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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2406.08658 |
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| _version_ | 1866929384598798336 |
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| author | Vural, Nuri Mert Erdogdu, Murat A. |
| author_facet | Vural, Nuri Mert Erdogdu, Murat A. |
| contents | While it is commonly observed in practice that pruning networks to a certain level of sparsity can improve the quality of the features, a theoretical explanation of this phenomenon remains elusive. In this work, we investigate this by demonstrating that a broad class of statistical models can be optimally learned using pruned neural networks trained with gradient descent, in high-dimensions.
We consider learning both single-index and multi-index models of the form $y = σ^*(\boldsymbol{V}^{\top} \boldsymbol{x}) + ε$, where $σ^*$ is a degree-$p$ polynomial, and $\boldsymbol{V} \in \mathbbm{R}^{d \times r}$ with $r \ll d$, is the matrix containing relevant model directions. We assume that $\boldsymbol{V}$ satisfies a certain $\ell_q$-sparsity condition for matrices and show that pruning neural networks proportional to the sparsity level of $\boldsymbol{V}$ improves their sample complexity compared to unpruned networks. Furthermore, we establish Correlational Statistical Query (CSQ) lower bounds in this setting, which take the sparsity level of $\boldsymbol{V}$ into account. We show that if the sparsity level of $\boldsymbol{V}$ exceeds a certain threshold, training pruned networks with a gradient descent algorithm achieves the sample complexity suggested by the CSQ lower bound. In the same scenario, however, our results imply that basis-independent methods such as models trained via standard gradient descent initialized with rotationally invariant random weights can provably achieve only suboptimal sample complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pruning is Optimal for Learning Sparse Features in High-Dimensions Vural, Nuri Mert Erdogdu, Murat A. Machine Learning While it is commonly observed in practice that pruning networks to a certain level of sparsity can improve the quality of the features, a theoretical explanation of this phenomenon remains elusive. In this work, we investigate this by demonstrating that a broad class of statistical models can be optimally learned using pruned neural networks trained with gradient descent, in high-dimensions. We consider learning both single-index and multi-index models of the form $y = σ^*(\boldsymbol{V}^{\top} \boldsymbol{x}) + ε$, where $σ^*$ is a degree-$p$ polynomial, and $\boldsymbol{V} \in \mathbbm{R}^{d \times r}$ with $r \ll d$, is the matrix containing relevant model directions. We assume that $\boldsymbol{V}$ satisfies a certain $\ell_q$-sparsity condition for matrices and show that pruning neural networks proportional to the sparsity level of $\boldsymbol{V}$ improves their sample complexity compared to unpruned networks. Furthermore, we establish Correlational Statistical Query (CSQ) lower bounds in this setting, which take the sparsity level of $\boldsymbol{V}$ into account. We show that if the sparsity level of $\boldsymbol{V}$ exceeds a certain threshold, training pruned networks with a gradient descent algorithm achieves the sample complexity suggested by the CSQ lower bound. In the same scenario, however, our results imply that basis-independent methods such as models trained via standard gradient descent initialized with rotationally invariant random weights can provably achieve only suboptimal sample complexity. |
| title | Pruning is Optimal for Learning Sparse Features in High-Dimensions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2406.08658 |