Functional Central Limit Theorem for Stochastic Gradient Descent

TL;DR

This paper establishes a functional central limit theorem for stochastic gradient descent, describing the asymptotic fluctuations of the algorithm's trajectory around the minimizer, including non-smooth cases like the geometric median.

Contribution

It introduces a novel functional CLT for SGD trajectories, capturing their temporal fluctuation structure under mild conditions, extending classical results to non-smooth settings.

Findings

Provides a diffusion limit for SGD trajectories

Applies to non-smooth convex optimization problems

Characterizes long-term fluctuations of SGD around the minimizer

Abstract

We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled trajectory. Our result characterizes the long-term fluctuations of the algorithm around the minimizer by providing a diffusion limit for the trajectory. In contrast with classical central limit theorems for the last iterate or Polyak-Ruppert averages, this functional result captures the temporal structure of the fluctuations and applies to non-smooth settings such as robust location estimation, including the geometric median.