$ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916849393860608 |
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| author | Yang, Jiang Zhao, Yuxiang Zhu, Quanhui |
| author_facet | Yang, Jiang Zhao, Yuxiang Zhu, Quanhui |
| contents | Understanding the training dynamics of deep neural networks (DNNs), particularly how they evolve low-dimensional features from high-dimensional data, remains a central challenge in deep learning theory. In this work, we introduce the concept of $ε$-rank, a novel metric quantifying the effective feature of neuron functions in the terminal hidden layer. Through extensive experiments across diverse tasks, we observe a universal staircase phenomenon: during training process implemented by the standard stochastic gradient descent methods, the decline of the loss function is accompanied by an increase in the $ε$-rank and exhibits a staircase pattern. Theoretically, we rigorously prove a negative correlation between the loss lower bound and $ε$-rank, demonstrating that a high $ε$-rank is essential for significant loss reduction. Moreover, numerical evidences show that within the same deep neural network, the $ε$-rank of the subsequent hidden layer is higher than that of the previous hidden layer. Based on these observations, to eliminate the staircase phenomenon, we propose a novel pre-training strategy on the initial hidden layer that elevates the $ε$-rank of the terminal hidden layer. Numerical experiments validate its effectiveness in reducing training time and improving accuracy across various tasks. Therefore, the newly introduced concept of $ε$-rank is a computable quantity that serves as an intrinsic effective metric characteristic for deep neural networks, providing a novel perspective for understanding the training dynamics of neural networks and offering a theoretical foundation for designing efficient training strategies in practical applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Yang, Jiang Zhao, Yuxiang Zhu, Quanhui Machine Learning Numerical Analysis Understanding the training dynamics of deep neural networks (DNNs), particularly how they evolve low-dimensional features from high-dimensional data, remains a central challenge in deep learning theory. In this work, we introduce the concept of $ε$-rank, a novel metric quantifying the effective feature of neuron functions in the terminal hidden layer. Through extensive experiments across diverse tasks, we observe a universal staircase phenomenon: during training process implemented by the standard stochastic gradient descent methods, the decline of the loss function is accompanied by an increase in the $ε$-rank and exhibits a staircase pattern. Theoretically, we rigorously prove a negative correlation between the loss lower bound and $ε$-rank, demonstrating that a high $ε$-rank is essential for significant loss reduction. Moreover, numerical evidences show that within the same deep neural network, the $ε$-rank of the subsequent hidden layer is higher than that of the previous hidden layer. Based on these observations, to eliminate the staircase phenomenon, we propose a novel pre-training strategy on the initial hidden layer that elevates the $ε$-rank of the terminal hidden layer. Numerical experiments validate its effectiveness in reducing training time and improving accuracy across various tasks. Therefore, the newly introduced concept of $ε$-rank is a computable quantity that serves as an intrinsic effective metric characteristic for deep neural networks, providing a novel perspective for understanding the training dynamics of neural networks and offering a theoretical foundation for designing efficient training strategies in practical applications. |
| title | $ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2412.05144 |