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Register a new userA Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case
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A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize a function $f:mathbb{R}^drightarrowmathbb{R}$ as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate univariate functions in Savarese et al. (2019), where it was shown that the required norm is determined by the L1-norm of the second derivative of the function. We extend the characterization to multivariate functions (i.e., networks with d input units), relating the required norm to the L1-norm of the Radon transform of a (d+1)/2-power Laplacian of the function. This characterization allows us to show that all functions in Sobolev spaces $W^{s,1}(mathbb{R})$, $sgeq d+1$, can be represented with bounded norm, to calculate the required norm for several specific functions, and to obtain a depth separation result. These results have important implications for understanding generalization performance and the distinction between neural networks and more traditional kernel learning.
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{em Hypernetworks} are architectures that produce the weights of a task-specific {em primary network}. A notable application of hypernetworks in the recent literature involves learning to output functional representations. In these scenarios, the hypernetwork learns a representation corresponding to the weights of a shallow MLP, which typically encodes shape or image information. While such representations have seen considerable success in practice, they remain lacking in the theoretical guarantees in the wide regime of the standard architectures. In this work, we study wide over-parameterized hypernetworks. We show that unlike typical architectures, infinitely wide hypernetworks do not guarantee convergence to a global minima under gradient descent. We further show that convexity can be achieved by increasing the dimensionality of the hypernetworks output, to represent wide MLPs. In the dually infinite-width regime, we identify the functional priors of these architectures by deriving their corresponding GP and NTK kernels, the latter of which we refer to as the {em hyperkernel}. As part of this study, we make a mathematical contribution by deriving tight bounds on high order Taylor expansion terms of standard fully connected ReLU networks.
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.
The recently-introduced class of ordinary differential equation networks (ODE-Nets) establishes a fruitful connection between deep learning and dynamical systems. In this work, we reconsider formulations of the weights as continuous-depth functions using linear combinations of basis functions. This perspective allows us to compress the weights through a change of basis, without retraining, while maintaining near state-of-the-art performance. In turn, both inference time and the memory footprint are reduced, enabling quick and rigorous adaptation between computational environments. Furthermore, our framework enables meaningful continuous-in-time batch normalization layers using function projections. The performance of basis function compression is demonstrated by applying continuous-depth models to (a) image classification tasks using convolutional units and (b) sentence-tagging tasks using transformer encoder units.
Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end -- rather than layerwise -- doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.
Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of H{o}lder functions on low-dimensional manifolds using deep ReLU networks. Suppose $n$ training data are sampled from a H{o}lder function in $mathcal{H}^{s,alpha}$ supported on a $d$-dimensional Riemannian manifold isometrically embedded in $mathbb{R}^D$, with sub-gaussian noise. A deep ReLU network architecture is designed to estimate the underlying function from the training data. The mean squared error of the empirical estimator is proved to converge in the order of $n^{-frac{2(s+alpha)}{2(s+alpha) + d}}log^3 n$. This result shows that deep ReLU networks give rise to a fast convergence rate depending on the data intrinsic dimension $d$, which is usually much smaller than the ambient dimension $D$. It therefore demonstrates the adaptivity of deep ReLU networks to low-dimensional geometric structures of data, and partially explains the power of deep ReLU networks in tackling high-dimensional data with low-dimensional geometric structures.
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