Residual connections significantly boost the performance of deep neural networks. However, there are few theoretical results that address the influence of residuals on the hypothesis complexity and the generalization ability of deep neural networks. This paper studies the influence of residual connections on the hypothesis complexity of the neural network in terms of the covering number of its hypothesis space. We prove that the upper bound of the covering number is the same as chain-like neural networks, if the total numbers of the weight matrices and nonlinearities are fixed, no matter whether they are in the residuals or not. This result demonstrates that residual connections may not increase the hypothesis complexity of the neural network compared with the chain-like counterpart. Based on the upper bound of the covering number, we then obtain an @math margin-based multi-class generalization bound for ResNet, as an exemplary case of any deep neural network with residual connections. Generalization guarantees for similar state-of-the-art neural network architectures, such as DenseNet and ResNeXt, are straight-forward. From our generalization bound, a practical implementation is summarized: to approach a good generalization ability, we need to use regularization terms to control the magnitude of the norms of weight matrices not to increase too much, which justifies the standard technique of weight decay.
explore the capacity measures for deep neural networks from a geometrical invariance viewpoint @cite_42 . They propose to use Fisher-Rao norm to measure the capacity of deep neural networks. Motivated by information geometry, they reveal the invariance property of the Fisher-Rao norm. The authors further establish some norm-comparison inequalities which demonstrate that the Fisher-Rao norm is an umbrella for many existing norm-based complexity measures. They also present experimental results to support their theoretical findings.
@cite_42
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We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity --- the Fisher-Rao norm --- that possesses desirable invariance properties and is motivated by Information Geometry. We discover an analytical characterization of the new capacity measure, through which we establish norm-comparison inequalities and further show that the new measure serves as an umbrella for several existing norm-based complexity measures. We discuss upper bounds on the generalization error induced by the proposed measure. Extensive numerical experiments on CIFAR-10 support our theoretical findings. Our theoretical analysis rests on a key structural lemma about partial derivatives of multi-layer rectifier networks. 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. This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation. Fisher information and natural gradient provided deep insights and powerful tools to artificial neural networks. However related analysis becomes more and more difficult as the learner's structure turns large and complex. This paper makes a preliminary step towards anew direction. We extract a local component from a large neural system, and define its relative Fisher information metric that describes accurately this small component, and is invariant to the other parts of the system. This concept is important because the geometry structure is much simplified and it can be easily applied to guide the learning of neural networks. We provide an analysis on a list of commonly used components, and demonstrate how to use this concept to further improve optimization.