Sharper Analysis of Sparsely Activated Wide Neural Networks with Trainable Biases

Hongru Yang, Ziyu Jiang, Ruizhe Zhang, Zhangyang Wang, Yingbin Liang

Keywords: Convergence analysis, sparse activation, neural tangent kernel, Rademacher complexity, generalization bound

TL;DR: We study convergence and generalization of training one-hidden-layer neural networks with sparse activation.

Abstract: This work studies training one-hidden-layer overparameterized ReLU networks via gradient descent in the neural tangent kernel (NTK) regime, where, differently from the previous works, the networks' biases are trainable and are initialized to some constant rather than zero. The tantalizing benefit of such initialization is that the neural network will provably have sparse activation pattern before, during and after training, which can enable fast training procedures and, therefore, reduce the training cost. The first set of results of this work characterize the convergence of the network's gradient descent dynamics. The required width is provided to ensure gradient descent can drive the training error towards zero in linear rate. The contribution over previous work is that not only the bias is allowed to be updated by gradient descent under our setting but also a finer analysis is given such that the required width to ensure the network's closeness to its NTK is improved. Secondly, the networks' generalization bound after training is provided. A width-sparsity dependence is presented which yields sparsity-dependent localized Rademacher complexity and same-as-previous (ignoring logarithmic factors) generalization bound. Up to our knowledge, this is the first sparsity-dependent generalization result via localized Rademacher complexity. As a by-product, if the bias initialization is chosen to be zero, the width requirement improves the previous bound for the shallow networks' generalization. Lastly, since the generalization bound has dependence on the smallest eigenvalue of the limiting NTK and the bounds from previous works yield vacuous generalization, this work further studies the least eigenvalue of the limiting NTK. Surprisingly, while it is not shown that trainable biases are necessary, trainable bias helps to identify a nice data-dependent region where a much finer analysis of the NTK's smallest eigenvalue can be conducted, which leads to a much sharper lower bound than the previously known worst-case bound and, consequently, a non-vacuous generalization bound. Experimental evaluation is provided to evaluate our results.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics

Submission Guidelines: Yes

Please Choose The Closest Area That Your Submission Falls Into: Theory (eg, control theory, learning theory, algorithmic game theory)