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.
study the sample complexity of deep neural networks and present upper bounds on the Rademacher complexity of the neural networks in terms of the norm of the weight matrix in each layer @cite_41 . Compared to previous works, these complexity bounds have improved dependence on the network depth, and under some additional assumptions, are fully independent of the network size (both depth and width). The upper bounds on the Rademacher complexity further lead to @math upper bounds on the generalization error of neural networks.
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We study the sample complexity of learning neural networks, by providing new bounds on their Rademacher complexity assuming norm constraints on the parameter matrix of each layer. Compared to previous work, these complexity bounds have improved dependence on the network depth, and under some additional assumptions, are fully independent of the network size (both depth and width). These results are derived using some novel techniques, which may be of independent interest. We study the sample complexity of learning neural networks, by providing new bounds on their Rademacher complexity assuming norm constraints on the parameter matrix of each layer. Compared to previous work, these bounds have improved dependence on the network depth, and under some additional assumptions, are fully independent of the network size (both depth and width). These results are derived using some novel techniques, which may be of independent interest. Many machine learning models are vulnerable to adversarial attacks; for example, adding adversarial perturbations that are imperceptible to humans can often make machine learning models produce wrong predictions with high confidence. Moreover, although we may obtain robust models on the training dataset via adversarial training, in some problems the learned models cannot generalize well to the test data. In this paper, we focus on @math attacks, and study the adversarially robust generalization problem through the lens of Rademacher complexity. For binary linear classifiers, we prove tight bounds for the adversarial Rademacher complexity, and show that the adversarial Rademacher complexity is never smaller than its natural counterpart, and it has an unavoidable dimension dependence, unless the weight vector has bounded @math norm. The results also extend to multi-class linear classifiers. For (nonlinear) neural networks, we show that the dimension dependence in the adversarial Rademacher complexity also exists. We further consider a surrogate adversarial loss for one-hidden layer ReLU network and prove margin bounds for this setting. Our results indicate that having @math norm constraints on the weight matrices might be a potential way to improve generalization in the adversarial setting. We demonstrate experimental results that validate our theoretical findings. People believe that depth plays an important role in success of deep neural networks (DNN). However, this belief lacks solid theoretical justifications as far as we know. We investigate role of depth from perspective of margin bound. In margin bound, expected error is upper bounded by empirical margin error plus Rademacher Average (RA) based capacity term. First, we derive an upper bound for RA of DNN, and show that it increases with increasing depth. This indicates negative impact of depth on test performance. Second, we show that deeper networks tend to have larger representation power (measured by Betti numbers based complexity) than shallower networks in multi-class setting, and thus can lead to smaller empirical margin error. This implies positive impact of depth. The combination of these two results shows that for DNN with restricted number of hidden units, increasing depth is not always good since there is a tradeoff between positive and negative impacts. These results inspire us to seek alternative ways to achieve positive impact of depth, e.g., imposing margin-based penalty terms to cross entropy loss so as to reduce empirical margin error without increasing depth. Our experiments show that in this way, we achieve significantly better test performance. The generalization error of deep neural networks via their classification margin is studied in this paper. Our approach is based on the Jacobian matrix of a deep neural network and can be applied to networks with arbitrary nonlinearities and pooling layers, and to networks with different architectures such as feed forward networks and residual networks. Our analysis leads to the conclusion that a bounded spectral norm of the network's Jacobian matrix in the neighbourhood of the training samples is crucial for a deep neural network of arbitrary depth and width to generalize well. This is a significant improvement over the current bounds in the literature, which imply that the generalization error grows with either the width or the depth of the network. Moreover, it shows that the recently proposed batch normalization and weight normalization reparametrizations enjoy good generalization properties, and leads to a novel network regularizer based on the network's Jacobian matrix. The analysis is supported with experimental results on the MNIST, CIFAR-10, LaRED, and ImageNet datasets. Recently, metric learning and similarity learning have attracted a large amount of interest. Many models and optimisation algorithms have been proposed. However, there is relatively little work on the generalization analysis of such methods. In this paper, we derive novel generalization bounds of metric and similarity learning. In particular, we first show that the generalization analysis reduces to the estimation of the Rademacher average over "sums-of-i.i.d." sample-blocks related to the specific matrix norm. Then, we derive generalization bounds for metric similarity learning with different matrix-norm regularisers by estimating their specific Rademacher complexities. Our analysis indicates that sparse metric similarity learning with @math -norm regularisation could lead to significantly better bounds than those with Frobenius-norm regularisation. Our novel generalization analysis develops and refines the techniques of U-statistics and Rademacher complexity analysis. We investigate the use of certain data-dependent estimates of the complexity of a function class, called Rademacher and Gaussian complexities. In a decision theoretic setting, we prove general risk bounds in terms of these complexities. We consider function classes that can be expressed as combinations of functions from basis classes and show how the Rademacher and Gaussian complexities of such a function class can be bounded in terms of the complexity of the basis classes. We give examples of the application of these techniques in finding data-dependent risk bounds for decision trees, neural networks and support vector machines. We investigate the use of certain data-dependent estimates of the complexity of a function class, called Rademacher and gaussian complexities. In a decision theoretic setting, we prove general risk bounds in terms of these complexities. We consider function classes that can be expressed as combinations of functions from basis classes and show how the Rademacher and gaussian complexities of such a function class can be bounded in terms of the complexity of the basis classes.We give examples of the application of these techniques in finding data-dependent risk bounds for decision trees, neural networks and support vector machines. We study the improper learning of multi-layer neural networks. Suppose that the neural network to be learned has @math hidden layers and that the @math -norm of the incoming weights of any neuron is bounded by @math . We present a kernel-based method, such that with probability at least @math , it learns a predictor whose generalization error is at most @math worse than that of the neural network. The sample complexity and the time complexity of the presented method are polynomial in the input dimension and in @math , where @math is a function depending on @math and on the activation function, independent of the number of neurons. The algorithm applies to both sigmoid-like activation functions and ReLU-like activation functions. It implies that any sufficiently sparse neural network is learnable in polynomial time. Compression is a key step to deploy large neural networks on resource-constrained platforms. As a popular compression technique, quantization constrains the number of distinct weight values and thus reducing the number of bits required to represent and store each weight. In this paper, we study the representation power of quantized neural networks. First, we prove the universal approximability of quantized ReLU networks. Then we provide upper bounds of storage size given the approximation error bound and the bit-width of weights for function-independent and function-dependent structures. To the best of the authors' knowledge, this is the first work on the universal approximability as well as the associated storage size bound of quantized neural networks. Recently, researchers in the artificial neural net- work field have focused their attention on connectionist models composed by several hidden layers. In fact, experimental results and heuristic considerations suggest that deep architectures are more suitable than shallow ones for modern applications, facing very complex problems, e.g., vision and human language understanding. However, the actual theoretical results supporting such a claim are still few and incomplete. In this paper, we propose a new approach to study how the depth of feedforward neural networks impacts on their ability in implementing high complexity functions. First, a new measure based on topological concepts is introduced, aimed at evaluating the complexity of the function implemented by a neural network, used for classification purposes. Then, deep and shallow neural architectures with com- mon sigmoidal activation functions are compared, by deriving upper and lower bounds on their complexity, and studying how the complexity depends on the number of hidden units and the used activation function. The obtained results seem to support the idea that deep networks actually implements functions of higher complexity, so that they are able, with the same number of resources, to address more difficult problems. We study the improper learning of multi-layer neural networks. Suppose that the neural network to be learned has k hidden layers and that the l1-norm of the incoming weights of any neuron is bounded by L. We present a kernel-based method, such that with probability at least 1 - δ, it learns a predictor whose generalization error is at most e worse than that of the neural network. The sample complexity and the time complexity of the presented method are polynomial in the input dimension and in (1 e, log(1 δ), F(k, L)), where F(k, L) is a function depending on (k, L) and on the activation function, independent of the number of neurons. The algorithm applies to both sigmoid-like activation functions and ReLU-like activation functions. It implies that any sufficiently sparse neural network is learnable in polynomial time. We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such compositional maps and contributes new theoretical results regarding the advantage of depth for neural networks with piecewise linear activation functions. In particular, our analysis is not specific to a single family of models, and as an example, we employ it for rectifier and maxout networks. We improve complexity bounds from pre-existing work and investigate the behavior of units in higher layers. This work characterizes the generalization ability of algorithms whose predictions are linear in the input vector. To this end, we provide sharp bounds for Rademacher and Gaussian complexities of (constrained) linear classes, which directly lead to a number of generalization bounds. This derivation provides simplified proofs of a number of corollaries including: risk bounds for linear prediction (including settings where the weight vectors are constrained by either L2 or L1 constraints), margin bounds (including both L2 and L1 margins, along with more general notions based on relative entropy), a proof of the PAC-Bayes theorem, and upper bounds on L2 covering numbers (with Lp norm constraints and relative entropy constraints). In addition to providing a unified analysis, the results herein provide some of the sharpest risk and margin bounds. Interestingly, our results show that the uniform convergence rates of empirical risk minimization algorithms tightly match the regret bounds of online learning algorithms for linear prediction, up to a constant factor of 2. Sample complexity results from computational learning theory, when applied to neural network learning for pattern classification problems, suggest that for good generalization performance the number of training examples should grow at least linearly with the number of adjustable parameters in the network. Results in this paper show that if a large neural network is used for a pattern classification problem and the learning algorithm finds a network with small weights that has small squared error on the training patterns, then the generalization performance depends on the size of the weights rather than the number of weights. For example, consider a two-layer feedforward network of sigmoid units, in which the sum of the magnitudes of the weights associated with each unit is bounded by A and the input dimension is n. We show that the misclassification probability is no more than a certain error estimate (that is related to squared error on the training set) plus A sup 3 spl radic ((log n) m) (ignoring log A and log m factors), where m is the number of training patterns. This may explain the generalization performance of neural networks, particularly when the number of training examples is considerably smaller than the number of weights. It also supports heuristics (such as weight decay and early stopping) that attempt to keep the weights small during training. The proof techniques appear to be useful for the analysis of other pattern classifiers: when the input domain is a totally bounded metric space, we use the same approach to give upper bounds on misclassification probability for classifiers with decision boundaries that are far from the training examples. We consider the problem of broadcasting in an unknown radio network modeled as a directed graph @math , where @math . In unknown networks, every node knows only its own label, while it is unaware of any other parameter of the network, including its neighborhood and even any upper bound on the number of nodes. We show an @math upper bound on the time complexity of deterministic broadcasting. This is an improvement over the currently best upper bound @math for arbitrary networks, thus shrinking exponentially the existing gap between the lower bound @math and the upper bound from @math to @math . 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. We introduce a formalism of localization for online learning problems, which, similarly to statistical learning theory, can be used to obtain fast rates. In particular, we introduce local sequential Rademacher complexities and other local measures. Based on the idea of relaxations for deriving algorithms, we provide a template method that takes advantage of localization. Furthermore, we build a general adaptive method that can take advantage of the suboptimality of the observed sequence. We illustrate the utility of the introduced concepts on several problems. Among them is a novel upper bound on regret in terms of classical Rademacher complexity when the data are i.i.d. We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on @math , for all @math . Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of @math . In more detail, we give an estimator that, for any @math and @math , draws @math samples from an unknown target log-concave density on @math , and outputs a hypothesis that (with high probability) is @math -close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of @math for this problem. We provide new theoretical insights on why over-parametrization is effective in learning neural networks. For a @math hidden node shallow network with quadratic activation and @math training data points, we show as long as @math , over-parametrization enables local search algorithms to find a optimal solution for general smooth and convex loss functions. Further, despite that the number of parameters may exceed the sample size, using theory of Rademacher complexity, we show with weight decay, the solution also generalizes well if the data is sampled from a regular distribution such as Gaussian. To prove when @math , the loss function has benign landscape properties, we adopt an idea from smoothed analysis, which may have other applications in studying loss surfaces of neural networks.