Generalization Guarantees of Gradient Descent for Multi-Layer Neural Networks Article Swipe

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Puyu Wang , Yunwen Lei , Di Wang , Yiming Ying , Ding‐Xuan Zhou · YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2305.16891

Recently, significant progress has been made in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on one-hidden-layer NNs and has not addressed the impact of different network scaling parameters. In this paper, we greatly extend the previous work \cite{lei2022stability,richards2021stability} by conducting a comprehensive stability and generalization analysis of GD for multi-layer NNs. For two-layer NNs, our results are established under general network scaling parameters, relaxing previous conditions. In the case of three-layer NNs, our technical contribution lies in demonstrating its nearly co-coercive property by utilizing a novel induction strategy that thoroughly explores the effects of over-parameterization. As a direct application of our general findings, we derive the excess risk rate of $O(1/\sqrt{n})$ for GD algorithms in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for under-parameterized and over-parameterized NNs trained by GD to attain the desired risk rate of $O(1/\sqrt{n})$. Moreover, we demonstrate that as the scaling parameter increases or the network complexity decreases, less over-parameterization is required for GD to achieve the desired error rates. Additionally, under a low-noise condition, we obtain a fast risk rate of $O(1/n)$ for GD in both two-layer and three-layer NNs.

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preprint

Language

en

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http://arxiv.org/abs/2305.16891

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https://arxiv.org/pdf/2305.16891

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green

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10

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https://openalex.org/W4378718232

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https://openalex.org/W4378718232

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DOI

https://doi.org/10.48550/arxiv.2305.16891

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Title

Generalization Guarantees of Gradient Descent for Multi-Layer Neural Networks

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Type

preprint

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Language

en

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Publication year

2023

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2023-05-26

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Authors

Puyu Wang, Yunwen Lei, Di Wang, Yiming Ying, Ding‐Xuan Zhou

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https://arxiv.org/abs/2305.16891

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https://arxiv.org/pdf/2305.16891

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Yes

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green

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https://arxiv.org/pdf/2305.16891

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Concepts

Parameterized complexity, Generalization, Gradient descent, Scaling, Artificial neural network, Layer (electronics), Stability (learning theory), Computer science, Algorithm, Applied mathematics, Mathematics, Artificial intelligence, Mathematical analysis, Machine learning, Materials science, Geometry, Composite material

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10

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