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.
Two recent works respectively by @cite_3 and @cite_11 provide upper bounds for the generalization error of chain-like deep neural networks. Specifically, @cite_3 proposes an @math spectral-normalized margin-based generalization bound by upper bounding the Rademacher complexity covering number of the hypothesis space through the divide-and-conquer strategy. Meanwhile, @cite_11 obtains a similar result under the PAC-bayesian framework. Our work is partially motivated by the analysis in @cite_3 .
@cite_3 @cite_11
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This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the mnist and cifar10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity. This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the MNIST and CIFAR10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity. 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. We present a formulation of deep learning that aims at producing a large margin classifier. The notion of margin, minimum distance to a decision boundary, has served as the foundation of several theoretically profound and empirically successful results for both classification and regression tasks. However, most large margin algorithms are applicable only to shallow models with a preset feature representation; and conventional margin methods for neural networks only enforce margin at the output layer. Such methods are therefore not well suited for deep networks. In this work, we propose a novel loss function to impose a margin on any chosen set of layers of a deep network (including input and hidden layers). Our formulation allows choosing any norm on the metric measuring the margin. We demonstrate that the decision boundary obtained by our loss has nice properties compared to standard classification loss functions. Specifically, we show improved empirical results on the MNIST, CIFAR-10 and ImageNet datasets on multiple tasks: generalization from small training sets, corrupted labels, and robustness against adversarial perturbations. The resulting loss is general and complementary to existing data augmentation (such as random adversarial input transform) and regularization techniques (such as weight decay, dropout, and batch norm). 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. 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. Past works have shown that, somewhat surprisingly, over-parametrization can help generalization in neural networks. Towards explaining this phenomenon, we adopt a margin-based perspective. We establish: 1) for multi-layer feedforward relu networks, the global minimizer of a weakly-regularized cross-entropy loss has the maximum normalized margin among all networks, 2) as a result, increasing the over-parametrization improves the normalized margin and generalization error bounds for two-layer networks. In particular, an infinite-size neural network enjoys the best generalization guarantees. The typical infinite feature methods are kernel methods; we compare the neural net margin with that of kernel methods and construct natural instances where kernel methods have much weaker generalization guarantees. We validate this gap between the two approaches empirically. Finally, this infinite-neuron viewpoint is also fruitful for analyzing optimization. We show that a perturbed gradient flow on infinite-size networks finds a global optimizer in polynomial time. We present a generalization bound for feedforward neural networks in terms of the product of the spectral norm of the layers and the Frobenius norm of the weights. The generalization bound is derived using a PAC-Bayes analysis. We present a generalization bound for feedforward neural networks in terms of the product of the spectral norm of the layers and the Frobenius norm of the weights. The generalization bound is derived using a PAC-Bayes analysis. One of the defining properties of deep learning is that models are chosen to have many more parameters than available training data. In light of this capacity for overfitting, it is remarkable that simple algorithms like SGD reliably return solutions with low test error. One roadblock to explaining these phenomena in terms of implicit regularization, structural properties of the solution, and or easiness of the data is that many learning bounds are quantitatively vacuous when applied to networks learned by SGD in this "deep learning" regime. Logically, in order to explain generalization, we need nonvacuous bounds. We return to an idea by Langford and Caruana (2001), who used PAC-Bayes bounds to compute nonvacuous numerical bounds on generalization error for stochastic two-layer two-hidden-unit neural networks via a sensitivity analysis. By optimizing the PAC-Bayes bound directly, we are able to extend their approach and obtain nonvacuous generalization bounds for deep stochastic neural network classifiers with millions of parameters trained on only tens of thousands of examples. We connect our findings to recent and old work on flat minima and MDL-based explanations of generalization. We perform an average case analysis of the generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on the order of or even larger than the number of examples in the dataset. Using random matrix theory and exact solutions in linear models, we derive the generalization error and training error dynamics of learning and analyze how they depend on the dimensionality of data and signal to noise ratio of the learning problem. We find that the dynamics of gradient descent learning naturally protect against overtraining and overfitting in large networks. Overtraining is worst at intermediate network sizes, when the effective number of free parameters equals the number of samples, and thus can be reduced by making a network smaller or larger. Additionally, in the high-dimensional regime, low generalization error requires starting with small initial weights. We then turn to non-linear neural networks, and show that making networks very large does not harm their generalization performance. On the contrary, it can in fact reduce overtraining, even without early stopping or regularization of any sort. We identify two novel phenomena underlying this behavior in overcomplete models: first, there is a frozen subspace of the weights in which no learning occurs under gradient descent; and second, the statistical properties of the high-dimensional regime yield better-conditioned input correlations which protect against overtraining. We demonstrate that naive application of worst-case theories such as Rademacher complexity are inaccurate in predicting the generalization performance of deep neural networks, and derive an alternative bound which incorporates the frozen subspace and conditioning effects and qualitatively matches the behavior observed in simulation. The traditional notion of generalization---i.e., learning a hypothesis whose empirical error is close to its true error---is surprisingly brittle. As has recently been noted in [DFH+15b], even if several algorithms have this guarantee in isolation, the guarantee need not hold if the algorithms are composed adaptively. In this paper, we study three notions of generalization---increasing in strength---that are robust to postprocessing and amenable to adaptive composition, and examine the relationships between them. We call the weakest such notion Robust Generalization. A second, intermediate, notion is the stability guarantee known as differential privacy. The strongest guarantee we consider we call Perfect Generalization. We prove that every hypothesis class that is PAC learnable is also PAC learnable in a robustly generalizing fashion, with almost the same sample complexity. It was previously known that differentially private algorithms satisfy robust generalization. In this paper, we show that robust generalization is a strictly weaker concept, and that there is a learning task that can be carried out subject to robust generalization guarantees, yet cannot be carried out subject to differential privacy. We also show that perfect generalization is a strictly stronger guarantee than differential privacy, but that, nevertheless, many learning tasks can be carried out subject to the guarantees of perfect generalization. We present a novel way of obtaining PAC-style bounds on the generalization error of learning algorithms, explicitly using their stability properties. A stable learner is one for which the learned solution does not change much with small changes in the training set. The bounds we obtain do not depend on any measure of the complexity of the hypothesis space (e.g. VC dimension) but rather depend on how the learning algorithm searches this space, and can thus be applied even when the VC dimension is infinite. We demonstrate that regularization networks possess the required stability property and apply our method to obtain new bounds on their generalization performance. We introduce a notion of algorithmic stability of learning algorithms---that we term ---that captures stability of the hypothesis output by the learning algorithm in the normed space of functions from which hypotheses are selected. The main result of the paper bounds the generalization error of any learning algorithm in terms of its argument stability. The bounds are based on martingale inequalities in the Banach space to which the hypotheses belong. We apply the general bounds to bound the performance of some learning algorithms based on empirical risk minimization and stochastic gradient descent. Neural networks have great success in many machine learning applications, but the fundamental learning theory behind them remains largely unsolved. Learning neural networks is NP-hard, but in practice, simple algorithms like stochastic gradient descent (SGD) often produce good solutions. Moreover, it is observed that overparameterization (that is, designing networks whose number of parameters is larger than statistically needed to perfectly fit the data) improves both optimization and generalization, appearing to contradict traditional learning theory. In this work, we extend the theoretical understanding of two and three-layer neural networks in the overparameterized regime. We prove that, using overparameterized neural networks, one can (improperly) learn some notable hypothesis classes, including two and three-layer neural networks with fewer parameters. Moreover, the learning process can be simply done by SGD or its variants in polynomial time using polynomially many samples. We also show that for a fixed sample size, the generalization error of the solution found by some SGD variant can be made almost independent of the number of parameters in the overparameterized network. 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. Leading classification methods such as support vector machines (SVMs) and their counterparts achieve strong generalization performance by maximizing the margin of separation between data classes. While the maximum margin approach has achieved promising performance, this article identifies its sensitivity to affine transformations of the data and to directions with large data spread. Maximum margin solutions may be misled by the spread of data and preferentially separate classes along large spread directions. This article corrects these weaknesses by measuring margin not in the absolute sense but rather only relative to the spread of data in any projection direction. Maximum relative margin corresponds to a data-dependent regularization on the classification function while maximum absolute margin corresponds to an l2 norm constraint on the classification function. Interestingly, the proposed improvements only require simple extensions to existing maximum margin formulations and preserve the computational efficiency of SVMs. Through the maximization of relative margin, surprising performance gains are achieved on real-world problems such as digit, text classification and on several other benchmark data sets. In addition, risk bounds are derived for the new formulation based on Rademacher averages. We consider the problem of learning an unknown large-margin halfspace in the context of parallel computation, giving both positive and negative results. As our main positive result, we give a parallel algorithm for learning a large-margin half-space, based on an algorithm of Nesterov's that performs gradient descent with a momentum term. We show that this algorithm can learn an unknown γ-margin halfspace over n dimensions using n ċ poly(1 γ) processors and running in time O(1 γ)+O(log n). In contrast, naive parallel algorithms that learn a γ-margin halfspace in time that depends polylogarithmically on n have an inverse quadratic running time dependence on the margin parameter γ. Our negative result deals with boosting, which is a standard approach to learning large-margin halfspaces. We prove that in the original PAC framework, in which a weak learning algorithm is provided as an oracle that is called by the booster, boosting cannot be parallelized. More precisely, we show that, if the algorithm is allowed to call the weak learner multiple times in parallel within a single boosting stage, this ability does not reduce the overall number of successive stages of boosting needed for learning by even a single stage. Our proof is information-theoretic and does not rely on unproven assumptions. Let M be an n? × n matrix of rank r, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm, which we call OptSpace, that reconstructs M from |E| = O(rn) observed entries with relative root mean square error 1 2 RMSE ? C(?) (nr |E|)1 2 with probability larger than 1 - 1 n3. Further, if r = O(1) and M is sufficiently unstructured, then OptSpace reconstructs it exactly from |E| = O(n log n) entries with probability larger than 1 - 1 n3. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|r log n), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices. We present a novel method to fuse the power of deep networks with the computational efficiency of geometric and probabilistic localization algorithms. In contrast to other methods that completely replace a classical visual estimator with a deep network, we propose an approach that uses a convolutional neural network to learn difficult-to-model corrections to the estimator from ground-truth training data. To this end, we derive a novel loss function for learning SE(3) corrections based on a matrix Lie groups approach, with a natural formulation for balancing translation and rotation errors. We use this loss to train a deep pose correction network (DPC-Net) that predicts corrections for a particular estimator, sensor and environment. Using the KITTI odometry dataset, we demonstrate significant improvements to the accuracy of a computationally-efficient sparse stereo visual odometry pipeline, that render it as accurate as a modern computationally-intensive dense estimator. Further, we show how DPC-Net can be used to mitigate the effect of poorly calibrated lens distortion parameters.