Statistical Guarantees for High-Dimensional Stochastic Gradient Descent Article Swipe
J. Jenny Li
,
Zhipeng Lou
,
Johannes Schmidt-Hieber
,
Wei Biao Wu
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2510.12013
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2510.12013
Stochastic Gradient Descent (SGD) and its Ruppert-Polyak averaged variant (ASGD) lie at the heart of modern large-scale learning, yet their theoretical properties in high-dimensional settings are rarely understood. In this paper, we provide rigorous statistical guarantees for constant learning-rate SGD and ASGD in high-dimensional regimes. Our key innovation is to transfer powerful tools from high-dimensional time series to online learning. Specifically, by viewing SGD as a nonlinear autoregressive process and adapting existing coupling techniques, we prove the geometric-moment contraction of high-dimensional SGD for constant learning rates, thereby establishing asymptotic stationarity of the iterates. Building on this, we derive the $q$-th moment convergence of SGD and ASGD for any $q\ge2$ in general $\ell^s$-norms, and, in particular, the $\ell^{\infty}$-norm that is frequently adopted in high-dimensional sparse or structured models. Furthermore, we provide sharp high-probability concentration analysis which entails the probabilistic bound of high-dimensional ASGD. Beyond closing a critical gap in SGD theory, our proposed framework offers a novel toolkit for analyzing a broad class of high-dimensional learning algorithms.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2510.12013
- https://arxiv.org/pdf/2510.12013
- OA Status
- green
- OpenAlex ID
- https://openalex.org/W4415257483
All OpenAlex metadata
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https://openalex.org/W4415257483Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2510.12013Digital Object Identifier
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Statistical Guarantees for High-Dimensional Stochastic Gradient DescentWork title
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preprintOpenAlex work type
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enPrimary language
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2025Year of publication
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2025-10-13Full publication date if available
- Authors
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J. Jenny Li, Zhipeng Lou, Johannes Schmidt-Hieber, Wei Biao WuList of authors in order
- Landing page
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https://arxiv.org/abs/2510.12013Publisher landing page
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https://arxiv.org/pdf/2510.12013Direct link to full text PDF
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2510.12013Direct OA link when available
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0Total citation count in OpenAlex
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