Neural networks have been used prominently in several machine learning and statistics applications. In general, the underlying optimization of neural networks is non-convex which makes their performance analysis challenging. In this paper, we take a novel approach to this problem by asking whether one can constrain neural network weights to make its optimization landscape have good theoretical properties while at the same time, be a good approximation for the unconstrained one. For two-layer neural networks, we provide affirmative answers to these questions by introducing Porcupine Neural Networks (PNNs) whose weight vectors are constrained to lie over a finite set of lines. We show that most local optima of PNN optimizations are global while we have a characterization of regions where bad local optimizers may exist. Moreover, our theoretical and empirical results suggest that an unconstrained neural network can be approximated using a polynomially-large PNN.
For any positive integer @math , there exist neural networks with @math layers, @math nodes per layer, and @math distinct parameters which can not be approximated by networks with @math layers unless they are exponentially large --- they must possess @math nodes. This result is proved here for a class of nodes termed "semi-algebraic gates" which includes the common choices of ReLU, maximum, indicator, and piecewise polynomial functions, therefore establishing benefits of depth against not just standard networks with ReLU gates, but also convolutional networks with ReLU and maximization gates, sum-product networks, and boosted decision trees (in this last case with a stronger separation: @math total tree nodes are required). Approximation properties of a class of artificial neural networks are established. It is shown that feedforward networks with one layer of sigmoidal nonlinearities achieve integrated squared error of order O(1 n), where n is the number of nodes. The approximated function is assumed to have a bound on the first moment of the magnitude distribution of the Fourier transform. The nonlinear parameters associated with the sigmoidal nodes, as well as the parameters of linear combination, are adjusted in the approximation. In contrast, it is shown that for series expansions with n terms, in which only the parameters of linear combination are adjusted, the integrated squared approximation error cannot be made smaller than order 1 n sup 2 d uniformly for functions satisfying the same smoothness assumption, where d is the dimension of the input to the function. For the class of functions examined, the approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings. > Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons needed by a deep network for a given degree of function approximation. First, we consider univariate functions on a bounded interval and require a neural network to achieve an approximation error of @math uniformly over the interval. We show that shallow networks (i.e., networks whose depth does not depend on @math ) require @math neurons while deep networks (i.e., networks whose depth grows with @math ) require @math neurons. We then extend these results to certain classes of important multivariate functions. Our results are derived for neural networks which use a combination of rectifier linear units (ReLUs) and binary step units, two of the most popular type of activation functions. Our analysis builds on a simple observation: the multiplication of two bits can be represented by a ReLU. 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. We establish @math and @math error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with @math and @math controls on their inner and outer parameters. With the squared ReLU ridge function, we show that the @math approximation error is inversely proportional to the inner layer @math sparsity and it need only be sublinear in the outer layer @math sparsity. Our constructions are obtained using a variant of the Jones-Barron probabilistic method, which can be interpreted as either stratified sampling with proportionate allocation or two-stage cluster sampling. We also provide companion error lower bounds that reveal near optimality of our constructions. Despite the sparsity assumptions, we showcase the richness and flexibility of these ridge combinations by defining a large family of functions, in terms of certain spectral conditions, that are particularly well approximated by them.  An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques. We consider the problem of approximating a smooth target function and its derivatives by networks involving superpositions and translations of a fixed activation function. The approximation is with respect to the sup-norm and the rate is shown to be of order O(n sup -1 2 ); that is, the rate is independent of the dimension d. The results apply to neural and wavelet networks and extend the work of Barren(see Proc. 7th Yale Workshop on Adaptive and Learning Systems, May, 1992, and ibid., vol.39, p.930, 1993). The approach involves probabilistic methods based on central limit theorems for empirical processes indexed by classes of functions. > Deep neural nets have caused a revolution in many classification tasks. A related ongoing revolution---also theoretically not understood---concerns their ability to serve as generative models for complicated types of data such as images and texts. These models are trained using ideas like variational autoencoders and Generative Adversarial Networks. We take a first cut at explaining the expressivity of multilayer nets by giving a sufficient criterion for a function to be approximable by a neural network with @math hidden layers. A key ingredient is Barron's Theorem Barron1993 , which gives a Fourier criterion for approximability of a function by a neural network with 1 hidden layer. We show that a composition of @math functions which satisfy certain Fourier conditions ("Barron functions") can be approximated by a @math -layer neural network. For probability distributions, this translates into a criterion for a probability distribution to be approximable in Wasserstein distance---a natural metric on probability distributions---by a neural network applied to a fixed base distribution (e.g., multivariate gaussian). Building up recent lower bound work, we also give an example function that shows that composition of Barron functions is more expressive than Barron functions alone. In this paper, we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the numbers of observations are fewer than the number of parameters in the model. We show that with the quadratic activations, the optimization landscape of training, such shallow neural networks, has certain favorable characteristics that allow globally optimal models to be found efficiently using a variety of local search heuristics. This result holds for an arbitrary training data of input output pairs. For differentiable activation functions, we also show that gradient descent, when suitably initialized, converges at a linear rate to a globally optimal model. This result focuses on a realizable model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted weight coefficients. Estimation of functions of @math variables is considered using ridge combinations of the form @math where the activation function @math is a function with bounded value and derivative. These include single-hidden layer neural networks, polynomials, and sinusoidal models. From a sample of size @math of possibly noisy values at random sites @math , the minimax mean square error is examined for functions in the closure of the @math hull of ridge functions with activation @math . It is shown to be of order @math to a fractional power (when @math is of smaller order than @math ), and to be of order @math to a fractional power (when @math is of larger order than @math ). Dependence on constraints @math and @math on the @math norms of inner parameter @math and outer parameter @math , respectively, is also examined. Also, lower and upper bounds on the fractional power are given. The heart of the analysis is development of information-theoretic packing numbers for these classes of functions. While the optimization problem behind deep neural networks is highly non-convex, it is frequently observed in practice that training deep networks seems possible without getting stuck in suboptimal points. It has been argued that this is the case as all local minima are close to being globally optimal. We show that this is (almost) true, in fact almost all local minima are globally optimal, for a fully connected network with squared loss and analytic activation function given that the number of hidden units of one layer of the network is larger than the number of training points and the network structure from this layer on is pyramidal.