We are interested in exploring the possibility and benefits of structure learning for deep models. As the first step, this paper investigates the matter for Restricted Boltzmann Machines (RBMs). We conduct the study with Replicated Softmax, a variant of RBMs for unsupervised text analysis. We present a method for learning what we call Sparse Boltzmann Machines, where each hidden unit is connected to a subset of the visible units instead of all of them. Empirical results show that the method yields models with significantly improved model fit and interpretability as compared with RBMs where each hidden unit is connected to all visible units.
Network pruning is also a potential way to optimize the structure of a neural network. Biased weight decay was the early approach to pruning. Later, Optimal Brain Damage and Optimal Brain Surgeon suggested that magnitude-based pruning may not be the best strategies and they proposed pruning methods based on the Hessian of the loss function. With respect to deep neural networks, @cite_2 proposed to compress a network through a three-step process: train, prune connections, and retrain. We call it redundancy pruning. In contrast, @cite_1 proposed to prune redundant neurons directly. They all reduced the number of parameters vastly with slight or even no performance loss. The drawback of network pruning is that the original networks should be large enough and hence some computation would be wasted on those unnecessary parameters during pre-training.
@cite_1 @cite_2
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To reduce the significant redundancy in deep Convolutional Neural Networks (CNNs), most existing methods prune neurons by only considering statistics of an individual layer or two consecutive layers (e.g., prune one layer to minimize the reconstruction error of the next layer), ignoring the effect of error propagation in deep networks. In contrast, we argue that it is essential to prune neurons in the entire neuron network jointly based on a unified goal: minimizing the reconstruction error of important responses in the "final response layer" (FRL), which is the second-to-last layer before classification, for a pruned network to retrain its predictive power. Specifically, we apply feature ranking techniques to measure the importance of each neuron in the FRL, and formulate network pruning as a binary integer optimization problem and derive a closed-form solution to it for pruning neurons in earlier layers. Based on our theoretical analysis, we propose the Neuron Importance Score Propagation (NISP) algorithm to propagate the importance scores of final responses to every neuron in the network. The CNN is pruned by removing neurons with least importance, and then fine-tuned to retain its predictive power. NISP is evaluated on several datasets with multiple CNN models and demonstrated to achieve significant acceleration and compression with negligible accuracy loss. To reduce the significant redundancy in deep Convolutional Neural Networks (CNNs), most existing methods prune neurons by only considering the statistics of an individual layer or two consecutive layers (e.g., prune one layer to minimize the reconstruction error of the next layer), ignoring the effect of error propagation in deep networks. In contrast, we argue that for a pruned network to retain its predictive power, it is essential to prune neurons in the entire neuron network jointly based on a unified goal: minimizing the reconstruction error of important responses in the "final response layer" (FRL), which is the second-to-last layer before classification. Specifically, we apply feature ranking techniques to measure the importance of each neuron in the FRL, formulate network pruning as a binary integer optimization problem, and derive a closed-form solution to it for pruning neurons in earlier layers. Based on our theoretical analysis, we propose the Neuron Importance Score Propagation (NISP) algorithm to propagate the importance scores of final responses to every neuron in the network. The CNN is pruned by removing neurons with least importance, and it is then fine-tuned to recover its predictive power. NISP is evaluated on several datasets with multiple CNN models and demonstrated to achieve significant acceleration and compression with negligible accuracy loss. Real-time application of deep learning algorithms is often hindered by high computational complexity and frequent memory accesses. Network pruning is a promising technique to solve this problem. However, pruning usually results in irregular network connections that not only demand extra representation efforts but also do not fit well on parallel computation. We introduce structured sparsity at various scales for convolutional neural networks: feature map-wise, kernel-wise, and intra-kernel strided sparsity. This structured sparsity is very advantageous for direct computational resource savings on embedded computers, in parallel computing environments, and in hardware-based systems. To decide the importance of network connections and paths, the proposed method uses a particle filtering approach. The importance weight of each particle is assigned by assessing the misclassification rate with a corresponding connectivity pattern. The pruned network is retrained to compensate for the losses due to pruning. While implementing convolutions as matrix products, we particularly show that intra-kernel strided sparsity with a simple constraint can significantly reduce the size of the kernel and feature map tensors. The proposed work shows that when pruning granularities are applied in combination, we can prune the CIFAR-10 network by more than 70 with less than a 1 loss in accuracy. 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.