Along with the rapid development of deep learning in practice, the theoretical explanations for its success become urgent. Generalization and expressivity are two widely used measurements to quantify theoretical behaviors of deep learning. The expressivity focuses on finding functions expressible by deep nets but cannot be approximated by shallow nets with the similar number of neurons. It usually implies the large capacity. The generalization aims at deriving fast learning rate for deep nets. It usually requires small capacity to reduce the variance. Different from previous studies on deep learning, pursuing either expressivity or generalization, we take both factors into account to explore the theoretical advantages of deep nets. For this purpose, we construct a deep net with two hidden layers possessing excellent expressivity in terms of localized and sparse approximation. Then, utilizing the well known covering number to measure the capacity, we find that deep nets possess excellent expressive power (measured by localized and sparse approximation) without enlarging the capacity of shallow nets. As a consequence, we derive near optimal learning rates for implementing empirical risk minimization (ERM) on the constructed deep nets. These results theoretically exhibit the advantage of deep nets from learning theory viewpoints.
In the recent work @cite_45 , a neural network with two hidden layers was developed for the learning purpose and the optimal learning rates of order @math were presented. It should be noticed that the main idea of the construction in @cite_45 is the local average argument rather than any optimization strategy such as ). Furthermore, @cite_45 's network architecture is a hybrid of feed-forward neural network (second hidden layer) and radial basis function networks (first hidden layer). The constructed network in the present paper is a standard deep net possessing the same network architectures in both hidden layers.
@cite_45
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In recent years, state-of-the-art methods in computer vision have utilized increasingly deep convolutional neural network architectures (CNNs), with some of the most successful models employing hundreds or even thousands of layers. A variety of pathologies such as vanishing exploding gradients make training such deep networks challenging. While residual connections and batch normalization do enable training at these depths, it has remained unclear whether such specialized architecture designs are truly necessary to train deep CNNs. In this work, we demonstrate that it is possible to train vanilla CNNs with ten thousand layers or more simply by using an appropriate initialization scheme. We derive this initialization scheme theoretically by developing a mean field theory for signal propagation and by characterizing the conditions for dynamical isometry, the equilibration of singular values of the input-output Jacobian matrix. These conditions require that the convolution operator be an orthogonal transformation in the sense that it is norm-preserving. We present an algorithm for generating such random initial orthogonal convolution kernels and demonstrate empirically that they enable efficient training of extremely deep architectures. Deep neural network architectures have recently produced excellent results in a variety of areas in artificial intelligence and visual recognition, well surpassing traditional shallow architectures trained using hand-designed features. The power of deep networks stems both from their ability to perform local computations followed by pointwise non-linearities over increasingly larger receptive fields, and from the simplicity and scalability of the gradient-descent training procedure based on backpropagation. An open problem is the inclusion of layers that perform global, structured matrix computations like segmentation (e.g. normalized cuts) or higher-order pooling (e.g. log-tangent space metrics defined over the manifold of symmetric positive definite matrices) while preserving the validity and efficiency of an end-to-end deep training framework. In this paper we propose a sound mathematical apparatus to formally integrate global structured computation into deep computation architectures. At the heart of our methodology is the development of the theory and practice of backpropagation that generalizes to the calculus of adjoint matrix variations. We perform segmentation experiments using the BSDS and MSCOCO benchmarks and demonstrate that deep networks relying on second-order pooling and normalized cuts layers, trained end-to-end using matrix backpropagation, outperform counterparts that do not take advantage of such global layers. The necessity of depth in efficient neural network learning has led to a family of designs referred to as very deep networks (e.g., GoogLeNet has 22 layers). As the depth increases even further, the need for appropriate tools to explore the space of hidden representations becomes paramount. For instance, beyond the gain in generalization, one may be interested in checking the change in class compositions as additional layers are added. Classical PCA or eigen-spectrum based global approaches do not model the complex inter-class relationships. In this work, we propose a novel decomposition referred to as multiresolution matrix factorization that models hierarchical and compositional structure in symmetric matrices. This new decomposition efficiently infers semantic relationships among deep representations of multiple classes, even when they are not explicitly trained to do so. We show that the proposed factorization is a valuable tool in understanding the landscape of hidden representations, in adapting existing architectures for new tasks and also for designing new architectures using interpretable, human-releatable, class-by-class relationships that we hope the network to learn. Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find @math on the training objective of DNNs in @math . We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter means the network width is sufficiently large: @math in @math , the number of layers and in @math , the number of samples. Our key technique is to derive that, in a sufficiently large neighborhood of the random initialization, the optimization landscape is almost-convex and semi-smooth even with ReLU activations. This implies an equivalence between over-parameterized neural networks and neural tangent kernel (NTK) in the finite (and polynomial) width setting. As concrete examples, starting from randomly initialized weights, we prove that SGD can attain 100 training accuracy in classification tasks, or minimize regression loss in linear convergence speed, with running time polynomial in @math . Our theory applies to the widely-used but non-smooth ReLU activation, and to any smooth and possibly non-convex loss functions. In terms of network architectures, our theory at least applies to fully-connected neural networks, convolutional neural networks (CNN), and residual neural networks (ResNet).