InRank: Incremental Low-Rank Learning
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913181362814976 |
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| author | Zhao, Jiawei Zhang, Yifei Chen, Beidi Schäfer, Florian Anandkumar, Anima |
| author_facet | Zhao, Jiawei Zhang, Yifei Chen, Beidi Schäfer, Florian Anandkumar, Anima |
| contents | The theory of greedy low-rank learning (GLRL) aims to explain the impressive generalization capabilities of deep learning. It proves that stochastic gradient-based training implicitly regularizes neural networks towards low-rank solutions through a gradual increase of the rank during training. However, there is a gap between theory and practice since GLRL requires an infinitesimal initialization of the weights, which is not practical due to the fact that it is a saddle point. In this work, we remove the assumption of infinitesimal initialization by focusing on cumulative weight updates. We prove the cumulative weight updates follow an incremental low-rank trajectory for arbitrary orthogonal initialization of weights in a three-layer linear network. Empirically, we demonstrate that our theory holds on a broad range of neural networks (e.g., transformers) and standard training algorithms (e.g., SGD, Adam). However, existing training algorithms do not exploit the low-rank property to improve computational efficiency as the networks are not parameterized in low-rank. To remedy this, we design a new training algorithm Incremental Low-Rank Learning (InRank), which explicitly expresses cumulative weight updates as low-rank matrices while incrementally augmenting their ranks during training. We evaluate InRank on GPT-2, and our results indicate that InRank achieves comparable prediction performance as the full-rank counterpart while requiring at most 33% of the total ranks throughout training. We also propose an efficient version of InRank that achieves a reduction of 37% in total training time and 36% in model size when training GPT-medium on WikiText-103 from scratch. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11250 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | InRank: Incremental Low-Rank Learning Zhao, Jiawei Zhang, Yifei Chen, Beidi Schäfer, Florian Anandkumar, Anima Machine Learning The theory of greedy low-rank learning (GLRL) aims to explain the impressive generalization capabilities of deep learning. It proves that stochastic gradient-based training implicitly regularizes neural networks towards low-rank solutions through a gradual increase of the rank during training. However, there is a gap between theory and practice since GLRL requires an infinitesimal initialization of the weights, which is not practical due to the fact that it is a saddle point. In this work, we remove the assumption of infinitesimal initialization by focusing on cumulative weight updates. We prove the cumulative weight updates follow an incremental low-rank trajectory for arbitrary orthogonal initialization of weights in a three-layer linear network. Empirically, we demonstrate that our theory holds on a broad range of neural networks (e.g., transformers) and standard training algorithms (e.g., SGD, Adam). However, existing training algorithms do not exploit the low-rank property to improve computational efficiency as the networks are not parameterized in low-rank. To remedy this, we design a new training algorithm Incremental Low-Rank Learning (InRank), which explicitly expresses cumulative weight updates as low-rank matrices while incrementally augmenting their ranks during training. We evaluate InRank on GPT-2, and our results indicate that InRank achieves comparable prediction performance as the full-rank counterpart while requiring at most 33% of the total ranks throughout training. We also propose an efficient version of InRank that achieves a reduction of 37% in total training time and 36% in model size when training GPT-medium on WikiText-103 from scratch. |
| title | InRank: Incremental Low-Rank Learning |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2306.11250 |