Pruning is Optimal for Learning Sparse Features in High-Dimensions

• URL: http://arxiv.org/abs/2406.08658v1
• Date: Wed, 12 Jun 2024 21:43:12 GMT
• Title: Pruning is Optimal for Learning Sparse Features in High-Dimensions
• Authors: Nuri Mert Vural, Murat A. Erdogdu,
• Abstract summary: We show that a class of statistical models can be optimally learned using pruned neural networks trained with gradient descent. We show that pruning neural networks proportional to the sparsity level of $boldsymbolV$ improves their sample complexity compared to unpruned networks.
• Score: 15.967123173054535
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $y = \sigma^*(\boldsymbol{V}^{\top} \boldsymbol{x}) + \epsilon$, where $\sigma^*$ 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.

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