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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.23362 |
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| _version_ | 1866917370459586560 |
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| author | Carrasco, Pablo Muñoz, Gonzalo |
| author_facet | Carrasco, Pablo Muñoz, Gonzalo |
| contents | The non-convex nature of trained neural networks has created significant obstacles in their incorporation into optimization models. In this context, Anderson et al. (2020) provided a framework to obtain the convex hull of the graph of a piecewise linear convex activation function composed with an affine function; this effectively convexifies activations such as the ReLU together with the affine transformation that precedes it. In this article, we contribute to this line of work by developing a recursive formula that yields a tight convexification for the composition of an activation with an affine function for a wide scope of activation functions, namely, convex or ``S-shaped". Our approach can be used to efficiently compute separating hyperplanes or determine that none exists in various settings, including non-polyhedral cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tightening convex relaxations of trained neural networks: a unified approach for convex and S-shaped activations Carrasco, Pablo Muñoz, Gonzalo Optimization and Control Machine Learning The non-convex nature of trained neural networks has created significant obstacles in their incorporation into optimization models. In this context, Anderson et al. (2020) provided a framework to obtain the convex hull of the graph of a piecewise linear convex activation function composed with an affine function; this effectively convexifies activations such as the ReLU together with the affine transformation that precedes it. In this article, we contribute to this line of work by developing a recursive formula that yields a tight convexification for the composition of an activation with an affine function for a wide scope of activation functions, namely, convex or ``S-shaped". Our approach can be used to efficiently compute separating hyperplanes or determine that none exists in various settings, including non-polyhedral cases. |
| title | Tightening convex relaxations of trained neural networks: a unified approach for convex and S-shaped activations |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2410.23362 |