arXiv subjects
Publications and source records attributed to Sophie Langer.
We show that deep Heaviside networks (DHNs) have limited expressiveness but that this can be overcome by including either skip connections or neurons with linear activation. We provide lower and upper bounds for the Vapnik-Chervonenkis (VC) dimensions and approximation rates of these network classes. As an application, we derive statistical convergence rates for DHN fits in the nonparametric regression model.
We analyze the landscape and training dynamics of diagonal linear networks in a linear regression task, with the network parameters being perturbed by isotropic normal noise during training. The addition of such noise may be interpreted as a stochastic form of sharpness-aware minimization (SAM) and we prove several results that relate its action on the underlying landscape and training dynamics to the sharpness of the loss. In particular, the noise induces a weighted mixture of fractional norm penalties on the network parameters, which forces the individual layers to balance at a fast rate and changes the underlying landscape to favor solutions that result from a shrinkage-thresholding operator applied to the true parameter. We show that balancing the layers equates to minimizing the average sharpness, as well as the trace of the Hessian matrix, among all possible factorizations of the same linear predictor. Further, we characterize how the noise level of the normal perturbations acts as a regularization parameter, with exact descriptions of its effect on shrinkage, thresholding, and balancing speed.
Finite-sample analyses of deep Q-learning typically treat replayed data as independent, even though it is sampled from temporally dependent state-action trajectories. We study the Deep Q-networks (DQN) algorithm under explicit dependence by modelling the minibatches used for updating the network as $τ$-mixing. We show that this assumption holds under certain dependence conditions on the underlying trajectories and the mechanism used to sample minibatches. Building on this observation, we extend statistical analyses of DQN with fully connected ReLU architectures to dependent data. We formulate each update as a nonparametric regression problem with $τ$-mixing observations and derive finite-sample risk bounds under this dependence structure. Our results show that temporal dependence leads to a degradation in the statistical rate by inducing an additional dimensionality penalty in the rate exponent, reflecting the reduced effective sample size of $τ$-mixing data. Moreover, we derive the sample complexity of DQN under $tau$-mixing from these risk bounds. Finally, we empirically demonstrate on standard Gymnasium environments that the independence assumption is systematically violated and that replay sampling yields approximately exponentially decaying correlations, supporting our theoretical framework.
We study approximation and statistical learning properties of deep ReLU networks under structural assumptions that mitigate the curse of dimensionality. We prove minimax-optimal uniform approximation rates for $s$-Hölder smooth functions defined on sets with low Minkowski dimension using fully connected networks with flexible width and depth, improving existing results by logarithmic factors even in classical full-dimensional settings. A key technical ingredient is a new memorization result for deep ReLU networks that enables efficient point fitting with dense architectures. We further introduce a class of compositional models in which each component function is smooth and acts on a domain of low intrinsic dimension. This framework unifies two common assumptions in the statistical learning literature, structural constraints on the target function and low dimensionality of the covariates, within a single model. We show that deep networks can approximate such functions at rates determined by the most difficult function in the composition. As an application, we derive improved convergence rates for empirical risk minimization in nonparametric regression that adapt to smoothness, compositional structure, and intrinsic dimensionality.
Diffusion models rely on a high-dimensional latent space of initial noise seeds, yet it remains unclear whether this space contains sufficient structure to predict properties of the generated samples, such as their classes. In this work, we investigate the emergence of latent structure through the lens of confidence scores assigned by a pre-trained classifier to generated samples. We show that while the latent space appears largely unstructured when considering all noise realizations, restricting attention to initial noise seeds that produce high-confidence samples reveals pronounced class separability. By comparing class predictability across noise subsets of varying confidence and examining the class separability of the latent space, we find evidence of class-relevant latent structure that becomes observable only under confidence-based filtering. As a practical implication, we discuss how confidence-based filtering enables conditional generation as an alternative to guidance-based methods.
The recent statistical theory of neural networks focuses on nonparametric denoising problems that treat randomness as additive noise. Variability in image classification datasets does, however, not originate from additive noise but from variation of the shape and other characteristics of the same object across different images. To address this problem, we introduce a tractable model for supervised image classification. While from the function estimation point of view, every pixel in an image is a variable, and large images lead to high-dimensional function recovery tasks suffering from the curse of dimensionality, increasing the number of pixels in the proposed image deformation model enhances the image resolution and makes the object classification problem easier. We introduce and theoretically analyze three approaches. Two methods combine image alignment with a one-nearest neighbor classifier. Under a separation condition, it is shown that perfect classification is possible. The third method fits a convolutional neural network (CNN) to the data. We derive a rate for the misclassification error that depends on the sample size and the complexity of the deformation class. An empirical study corroborates the theoretical findings.
Acceleration for non-convex functions is a fundamental challenge in optimisation. We revisit star-convex functions, which are strictly unimodal on all lines through a minimizer. [1] accelerate unconstrained star-convex minimization of functions that are smooth with respect to the Euclidean norm. To do so, they add a certain binary search step to gradient descent. In this paper, we accelerate unconstrained star-convex minimization of functions that are weakly smooth with respect to an arbitrary norm. We add a binary search step to mirror descent, generalize the approach and refine its complexity analysis. We prove that our algorithms have sharp convergence rates for star-convex functions with $α$-Holder continuous gradients and demonstrate that our rates are nearly optimal for $p$-norms. [1] Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond, Hinder Oliver and Sidford Aaron and Sohoni Nimit
We study the generalization capabilities of Group Convolutional Neural Networks (GCNNs) with ReLU activation function by deriving upper and lower bounds for their Vapnik-Chervonenkis (VC) dimension. Specifically, we analyze how factors such as the number of layers, weights, and input dimension affect the VC dimension. We further compare the derived bounds to those known for other types of neural networks. Our findings extend previous results on the VC dimension of continuous GCNNs with two layers, thereby providing new insights into the generalization properties of GCNNs, particularly regarding the dependence on the input resolution of the data.
Convolutional neural networks (CNNs) trained with cross-entropy loss have proven to be extremely successful in classifying images. In recent years, much work has been done to also improve the theoretical understanding of neural networks. Nevertheless, it seems limited when these networks are trained with cross-entropy loss, mainly because of the unboundedness of the target function. In this paper, we aim to fill this gap by analyzing the rate of the excess risk of a CNN classifier trained by cross-entropy loss. Under suitable assumptions on the smoothness and structure of the a posteriori probability, it is shown that these classifiers achieve a rate of convergence which is independent of the dimension of the image. These rates are in line with the practical observations about CNNs.
We investigate the statistical behavior of gradient descent iterates with dropout in the linear regression model. In particular, non-asymptotic bounds for the convergence of expectations and covariance matrices of the iterates are derived. The results shed more light on the widely cited connection between dropout and l2-regularization in the linear model. We indicate a more subtle relationship, owing to interactions between the gradient descent dynamics and the additional randomness induced by dropout. Further, we study a simplified variant of dropout which does not have a regularizing effect and converges to the least squares estimator
In this paper we analyze the $L_2$ error of neural network regression estimates with one hidden layer. Under the assumption that the Fourier transform of the regression function decays suitably fast, we show that an estimate, where all initial weights are chosen according to proper uniform distributions and where the weights are learned by gradient descent, achieves a rate of convergence of $1/\sqrt{n}$ (up to a logarithmic factor). Our statistical analysis implies that the key aspect behind this result is the proper choice of the initial inner weights and the adjustment of the outer weights via gradient descent. This indicates that we can also simply use linear least squares to choose the outer weights. We prove a corresponding theoretical result and compare our new linear least squares neural network estimate with standard neural network estimates via simulated data. Our simulations show that our theoretical considerations lead to an estimate with an improved performance in many cases.
Learning the Green's function using deep learning models enables to solve different classes of partial differential equations. A practical limitation of using deep learning for the Green's function is the repeated computationally expensive Monte-Carlo integral approximations. We propose to learn the Green's function by low-rank decomposition, which results in a novel architecture to remove redundant computations by separate learning with domain data for evaluation and Monte-Carlo samples for integral approximation. Using experiments we show that the proposed method improves computational time compared to MOD-Net while achieving comparable accuracy compared to both PINNs and MOD-Net.
Estimation of a regression function from independent and identically distributed data is considered. The $L_2$ error with integration with respect to the distribution of the predictor variable is used as the error criterion. The rate of convergence of least squares estimates based on fully connected spaces of deep neural networks with ReLU activation function is analyzed for smooth regression functions. It is shown that in case that the distribution of the predictor variable is concentrated on a manifold, these estimates achieve a rate of convergence which depends on the dimension of the manifold and not on the number of components of the predictor variable.
This article contributes to the current statistical theory of deep neural networks (DNNs). It was shown that DNNs are able to circumvent the so--called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. In most of those results the tuning parameter is the sparsity of the network, which describes the number of non-zero weights in the network. This constraint seemed to be the key factor for the good rate of convergence results. Recently, the assumption was disproved. In particular, it was shown that simple fully connected DNNs can achieve the same rate of convergence. Those fully connected DNNs are based on the unbounded ReLU activation function. In this article we extend the results to smooth activation functions, i.e., to the sigmoid activation function. It is shown that estimators based on fully connected DNNs with sigmoid activation function also achieve the minimax rates of convergence (up to $\ln n$-factors). In our result the number of hidden layers is fixed, the number of neurons per layer tends to infinity for sample size tending to infinity and a bound for the weights in the network is given.
We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any $d$-dimensional, smooth function on a compact set with a rate of order $W^{-p/d}$, where $W$ is the number of nonzero weights in the network and $p$ is the smoothness of the function. Unfortunately, these rates only hold for a special class of sparsely connected DNNs. We ask ourselves if we can show the same approximation rate for a simpler and more general class, i.e., DNNs which are only defined by its width and depth. In this article we show that DNNs with fixed depth and a width of order $M^d$ achieve an approximation rate of $M^{-2p}$. As a conclusion we quantitatively characterize the approximation power of DNNs in terms of the overall weights $W_0$ in the network and show an approximation rate of $W_0^{-p/d}$. This more general result finally helps us to understand which network topology guarantees a special target accuracy.
Recent results in nonparametric regression show that deep learning, i.e., neural network estimates with many hidden layers, are able to circumvent the so-called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. One key feature of the neural networks used in these results is that their network architecture has a further constraint, namely the network sparsity. In this paper we show that we can get similar results also for least squares estimates based on simple fully connected neural networks with ReLU activation functions. Here either the number of neurons per hidden layer is fixed and the number of hidden layers tends to infinity suitably fast for sample size tending to infinity, or the number of hidden layers is bounded by some logarithmic factor in the sample size and the number of neurons per hidden layer tends to infinity suitably fast for sample size tending to infinity. The proof is based on new approximation results concerning deep neural networks.
Deep neural networks (DNNs) achieve impressive results for complicated tasks like object detection on images and speech recognition. Motivated by this practical success, there is now a strong interest in showing good theoretical properties of DNNs. To describe for which tasks DNNs perform well and when they fail, it is a key challenge to understand their performance. The aim of this paper is to contribute to the current statistical theory of DNNs. We apply DNNs on high dimensional data and we show that the least squares regression estimates using DNNs are able to achieve dimensionality reduction in case that the regression function has locally low dimensionality. Consequently, the rate of convergence of the estimate does not depend on its input dimension $d$, but on its local dimension $d^*$ and the DNNs are able to circumvent the curse of dimensionality in case that $d^*$ is much smaller than $d$. In our simulation study we provide numerical experiments to support our theoretical result and we compare our estimate with other conventional nonparametric regression estimates. The performance of our estimates is also validated in experiments with real data.