Nonparametric Classification on Low Dimensional Manifolds using Overparameterized Convolutional Residual Networks

Convolutional residual neural networks (ConvResNets), though overparameterized, can achieve remarkable prediction performance in practice, which cannot be well explained by conventional wisdom. To bridge this gap, we study the performance of ConvResNeXts, which cover ConvResNets as a special case, trained with weight decay from the perspective of nonparametric classification. Our analysis allows for infinitely many building blocks in ConvResNeXts, and shows that weight decay implicitly enforces sparsity on these blocks. Specifically, we consider a smooth target function supported on a low-dimensional manifold, then prove that ConvResNeXts can adapt to the function smoothness and low-dimensional structures and efficiently learn the function without suffering from the curse of dimensionality. Our findings partially justify the advantage of overparameterized ConvResNeXts over conventional machine learning models.

Why Quantization Improves Generalization: NTK of Binary Weight Neural Networks

Quantized neural networks have drawn a lot of attention as they reduce the space and computational complexity during the inference. Moreover, there has been folklore that quantization acts as an implicit regularizer and thus can improve the generalizability of neural networks, yet no existing work formalizes this interesting folklore. In this paper, we take the binary weights in a neural network as random variables under stochastic rounding, and study the distribution propagation over different layers in the neural network. We propose a quasi neural network to approximate the distribution propagation, which is a neural network with continuous parameters and smooth activation function. We derive the neural tangent kernel (NTK) for this quasi neural network, and show that the eigenvalue of NTK decays at approximately exponential rate, which is comparable to that of Gaussian kernel with randomized scale. This in turn indicates that the Reproducing Kernel Hilbert Space (RKHS) of a binary weight neural network covers a strict subset of functions compared with the one with real value weights. We use experiments to verify that the quasi neural network we proposed can well approximate binary weight neural network. Furthermore, binary weight neural network gives a lower generalization gap compared with real value weight neural network, which is similar to the difference between Gaussian kernel and Laplace kernel.

Deep Learning meets Nonparametric Regression: Are Weight-Decayed DNNs Locally Adaptive?

We study the theory of neural network (NN) from the lens of classical nonparametric regression problems with a focus on NN's ability to adaptively estimate functions with heterogeneous smoothness -- a property of functions in Besov or Bounded Variation (BV) classes. Existing work on this problem requires tuning the NN architecture based on the function spaces and sample sizes. We consider a "Parallel NN" variant of deep ReLU networks and show that the standard weight decay is equivalent to promoting the $\ell_p$-sparsity ($0<p<1$) of the coefficient vector of an end-to-end learned function bases, i.e., a dictionary. Using this equivalence, we further establish that by tuning only the weight decay, such Parallel NN achieves an estimation error arbitrarily close to the minimax rates for both the Besov and BV classes. Notably, it gets exponentially closer to minimax optimal as the NN gets deeper. Our research sheds new lights on why depth matters and how NNs are more powerful than kernel methods.

3U-EdgeAI: Ultra-Low Memory Training, Ultra-Low BitwidthQuantization, and Ultra-Low Latency Acceleration

The deep neural network (DNN) based AI applications on the edge require both low-cost computing platforms and high-quality services. However, the limited memory, computing resources, and power budget of the edge devices constrain the effectiveness of the DNN algorithms. Developing edge-oriented AI algorithms and implementations (e.g., accelerators) is challenging. In this paper, we summarize our recent efforts for efficient on-device AI development from three aspects, including both training and inference. First, we present on-device training with ultra-low memory usage. We propose a novel rank-adaptive tensor-based tensorized neural network model, which offers orders-of-magnitude memory reduction during training. Second, we introduce an ultra-low bitwidth quantization method for DNN model compression, achieving the state-of-the-art accuracy under the same compression ratio. Third, we introduce an ultra-low latency DNN accelerator design, practicing the software/hardware co-design methodology. This paper emphasizes the importance and efficacy of training, quantization and accelerator design, and calls for more research breakthroughs in the area for AI on the edge.

Active Subspace of Neural Networks: Structural Analysis and Universal Attacks

Active subspace is a model reduction method widely used in the uncertainty quantification community. In this paper, we propose analyzing the internal structure and vulnerability and deep neural networks using active subspace. Firstly, we employ the active subspace to measure the number of "active neurons" at each intermediate layer and reduce the number of neurons from several thousands to several dozens. This motivates us to change the network structure and to develop a new and more compact network, referred to as {ASNet}, that has significantly fewer model parameters. Secondly, we propose analyzing the vulnerability of a neural network using active subspace and finding an additive universal adversarial attack vector that can misclassify a dataset with a high probability. Our experiments on CIFAR-10 show that ASNet can achieve 23.98$\times$ parameter and 7.30$\times$ flops reduction. The universal active subspace attack vector can achieve around 20% higher attack ratio compared with the existing approach in all of our numerical experiments. The PyTorch codes for this paper are available online.

A Unified Framework of DNN Weight Pruning and Weight Clustering/Quantization Using ADMM

Many model compression techniques of Deep Neural Networks (DNNs) have been investigated, including weight pruning, weight clustering and quantization, etc. Weight pruning leverages the redundancy in the number of weights in DNNs, while weight clustering/quantization leverages the redundancy in the number of bit representations of weights. They can be effectively combined in order to exploit the maximum degree of redundancy. However, there lacks a systematic investigation in literature towards this direction. In this paper, we fill this void and develop a unified, systematic framework of DNN weight pruning and clustering/quantization using Alternating Direction Method of Multipliers (ADMM), a powerful technique in optimization theory to deal with non-convex optimization problems. Both DNN weight pruning and clustering/quantization, as well as their combinations, can be solved in a unified manner. For further performance improvement in this framework, we adopt multiple techniques including iterative weight quantization and retraining, joint weight clustering training and centroid updating, weight clustering retraining, etc. The proposed framework achieves significant improvements both in individual weight pruning and clustering/quantization problems, as well as their combinations. For weight pruning alone, we achieve 167x weight reduction in LeNet-5, 24.7x in AlexNet, and 23.4x in VGGNet, without any accuracy loss. For the combination of DNN weight pruning and clustering/quantization, we achieve 1,910x and 210x storage reduction of weight data on LeNet-5 and AlexNet, respectively, without accuracy loss. Our codes and models are released at the link http://bit.ly/2D3F0np

Progressive Weight Pruning of Deep Neural Networks using ADMM

Deep neural networks (DNNs) although achieving human-level performance in many domains, have very large model size that hinders their broader applications on edge computing devices. Extensive research work have been conducted on DNN model compression or pruning. However, most of the previous work took heuristic approaches. This work proposes a progressive weight pruning approach based on ADMM (Alternating Direction Method of Multipliers), a powerful technique to deal with non-convex optimization problems with potentially combinatorial constraints. Motivated by dynamic programming, the proposed method reaches extremely high pruning rate by using partial prunings with moderate pruning rates. Therefore, it resolves the accuracy degradation and long convergence time problems when pursuing extremely high pruning ratios. It achieves up to 34 times pruning rate for ImageNet dataset and 167 times pruning rate for MNIST dataset, significantly higher than those reached by the literature work. Under the same number of epochs, the proposed method also achieves faster convergence and higher compression rates. The codes and pruned DNN models are released in the link bit.ly/2zxdlss

ADAM-ADMM: A Unified, Systematic Framework of Structured Weight Pruning for DNNs

Weight pruning methods of deep neural networks (DNNs) have been demonstrated to achieve a good model pruning ratio without loss of accuracy, thereby alleviating the significant computation/storage requirements of large-scale DNNs. Structured weight pruning methods have been proposed to overcome the limitation of irregular network structure and demonstrated actual GPU acceleration. However, the pruning ratio (degree of sparsity) and GPU acceleration are limited (to less than 50%) when accuracy needs to be maintained. In this work, we overcome pruning ratio and GPU acceleration limitations by proposing a unified, systematic framework of structured weight pruning for DNNs, named ADAM-ADMM (Adaptive Moment Estimation-Alternating Direction Method of Multipliers). It is a framework that can be used to induce different types of structured sparsity, such as filter-wise, channel-wise, and shape-wise sparsity, as well non-structured sparsity. The proposed framework incorporates stochastic gradient descent with ADMM, and can be understood as a dynamic regularization method in which the regularization target is analytically updated in each iteration. A significant improvement in weight pruning ratio is achieved without loss of accuracy, along with fast convergence rate. With a small sparsity degree of 33% on the convolutional layers, we achieve 1.64% accuracy enhancement for the AlexNet (CaffeNet) model. This is obtained by mitigation of overfitting. Without loss of accuracy on the AlexNet model, we achieve 2.6 times and 3.65 times average measured speedup on two GPUs, clearly outperforming the prior work. The average speedups reach 2.77 times and 7.5 times when allowing a moderate accuracy loss of 2%. In this case the model compression for convolutional layers is 13.2 times, corresponding to 10.5 times CPU speedup. Our models and codes are released at https://github.com/KaiqiZhang/ADAM-ADMM

A Systematic DNN Weight Pruning Framework using Alternating Direction Method of Multipliers

Weight pruning methods for deep neural networks (DNNs) have been investigated recently, but prior work in this area is mainly heuristic, iterative pruning, thereby lacking guarantees on the weight reduction ratio and convergence time. To mitigate these limitations, we present a systematic weight pruning framework of DNNs using the alternating direction method of multipliers (ADMM). We first formulate the weight pruning problem of DNNs as a nonconvex optimization problem with combinatorial constraints specifying the sparsity requirements, and then adopt the ADMM framework for systematic weight pruning. By using ADMM, the original nonconvex optimization problem is decomposed into two subproblems that are solved iteratively. One of these subproblems can be solved using stochastic gradient descent, the other can be solved analytically. Besides, our method achieves a fast convergence rate. The weight pruning results are very promising and consistently outperform the prior work. On the LeNet-5 model for the MNIST data set, we achieve 71.2 times weight reduction without accuracy loss. On the AlexNet model for the ImageNet data set, we achieve 21 times weight reduction without accuracy loss. When we focus on the convolutional layer pruning for computation reductions, we can reduce the total computation by five times compared with the prior work (achieving a total of 13.4 times weight reduction in convolutional layers). Our models and codes are released at https://github.com/KaiqiZhang/admm-pruning