Capacity Control of ReLU Neural Networks by Basis-path Norm

Recently, path norm was proposed as a new capacity measure for neural networks with Rectified Linear Unit (ReLU) activation function, which takes the rescaling-invariant property of ReLU into account. It has been shown that the generalization error bound in terms of the path norm explains the empirical generalization behaviors of the ReLU neural networks better than that of other capacity measures. Moreover, optimization algorithms which take path norm as the regularization term to the loss function, like Path-SGD, have been shown to achieve better generalization performance. However, the path norm counts the values of all paths, and hence the capacity measure based on path norm could be improperly influenced by the dependency among different paths. It is also known that each path of a ReLU network can be represented by a small group of linearly independent basis paths with multiplication and division operation, which indicates that the generalization behavior of the network only depends on only a few basis paths. Motivated by this, we propose a new norm \emph{Basis-path Norm} based on a group of linearly independent paths to measure the capacity of neural networks more accurately. We establish a generalization error bound based on this basis path norm, and show it explains the generalization behaviors of ReLU networks more accurately than previous capacity measures via extensive experiments. In addition, we develop optimization algorithms which minimize the empirical risk regularized by the basis-path norm. Our experiments on benchmark datasets demonstrate that the proposed regularization method achieves clearly better performance on the test set than the previous regularization approaches

Towards Understanding Generalization of Deep Learning: Perspective of Loss Landscapes

It is widely observed that deep learning models with learned parameters generalize well, even with much more model parameters than the number of training samples. We systematically investigate the underlying reasons why deep neural networks often generalize well, and reveal the difference between the minima (with the same training error) that generalize well and those they don't. We show that it is the characteristics the landscape of the loss function that explains the good generalization capability. For the landscape of loss function for deep networks, the volume of basin of attraction of good minima dominates over that of poor minima, which guarantees optimization methods with random initialization to converge to good minima. We theoretically justify our findings through analyzing 2-layer neural networks; and show that the low-complexity solutions have a small norm of Hessian matrix with respect to model parameters. For deeper networks, extensive numerical evidence helps to support our arguments

Sample Compression, Support Vectors, and Generalization in Deep Learning

Even though Deep Neural Networks (DNNs) are widely celebrated for their practical performance, they possess many intriguing properties related to depth that are difficult to explain both theoretically and intuitively. Understanding how weights in deep networks coordinate together across layers to form useful learners has proven challenging, in part because the repeated composition of nonlinearities has proved intractable. This paper presents a reparameterization of DNNs as a linear function of a feature map that is locally independent of the weights. This feature map transforms depth-dependencies into simple tensor products and maps each input to a discrete subset of the feature space. Then, using a max-margin assumption, the paper develops a sample compression representation of the neural network in terms of the discrete activation state of neurons induced by s ``support vectors". The paper shows that the number of support vectors s relates with learning guarantees for neural networks through sample compression bounds, yielding a sample complexity of O(ns/epsilon) for networks with n neurons. Finally, the number of support vectors s is found to monotonically increase with width and label noise but decrease with depth.Comment: 15 pages, 10 figure

Data-Dependent Path Normalization in Neural Networks

We propose a unified framework for neural net normalization, regularization and optimization, which includes Path-SGD and Batch-Normalization and interpolates between them across two different dimensions. Through this framework we investigate issue of invariance of the optimization, data dependence and the connection with natural gradients.Comment: 17 pages, 3 figure

Towards moderate overparameterization: global convergence guarantees for training shallow neural networks

Many modern neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Sufficiently overparameterized neural network architectures in principle have the capacity to fit any set of labels including random noise. However, given the highly nonconvex nature of the training landscape it is not clear what level and kind of overparameterization is required for first order methods to converge to a global optima that perfectly interpolate any labels. A number of recent theoretical works have shown that for very wide neural networks where the number of hidden units is polynomially large in the size of the training data gradient descent starting from a random initialization does indeed converge to a global optima. However, in practice much more moderate levels of overparameterization seems to be sufficient and in many cases overparameterized models seem to perfectly interpolate the training data as soon as the number of parameters exceed the size of the training data by a constant factor. Thus there is a huge gap between the existing theoretical literature and practical experiments. In this paper we take a step towards closing this gap. Focusing on shallow neural nets and smooth activations, we show that (stochastic) gradient descent when initialized at random converges at a geometric rate to a nearby global optima as soon as the square-root of the number of network parameters exceeds the size of the training data. Our results also benefit from a fast convergence rate and continue to hold for non-differentiable activations such as Rectified Linear Units (ReLUs)

Training wide residual networks for deployment using a single bit for each weight

For fast and energy-efficient deployment of trained deep neural networks on resource-constrained embedded hardware, each learned weight parameter should ideally be represented and stored using a single bit. Error-rates usually increase when this requirement is imposed. Here, we report large improvements in error rates on multiple datasets, for deep convolutional neural networks deployed with 1-bit-per-weight. Using wide residual networks as our main baseline, our approach simplifies existing methods that binarize weights by applying the sign function in training; we apply scaling factors for each layer with constant unlearned values equal to the layer-specific standard deviations used for initialization. For CIFAR-10, CIFAR-100 and ImageNet, and models with 1-bit-per-weight requiring less than 10 MB of parameter memory, we achieve error rates of 3.9%, 18.5% and 26.0% / 8.5% (Top-1 / Top-5) respectively. We also considered MNIST, SVHN and ImageNet32, achieving 1-bit-per-weight test results of 0.27%, 1.9%, and 41.3% / 19.1% respectively. For CIFAR, our error rates halve previously reported values, and are within about 1% of our error-rates for the same network with full-precision weights. For networks that overfit, we also show significant improvements in error rate by not learning batch normalization scale and offset parameters. This applies to both full precision and 1-bit-per-weight networks. Using a warm-restart learning-rate schedule, we found that training for 1-bit-per-weight is just as fast as full-precision networks, with better accuracy than standard schedules, and achieved about 98%-99% of peak performance in just 62 training epochs for CIFAR-10/100. For full training code and trained models in MATLAB, Keras and PyTorch see https://github.com/McDonnell-Lab/1-bit-per-weight/ .Comment: Published as a conference paper at ICLR 201

Deep Neural Networks

Deep Neural Networks (DNNs) are universal function approximators providing state-of- the-art solutions on wide range of applications. Common perceptual tasks such as speech recognition, image classification, and object tracking are now commonly tackled via DNNs. Some fundamental problems remain: (1) the lack of a mathematical framework providing an explicit and interpretable input-output formula for any topology, (2) quantification of DNNs stability regarding adversarial examples (i.e. modified inputs fooling DNN predictions whilst undetectable to humans), (3) absence of generalization guarantees and controllable behaviors for ambiguous patterns, (4) leverage unlabeled data to apply DNNs to domains where expert labeling is scarce as in the medical field. Answering those points would provide theoretical perspectives for further developments based on a common ground. Furthermore, DNNs are now deployed in tremendous societal applications, pushing the need to fill this theoretical gap to ensure control, reliability, and interpretability.Comment: Technical Repor

Deep Semi-Random Features for Nonlinear Function Approximation

We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer models with semi-random features, we prove with no unrealistic assumptions that the model classes contain an arbitrarily good function as the width increases (universality), and despite non-convexity, we can find such a good function (optimization theory) that generalizes to unseen new data (generalization bound). For deep models, with no unrealistic assumptions, we prove universal approximation ability, a lower bound on approximation error, a partial optimization guarantee, and a generalization bound. Depending on the problems, the generalization bound of deep semi-random features can be exponentially better than the known bounds of deep ReLU nets; our generalization error bound can be independent of the depth, the number of trainable weights as well as the input dimensionality. In experiments, we show that semi-random features can match the performance of neural networks by using slightly more units, and it outperforms random features by using significantly fewer units. Moreover, we introduce a new implicit ensemble method by using semi-random features.Comment: AAAI 2018 - Extended versio

Approximation and Estimation for High-Dimensional Deep Learning Networks

It has been experimentally observed in recent years that multi-layer artificial neural networks have a surprising ability to generalize, even when trained with far more parameters than observations. Is there a theoretical basis for this? The best available bounds on their metric entropy and associated complexity measures are essentially linear in the number of parameters, which is inadequate to explain this phenomenon. Here we examine the statistical risk (mean squared predictive error) of multi-layer networks with $\ell^1$-type controls on their parameters and with ramp activation functions (also called lower-rectified linear units). In this setting, the risk is shown to be upper bounded by $[(L^3 \log d)/n]^{1/2}$, where $d$ is the input dimension to each layer, $L$ is the number of layers, and $n$ is the sample size. In this way, the input dimension can be much larger than the sample size and the estimator can still be accurate, provided the target function has such $\ell^1$ controls and that the sample size is at least moderately large compared to $L^3\log d$. The heart of the analysis is the development of a sampling strategy that demonstrates the accuracy of a sparse covering of deep ramp networks. Lower bounds show that the identified risk is close to being optimal

What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory

We develop a variational framework to understand the properties of functions learned by deep neural networks with ReLU activation functions fit to data. We propose a new function space, which is reminiscent of classical bounded variation spaces, that captures the compositional structure associated with deep neural networks. We derive a representer theorem showing that deep ReLU networks are solutions to regularized data fitting problems in this function space. The function space consists of compositions of functions from the (non-reflexive) Banach spaces of second-order bounded variation in the Radon domain. These are Banach spaces with sparsity-promoting norms, giving insight into the role of sparsity in deep neural networks. The neural network solutions have skip connections and rank bounded weight matrices, providing new theoretical support for these common architectural choices. The variational problem we study can be recast as a finite-dimensional neural network training problem with regularization schemes related to the notions of weight decay and path-norm regularization. Finally, our analysis builds on techniques from variational spline theory, providing new connections between deep neural networks and splines