Convergence of Stochastic Gradient Descent with mini-batching and infinite variance
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915991575855104 |
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| author | Glowacki, Bartosz Kulik, Rafal Soulier, Philippe |
| author_facet | Glowacki, Bartosz Kulik, Rafal Soulier, Philippe |
| contents | Stochastic gradient descent (SGD) with mini-batching is a standard tool in large-scale optimization, yet its theoretical properties under heavy-tailed gradient noise remain largely unexplored. In this paper we study SGD with increasing batch sizes when the gradient noise belongs to the domain of attraction of an $α$-stable law with $α\in(1,2)$. Building on existing results for the finite-variance regime and for heavy-tailed SGD without batching, we establish three main results. First, we derive $L^p$ moment bounds for the SGD error and show that increasing batch sizes lead to faster convergence rates. In particular, batching enables convergence in probability even for a constant stepsize. Second, we prove that the properly normalized SGD iterates converge in distribution to the stationary law of an Ornstein-Uhlenbeck process driven by an $α$-stable Lévy process. Third, for Polyak-Ruppert averaging we obtain a stable limit theorem with a normalization that explicitly depends on the batch-size schedule. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07184 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence of Stochastic Gradient Descent with mini-batching and infinite variance Glowacki, Bartosz Kulik, Rafal Soulier, Philippe Probability Stochastic gradient descent (SGD) with mini-batching is a standard tool in large-scale optimization, yet its theoretical properties under heavy-tailed gradient noise remain largely unexplored. In this paper we study SGD with increasing batch sizes when the gradient noise belongs to the domain of attraction of an $α$-stable law with $α\in(1,2)$. Building on existing results for the finite-variance regime and for heavy-tailed SGD without batching, we establish three main results. First, we derive $L^p$ moment bounds for the SGD error and show that increasing batch sizes lead to faster convergence rates. In particular, batching enables convergence in probability even for a constant stepsize. Second, we prove that the properly normalized SGD iterates converge in distribution to the stationary law of an Ornstein-Uhlenbeck process driven by an $α$-stable Lévy process. Third, for Polyak-Ruppert averaging we obtain a stable limit theorem with a normalization that explicitly depends on the batch-size schedule. |
| title | Convergence of Stochastic Gradient Descent with mini-batching and infinite variance |
| topic | Probability |
| url | https://arxiv.org/abs/2605.07184 |