Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem

TL;DR

This paper establishes a central limit theorem for stochastic gradient descent in continuous time, providing insights into its convergence behavior for both convex and non-convex functions.

Contribution

It introduces a CLT for SGD in continuous time, extending understanding of its asymptotic distribution and convergence rates in complex settings.

Findings

Proves a CLT for strongly convex functions.

Establishes an $L^{p}$ convergence rate in the strongly convex case.

Extends analysis to certain non-convex functions.

Abstract

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. The parameter updates occur in continuous time and satisfy a stochastic differential equation. This paper analyzes the asymptotic convergence rate of the SGDCT algorithm by proving a central limit theorem (CLT) for strongly convex objective functions and, under slightly stronger conditions, for non-convex objective functions as well. An $L^{p}$ convergence rate is also proven for the algorithm in the strongly convex case. The mathematical analysis lies at the intersection of stochastic analysis and statistical learning.