Hongru Yang

Hongru Yang, Zhangyang Wang

Motivated by both theory and practice, we study how random pruning of the weights affects a neural network's neural tangent kernel (NTK). In particular, this work establishes an equivalence of the NTKs between a fully-connected neural network and its randomly pruned version. The equivalence is established under two cases. The first main result studies the infinite-width asymptotic. It is shown that given a pruning probability, for fully-connected neural networks with the weights randomly pruned at the initialization, as the width of each layer grows to infinity sequentially, the NTK of the pruned neural network converges to the limiting NTK of the original network with some extra scaling. If the network weights are rescaled appropriately after pruning, this extra scaling can be removed. The second main result considers the finite-width case. It is shown that to ensure the NTK's closeness to the limit, the dependence of width on the sparsity parameter is asymptotically linear, as the NTK's gap to its limit goes down to zero. Moreover, if the pruning probability is set to zero (i.e., no pruning), the bound on the required width matches the bound for fully-connected neural networks in previous works up to logarithmic factors. The proof of this result requires developing a novel analysis of a network structure which we called \textit{mask-induced pseudo-networks}. Experiments are provided to evaluate our results.

Hongru Yang, Ziyu Jiang, Ruizhe Zhang et al.

We study training one-hidden-layer ReLU networks in the neural tangent kernel (NTK) regime, where the networks' biases are initialized to some constant rather than zero. We prove that under such initialization, the neural network will have sparse activation throughout the entire training process, which enables fast training procedures via some sophisticated computational methods. With such initialization, we show that the neural networks possess a different limiting kernel which we call \textit{bias-generalized NTK}, and we study various properties of the neural networks with this new kernel. We first characterize the gradient descent dynamics. In particular, we show that the network in this case can achieve as fast convergence as the dense network, as opposed to the previous work suggesting that the sparse networks converge slower. In addition, our result improves the previous required width to ensure convergence. Secondly, we study the networks' generalization: we show a width-sparsity dependence, which yields a sparsity-dependent Rademacher complexity and generalization bound. To our knowledge, this is the first sparsity-dependent generalization result via Rademacher complexity. Lastly, we study the smallest eigenvalue of this new kernel. We identify a data-dependent region where we can derive a much sharper lower bound on the NTK's smallest eigenvalue than the worst-case bound previously known. This can lead to improvement in the generalization bound.

Hongru Yang, Yingbin Liang, Xiaojie Guo et al.

It has been observed in practice that applying pruning-at-initialization methods to neural networks and training the sparsified networks can not only retain the testing performance of the original dense models, but also sometimes even slightly boost the generalization performance. Theoretical understanding for such experimental observations are yet to be developed. This work makes the first attempt to study how different pruning fractions affect the model's gradient descent dynamics and generalization. Specifically, this work considers a classification task for overparameterized two-layer neural networks, where the network is randomly pruned according to different rates at the initialization. It is shown that as long as the pruning fraction is below a certain threshold, gradient descent can drive the training loss toward zero and the network exhibits good generalization performance. More surprisingly, the generalization bound gets better as the pruning fraction gets larger. To complement this positive result, this work further shows a negative result: there exists a large pruning fraction such that while gradient descent is still able to drive the training loss toward zero (by memorizing noise), the generalization performance is no better than random guessing. This further suggests that pruning can change the feature learning process, which leads to the performance drop of the pruned neural network.

Hongru Yang, Bhavya Kailkhura, Zhangyang Wang et al.

Understanding the training dynamics of transformers is important to explain the impressive capabilities behind large language models. In this work, we study the dynamics of training a shallow transformer on a task of recognizing co-occurrence of two designated words. In the literature of studying training dynamics of transformers, several simplifications are commonly adopted such as weight reparameterization, attention linearization, special initialization, and lazy regime. In contrast, we analyze the gradient flow dynamics of simultaneously training three attention matrices and a linear MLP layer from random initialization, and provide a framework of analyzing such dynamics via a coupled dynamical system. We establish near minimum loss and characterize the attention model after training. We discover that gradient flow serves as an inherent mechanism that naturally divide the training process into two phases. In Phase 1, the linear MLP quickly aligns with the two target signals for correct classification, whereas the softmax attention remains almost unchanged. In Phase 2, the attention matrices and the MLP evolve jointly to enlarge the classification margin and reduce the loss to a near minimum value. Technically, we prove a novel property of the gradient flow, termed \textit{automatic balancing of gradients}, which enables the loss values of different samples to decrease almost at the same rate and further facilitates the proof of near minimum training loss. We also conduct experiments to verify our theoretical results.