This study analyzes the Fisher information matrix (FIM) by applying mean-field theory to deep neural networks with random weights. We theoretically find novel statistics of the FIM, which are universal among a wide class of deep networks with any number of layers and various activation functions. Although most of the FIM's eigenvalues are close to zero, the maximum eigenvalue takes on a huge value and the eigenvalue distribution has an extremely long tail. These statistics suggest that the shape of a loss landscape is locally flat in most dimensions, but strongly distorted in the other dimensions. Moreover, our theory of the FIM leads to quantitative evaluation of learning in deep networks. First, the maximum eigenvalue enables us to estimate an appropriate size of a learning rate for steepest gradient methods to converge. Second, the flatness induced by the small eigenvalues is connected to generalization ability through a norm-based capacity measure.
Despite its importance in statistics and machine learning, study on the FIM for neural networks has been limited so far. This is because layer-by-layer nonlinear maps and huge parameter dimensions make it difficult to take analysis any further. Degeneracy of the eigenvalues of the FIM has been found in certain parameter regions @cite_3 . To understand the loss landscape, has utilized random matrix theory and obtained the spectrum of FIM and Hessian under several assumptions, although the analysis is limited to special types of shallow networks. In contrast, this paper is the first attempt to apply the mean field approach, which overcomes the difficulties above and enables us to identify universal properties of the FIM in various types of DNNs.
@cite_3
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The Fisher information matrix (FIM) is a fundamental quantity to represent the characteristics of a stochastic model, including deep neural networks (DNNs). The present study reveals novel statistics of FIM that are universal among a wide class of DNNs. To this end, we use random weights and large width limits, which enables us to utilize mean field theories. We investigate the asymptotic statistics of the FIM’s eigenvalues and reveal that most of them are close to zero while the maximum eigenvalue takes a huge value. Because the landscape of the parameter space is defined by the FIM, it is locally flat in most dimensions, but strongly distorted in others. Moreover, we demonstrate the potential usage of the derived statistics in learning strategies. First, small eigenvalues that induce flatness can be connected to a norm-based capacity measure of generalization ability. Second, the maximum eigenvalue that induces the distortion enables us to quantitatively estimate an appropriately sized learning rate for gradient methods to converge. Previous theoretical work on deep learning and neural network optimization tend to focus on avoiding saddle points and local minima. However, the practical observation is that, at least in the case of the most successful Deep Convolutional Neural Networks (DCNNs), practitioners can always increase the network size to fit the training data (an extreme example would be [1]). The most successful DCNNs such as VGG and ResNets are best used with a degree of "overparametrization". In this work, we characterize with a mix of theory and experiments, the landscape of the empirical risk of overparametrized DCNNs. We first prove in the regression framework the existence of a large number of degenerate global minimizers with zero empirical error (modulo inconsistent equations). The argument that relies on the use of Bezout theorem is rigorous when the RELUs are replaced by a polynomial nonlinearity (which empirically works as well). As described in our Theory III [2] paper, the same minimizers are degenerate and thus very likely to be found by SGD that will furthermore select with higher probability the most robust zero-minimizer. We further experimentally explored and visualized the landscape of empirical risk of a DCNN on CIFAR-10 during the entire training process and especially the global minima. Finally, based on our theoretical and experimental results, we propose an intuitive model of the landscape of DCNN's empirical loss surface, which might not be as complicated as people commonly believe. Optimizing deep neural networks (DNNs) often suffers from the ill-conditioned problem. We observe that the scaling-based weight space symmetry property in rectified nonlinear network will cause this negative effect. Therefore, we propose to constrain the incoming weights of each neuron to be unit-norm, which is formulated as an optimization problem over Oblique manifold. A simple yet efficient method referred to as projection based weight normalization (PBWN) is also developed to solve this problem. PBWN executes standard gradient updates, followed by projecting the updated weight back to Oblique manifold. This proposed method has the property of regularization and collaborates well with the commonly used batch normalization technique. We conduct comprehensive experiments on several widely-used image datasets including CIFAR-10, CIFAR-100, SVHN and ImageNet for supervised learning over the state-of-the-art convolutional neural networks, such as Inception, VGG and residual networks. The results show that our method is able to improve the performance of DNNs with different architectures consistently. We also apply our method to Ladder network for semi-supervised learning on permutation invariant MNIST dataset, and our method outperforms the state-of-the-art methods: we obtain test errors as 2.52 , 1.06 , and 0.91 with only 20, 50, and 100 labeled samples, respectively. Despite the state-of-the-art accuracy of Deep Neural Networks (DNN) in various classification problems, their deployment onto resource constrained edge computing devices remains challenging, due to their large size and complexity. Several recent studies have reported remarkable results in reducing this complexity through quantization of DNN models. However, these studies usually do not consider the change in loss when performing quantization, nor do they take the disparate importance of DNN connections to the accuracy into account. We address these issues in this paper by proposing a new method, called adaptive quantization, which simplifies a trained DNN model by finding a unique, optimal precision for each connection weight such that the increase in loss is minimized. The optimization problem at the core of this method iteratively uses the loss function gradient to determine an error margin for each weight and assign it a precision accordingly. Since this problem uses linear functions, it is computationally cheap and, as we will show, has a closed-form approximate solution. Experiments on MNIST, CIFAR, and SVHN datasets showed that the proposed method can achieve near or better than state-of-the-art reduction in model size with similar error rate. Furthermore, it can achieve compressions close to floating-point model compression methods without loss of accuracy.