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.

We study an approach to learning pruning masks by optimizing the expected loss of stochastic pruning masks, i.e., masks which zero out each weight independently with some weight-specific probability. We analyze the training dynamics of the induced stochastic predictor in the setting of linear regression, and observe a data-adaptive L1 regularization term, in contrast to the dataadaptive L2 regularization term known to underlie dropout in linear regression. We also observe a preference to prune weights that are less well-aligned with the data labels. We evaluate probabilistic fine-tuning for optimizing stochastic pruning masks for neural networks, starting from masks produced by several baselines. In each case, we see improvements in test error over baselines, even after we threshold fine-tuned stochastic pruning masks. Finally, since a stochastic pruning mask induces a stochastic neural network, we consider training the weights and/or pruning probabilities simultaneously to minimize a PAC-Bayes bound on generalization error. Using data-dependent priors, we obtain a selfbounded learning algorithm with strong performance and numerically tight bounds. In the linear model, we show that a PAC-Bayes generalization error bound is controlled by the magnitude of the change in feature alignment between the 'prior' and 'posterior' data.

Although deep Convolutional Neural Network (CNN) has shown better performance in various computer vision tasks, its application is restricted by a significant increase in storage and computation. Among CNN simplification techniques, parameter pruning is a promising approach which aims at reducing the number of weights of various layers without intensively reducing the original accuracy. In this paper, we propose a novel progressive parameter pruning method, named Structured Probabilistic Pruning (SPP), which effectively prunes weights of convolutional layers in a probabilistic manner. Specifically, unlike existing deterministic pruning approaches, where unimportant weights are permanently eliminated, SPP introduces a pruning probability for each weight, and pruning is guided by sampling from the pruning probabilities. A mechanism is designed to increase and decrease pruning probabilities based on importance criteria for the training process. Experiments show that, with 4x speedup, SPP can accelerate AlexNet with only 0.3% loss of top-5 accuracy and VGG-16 with 0.8% loss of top-5 accuracy in ImageNet classification. Moreover, SPP can be directly applied to accelerate multi-branch CNN networks, such as ResNet, without specific adaptations. Our 2x speedup ResNet-50 only suffers 0.8% loss of top-5 accuracy on ImageNet. We further prove the effectiveness of our method on transfer learning task on Flower-102 dataset with AlexNet.