Kaiqi Zhang

Kaiqi Zhang, Ming Yin, Yu-Xiang Wang · princeton

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.

Kaiqi Zhang, Yu-Xiang Wang

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 size. We consider a "Parallel NN" variant of deep ReLU networks and show that the standard $\ell_2$ regularization is equivalent to promoting the $\ell_p$-sparsity ($0<p<1$) in 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 regularization factor, 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.

Zixuan Zhang, Kaiqi Zhang, Minshuo Chen et al.

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.

Tianyun Zhang, Shaokai Ye, Kaiqi Zhang et al.

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

Dan Qiao, Kaiqi Zhang, Esha Singh et al.

We study the generalization of two-layer ReLU neural networks in a univariate nonparametric regression problem with noisy labels. This is a problem where kernels (\emph{e.g.} NTK) are provably sub-optimal and benign overfitting does not happen, thus disqualifying existing theory for interpolating (0-loss, global optimal) solutions. We present a new theory of generalization for local minima that gradient descent with a constant learning rate can \emph{stably} converge to. We show that gradient descent with a fixed learning rate $η$ can only find local minima that represent smooth functions with a certain weighted \emph{first order total variation} bounded by $1/η- 1/2 + \widetilde{O}(σ+ \sqrt{\mathrm{MSE}})$ where $σ$ is the label noise level, $\mathrm{MSE}$ is short for mean squared error against the ground truth, and $\widetilde{O}(\cdot)$ hides a logarithmic factor. Under mild assumptions, we also prove a nearly-optimal MSE bound of $\widetilde{O}(n^{-4/5})$ within the strict interior of the support of the $n$ data points. Our theoretical results are validated by extensive simulation that demonstrates large learning rate training induces sparse linear spline fits. To the best of our knowledge, we are the first to obtain generalization bound via minima stability in the non-interpolation case and the first to show ReLU NNs without regularization can achieve near-optimal rates in nonparametric regression.

Shaokai Ye, Tianyun Zhang, Kaiqi Zhang et al.

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