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Register a new userOrthogonal Weight Normalization: Solution to Optimization over Multiple Dependent Stiefel Manifolds in Deep Neural Networks
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Lei Huang
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Orthogonal matrix has shown advantages in training Recurrent Neural Networks (RNNs), but such matrix is limited to be square for the hidden-to-hidden transformation in RNNs. In this paper, we generalize such square orthogonal matrix to orthogonal rectangular matrix and formulating this problem in feed-forward Neural Networks (FNNs) as Optimization over Multiple Dependent Stiefel Manifolds (OMDSM). We show that the rectangular orthogonal matrix can stabilize the distribution of network activations and regularize FNNs. We also propose a novel orthogonal weight normalization method to solve OMDSM. Particularly, it constructs orthogonal transformation over proxy parameters to ensure the weight matrix is orthogonal and back-propagates gradient information through the transformation during training. To guarantee stability, we minimize the distortions between proxy parameters and canonical weights over all tractable orthogonal transformations. In addition, we design an orthogonal linear module (OLM) to learn orthogonal filter banks in practice, which can be used as an alternative to standard linear module. Extensive experiments demonstrate that by simply substituting OLM for standard linear module without revising any experimental protocols, our method largely improves the performance of the state-of-the-art networks, including Inception and residual networks on CIFAR and ImageNet datasets. In particular, we have reduced the test error of wide residual network on CIFAR-100 from 20.04% to 18.61% with such simple substitution. Our code is available online for result reproduction.
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We introduce an efficient approach for optimization over orthogonal groups on highly parallel computation units such as GPUs or TPUs. As in earlier work, we parametrize an orthogonal matrix as a product of Householder reflections. However, to overcome low parallelization capabilities of computing Householder reflections sequentially, we propose employing an accumulation scheme called the compact WY (or CWY) transform -- a compact parallelization-friendly matrix representation for the series of Householder reflections. We further develop a novel Truncated CWY (or T-CWY) approach for Stiefel manifold parametrization which has a competitive complexity and, again, yields benefits when computed on GPUs and TPUs. We prove that our CWY and T-CWY methods lead to convergence to a stationary point of the training objective when coupled with stochastic gradient descent. We apply our methods to train recurrent neural network architectures in the tasks of neural machine translation and video prediction.
We propose a unified framework for neural net normalization, regularization and optimization, which includes Path-SGD and Batch-Normalization and interpolates between them across two different dimensions. Through this framework we investigate issue of invariance of the optimization, data dependence and the connection with natural gradients.
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
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This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where $qle p^*$, and $1/p+1/p^{*}=1$, we discuss properties of a width-independent capacity control, which only depends on depth by a square root term. We further analyze the approximation properties of $L_{p,q}$ weight normalized deep neural networks. In particular, for an $L_{1,infty}$ weight normalized network, the approximation error can be controlled by the $L_1$ norm of the output layer, and the corresponding generalization error only depends on the architecture by the square root of the depth.
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