# Functional Central Limit Theorem for Stochastic Gradient Descent

arXiv: https://arxiv.org/abs/2602.15538 · HTML: https://mlsanity.com/paper/2602.15538/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 30 references that are in the map.
- First posted on arXiv 17 Feb 2026.

## Builds on (most cited first, 4 of 4)

- [Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations](https://mlsanity.com/paper/1811.01558/) (2018)
- [Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)](https://mlsanity.com/paper/1306.2119/) (2013)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Limit Theorems for Stochastic Gradient Descent with Infinite Variance](https://mlsanity.com/paper/2410.16340/) (2024)

## Built on by (most cited first, 1 of 1)

- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)

## Abstract (arXiv)

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.