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.
conduct systematic experiments to explore the generalization ability of deep neural networks @cite_43 . They show that neural networks can almost perfectly fit the training data even when the training labels are random. This paper attracts the community of learning theory to the important topic that how to theoretically interpret the success of deep neural networks.
@cite_43
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Despite their massive size, successful deep artificial neural networks can exhibit a remarkably small difference between training and test performance. Conventional wisdom attributes small generalization error either to properties of the model family, or to the regularization techniques used during training. Through extensive systematic experiments, we show how these traditional approaches fail to explain why large neural networks generalize well in practice. Specifically, our experiments establish that state-of-the-art convolutional networks for image classification trained with stochastic gradient methods easily fit a random labeling of the training data. This phenomenon is qualitatively unaffected by explicit regularization, and occurs even if we replace the true images by completely unstructured random noise. We corroborate these experimental findings with a theoretical construction showing that simple depth two neural networks already have perfect finite sample expressivity as soon as the number of parameters exceeds the number of data points as it usually does in practice. We interpret our experimental findings by comparison with traditional models. Despite their massive size, successful deep artificial neural networks can exhibit a remarkably small difference between training and test performance. Conventional wisdom attributes small generalization error either to properties of the model family, or to the regularization techniques used during training. @PARASPLIT Through extensive systematic experiments, we show how these traditional approaches fail to explain why large neural networks generalize well in practice. Specifically, our experiments establish that state-of-the-art convolutional networks for image classification trained with stochastic gradient methods easily fit a random labeling of the training data. This phenomenon is qualitatively unaffected by explicit regularization, and occurs even if we replace the true images by completely unstructured random noise. We corroborate these experimental findings with a theoretical construction showing that simple depth two neural networks already have perfect finite sample expressivity as soon as the number of parameters exceeds the number of data points as it usually does in practice. @PARASPLIT We interpret our experimental findings by comparison with traditional models. Recent studies have discovered that deep networks are capable of memorizing the entire data even when the labels are completely random. Since deep models are trained on big data where labels are often noisy, the ability to overfit noise can lead to poor performance. To overcome the overfitting on corrupted training data, we propose a novel technique to regularize deep networks in the data dimension. This is achieved by learning a neural network called MentorNet to supervise the training of the base network, namely, StudentNet. Our work is inspired by curriculum learning and advances the theory by learning a curriculum from data by neural networks. We demonstrate the efficacy of MentorNet on several benchmarks. Comprehensive experiments show that it is able to significantly improve the generalization performance of the state-of-the-art deep networks on corrupted training data. The success of recent deep convolutional neural networks (CNNs) depends on learning hidden representations that can summarize the important factors of variation behind the data. In this work, we describe Network Dissection, a method that interprets networks by providing meaningful labels to their individual units. The proposed method quantifies the interpretability of CNN representations by evaluating the alignment between individual hidden units and visual semantic concepts. By identifying the best alignments, units are given interpretable labels ranging from colors, materials, textures, parts, objects and scenes. The method reveals that deep representations are more transparent and interpretable than they would be under a random equivalently powerful basis. We apply our approach to interpret and compare the latent representations of several network architectures trained to solve a wide range of supervised and self-supervised tasks. We then examine factors affecting the network interpretability such as the number of the training iterations, regularizations, different initialization parameters, as well as networks depth and width. Finally we show that the interpreted units can be used to provide explicit explanations of a given CNN prediction for an image. Our results highlight that interpretability is an important property of deep neural networks that provides new insights into what hierarchical structures can learn.