Functional Central Limit Theorem for Stochastic Gradient Descent

Kessang Flamand; Victor-Emmanuel Brunel

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arXiv · 2602.15538

Functional Central Limit Theorem for Stochastic Gradient Descent

Kessang Flamand·Victor-Emmanuel Brunel

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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.

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Kessang Flamand, Victor-Emmanuel Brunel. 2026-02-17. Functional Central Limit Theorem for Stochastic Gradient Descent. https://arxiv.org/abs/2602.15538

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