Functional Central Limit Theorem for Stochastic Gradient Descent
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918342924697600 |
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| author | Flamand, Kessang Brunel, Victor-Emmanuel |
| author_facet | Flamand, Kessang Brunel, Victor-Emmanuel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15538 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Functional Central Limit Theorem for Stochastic Gradient Descent Flamand, Kessang Brunel, Victor-Emmanuel Machine Learning Optimization and Control 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. |
| title | Functional Central Limit Theorem for Stochastic Gradient Descent |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2602.15538 |