Quantitative Fluctuation Analysis for Continuous-Time Stochastic Gradient Descent via Malliavin Calculus

TL;DR

This paper establishes a quantitative central limit theorem for continuous-time stochastic gradient descent using Malliavin calculus, providing explicit convergence rates and insights into how learning rate affects convergence.

Contribution

It introduces a novel quantitative CLT for continuous-time SGD, deriving explicit convergence rates via Malliavin calculus tools, which was not previously available.

Findings

Convergence rate depends on the learning rate magnitude.

Smaller learning rates slow down convergence.

Numerical experiments confirm theoretical predictions.

Abstract

In this paper, we establish a Quantitative Central Limit Theorem ({\sc qclt}) for the Stochastic Gradient Descent in Continuous Time ({\sc sgdct}) algorithm, whose parameter updates are governed by a stochastic differential equation. We derive an explicit rate at which the {\sc sgdct} iterates converge, in the Wasserstein metric, to a critical point of the objective function. This rate is driven primarily by the magnitude of the learning rate: for a fixed convexity constant of the objective function, smaller learning rates lead to slower convergence. Our approach relies on tools from Malliavin calculus. In particular, we apply a second-order Poincar\'e inequality and obtain explicit bounds by estimating the first- and second-order Malliavin derivatives separately. Controlling the second-order derivative requires several delicate calculations and a careful sequence of decompositions in order to achieve sharp estimates. We complement the theoretical results with several numerical experiments that illustrate the predicted convergence behavior.