Greedy Pruning with Group Lasso Provably Generalizes for Matrix Sensing and Neural Networks with Quadratic Activations

Abstract

Pruning schemes have been widely used in practice to reduce the complexity of trained models with a massive number of parameters. Several practical studies have shown that pruning an overparameterized model and fine-tuning generalizes well to new samples. Although the above pipeline, which we refer to as pruning + fine-tuning, has been extremely successful in lowering the complexity of trained models, there is very little known about the theory behind this success. In this paper we address this issue by investigating the pruning + fine-tuning framework on the overparameterized matrix sensing problem, with the ground truth denoted \(U_\star \in \mathbb{R}^{d \times r}\) and the overparameterized model \(U \in \mathbb{R}^{d \times k}\) with \(k \gg r\). We study the approximate local minima of the empirical mean square error, augmented with a smooth version of a group Lasso regularizer, \(\sum_{i=1}^k \| U e_i \|_2\) and show that pruning the low \(\ell_2\)-norm columns results in a solution \(U_{\text{prune}}\) which has the minimum number of columns \(r\), yet is close to the ground truth in training loss. Initializing the subsequent fine-tuning phase from \(U_{\text{prune}}\), the resulting solution converges linearly to a generalization error of \(O(\sqrt{rd/n})\) ignoring lower order terms, which is statistically optimal. While our analysis provides insights into the role of regularization in pruning, we also show that running gradient descent in the absence of regularization results in models which {are not suitable for greedy pruning}, i.e., many columns could have their \(\ell_2\) norm comparable to that of the maximum. Lastly, we extend our results for the training and pruning of two-layer neural networks with quadratic activation functions. Our results provide the first rigorous insights on why greedy pruning + fine-tuning leads to smaller models which also generalize well.