmic factors.

Abstract

We consider the problem of minimizing a $d$-dimensional Lipschitz convex function using a stochastic gradient oracle. We introduce and motivate a setting where the noise of the stochastic gradient is isotropic in that it is bounded in every direction with high probability. We then develop an algorithm for this setting which improves upon prior results by a factor of $d$ in certain regimes, and as a corollary, achieves a new state-of-the-art complexity for sub-exponential noise. We give matching lower bounds (up to polylogarithmic factors) for both results. Additionally, we develop an efficient quantum isotropifier, a quantum algorithm which converts a variance-bounded quantum sampling oracle into one that outputs an unbiased estimate with isotropic error. Combining our results, we obtain improved dimension-dependent rates for quantum stochastic convex optimization.