# Functional Central Limit Theorem for Stochastic Gradient Descent

**Authors:** Kessang Flamand, Victor-Emmanuel Brunel  
**arXiv:** [2602.15538](https://arxiv.org/abs/2602.15538)  
**Date:** 2026-02-17  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2602.15538v1/

## Key points

- Investigating asymptotic behavior of trajectory generated by stochastic gradient descent algorithm for convex objective function
- Establishing functional central limit theorem under mild regularity conditions
- Shedding light on long-term fluctuations around minimizer and providing diffusion limit for trajectory
- Extending findings to non-smooth scenarios including robust location estimation and geometric median calculations
- Convergence in distribution towards Gaussian distribution with asymptotic variance ∆ as n approaches infinity and θn represents measurable minimizer of empirical risk in d-dimensional space
- Comparing asymptotic variance ∆ to matrix Σ to gain insights into their relationship and properties
- Introducing asymptotic stochastic process Y as centered Gaussian process with estimates for its norm on bounded intervals
- Establishing bounds on expected supremum of Yt within specified time intervals T through Theorem 4 and Remark 2
- Highlighting similarities between Y and Brownian motion processes when appropriately rescaled by diffusion coefficients

## Summary

We investigate the asymptotic behavior of the trajectory generated by the stochastic gradient descent algorithm when applied to a convex objective function. By assuming mild regularity conditions, we establish a functional central limit theorem for the appropriately rescaled trajectory. This result sheds light on the long-term fluctuations exhibited by the algorithm around the minimizer and provides a diffusion limit for the trajectory. Our findings differ from traditional central limit theorems that focus on specific iterates or averages and extend to non-smooth scenarios like robust location estimation. This includes cases involving geometric median calculations. Furthermore, our analysis reveals that as n approaches infinity and θn represents a measurable minimizer of empirical risk in d-dimensional space, there is convergence in distribution towards a Gaussian distribution with an asymptotic variance denoted by ∆. Comparing this variance to another matrix Σ provides insights into their relationship and respective properties. Additionally, we introduce an asymptotic stochastic process Y characterized as a centered Gaussian process with estimates for its norm on bounded intervals. Through Theorem 4 and Remark 2, we establish bounds on the expected supremum of Yt within specified time intervals T. These results highlight similarities between Y and Brownian motion processes when appropriately rescaled by their respective diffusion coefficients. Ultimately, our study contributes valuable insights into understanding the behavior and fluctuations of algorithms like stochastic gradient descent in optimization problems across various settings.