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Register a new userCharacterization of the Variation Spaces Corresponding to Shallow Neural Networks
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We consider the variation space corresponding to a dictionary of functions in $L^2(Omega)$ and present the basic theory of approximation in these spaces. Specifically, we compare the definition based on integral representations with the definition in terms of convex hulls. We show that in many cases, including the dictionaries corresponding to shallow ReLU$^k$ networks and a dictionary of decaying Fourier modes, that the two definitions coincide. We also give a partial characterization of the variation space for shallow ReLU$^k$ networks and show that the variation space with respect to the dictionary of decaying Fourier modes corresponds to the Barron spectral space.
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We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem whether these rates are sharp. In this article, we provide a solution to this problem by proving sharp lower bounds on the approximation rates for shallow neural networks, which are obtained by lower bounding the $L^2$-metric entropy of the convex hull of the neural network basis functions. In addition, our methods also give sharp lower bounds on the Kolmogorov $n$-widths of this convex hull, which show that the variation spaces corresponding to shallow neural networks cannot be efficiently approximated by linear methods. These lower bounds apply to both sigmoidal activation functions with bounded variation and to activation functions which are a power of the ReLU. Our results also quantify how much stronger the Barron spectral norm is than the variation norm and, combined with previous results, give the asymptotics of the $L^infty$-metric entropy up to logarithmic factors in the case of the ReLU activation function.
We consider the teacher-student setting of learning shallow neural networks with quadratic activations and planted weight matrix $W^*inmathbb{R}^{mtimes d}$, where $m$ is the width of the hidden layer and $dle m$ is the data dimension. We study the optimization landscape associated with the empirical and the population squared risk of the problem. Under the assumption the planted weights are full-rank we obtain the following results. First, we establish that the landscape of the empirical risk admits an energy barrier separating rank-deficient $W$ from $W^*$: if $W$ is rank deficient, then its risk is bounded away from zero by an amount we quantify. We then couple this result by showing that, assuming number $N$ of samples grows at least like a polynomial function of $d$, all full-rank approximate stationary points of the empirical risk are nearly global optimum. These two results allow us to prove that gradient descent, when initialized below the energy barrier, approximately minimizes the empirical risk and recovers the planted weights in polynomial-time. Next, we show that initializing below this barrier is in fact easily achieved when the weights are randomly generated under relatively weak assumptions. We show that provided the network is sufficiently overparametrized, initializing with an appropriate multiple of the identity suffices to obtain a risk below the energy barrier. At a technical level, the last result is a consequence of the semicircle law for the Wishart ensemble and could be of independent interest. Finally, we study the minimizers of the empirical risk and identify a simple necessary and sufficient geometric condition on the training data under which any minimizer has necessarily zero generalization error. We show that as soon as $Nge N^*=d(d+1)/2$, randomly generated data enjoys this geometric condition almost surely, while that ceases to be true if $N<N^*$.
We consider shallow (single hidden layer) neural networks and characterize their performance when trained with stochastic gradient descent as the number of hidden units $N$ and gradient descent steps grow to infinity. In particular, we investigate the effect of different scaling schemes, which lead to different normalizations of the neural network, on the networks statistical output, closing the gap between the $1/sqrt{N}$ and the mean-field $1/N$ normalization. We develop an asymptotic expansion for the neural networks statistical output pointwise with respect to the scaling parameter as the number of hidden units grows to infinity. Based on this expansion, we demonstrate mathematically that to leading order in $N$, there is no bias-variance trade off, in that both bias and variance (both explicitly characterized) decrease as the number of hidden units increases and time grows. In addition, we show that to leading order in $N$, the variance of the neural networks statistical output decays as the implied normalization by the scaling parameter approaches the mean field normalization. Numerical studies on the MNIST and CIFAR10 datasets show that test and train accuracy monotonically improve as the neural networks normalization gets closer to the mean field normalization.
Dropout is a regularisation technique in neural network training where unit activations are randomly set to zero with a given probability emph{independently}. In this work, we propose a generalisation of dropout and other multiplicative noise injection schemes for shallow and deep neural networks, where the random noise applied to different units is not independent but follows a joint distribution that is either fixed or estimated during training. We provide theoretical insights on why such adaptive structured noise injection (ASNI) may be relevant, and empirically confirm that it helps boost the accuracy of simple feedforward and convolutional neural networks, disentangles the hidden layer representations, and leads to sparser representations. Our proposed method is a straightforward modification of the classical dropout and does not require additional computational overhead.
Neural networks have excelled at regression and classification problems when the input space consists of scalar variables. As a result of this proficiency, several popular packages have been developed that allow users to easily fit these kinds of models. However, the methodology has excluded the use of functional covariates and to date, there exists no software that allows users to build deep learning models with this generalized input space. To the best of our knowledge, the functional neural network (FuncNN) library is the first such package in any programming language; the library has been developed for R and is built on top of the keras architecture. Throughout this paper, several functions are introduced that provide users an avenue to easily build models, generate predictions, and run cross-validations. A summary of the underlying methodology is also presented. The ultimate contribution is a package that provides a set of general modelling and diagnostic tools for data problems in which there exist both functional and scalar covariates.
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