Related papers: Some Theoretical Results on Layerwise Effective Di…
We establish convergence of the training dynamics of residual neural networks (ResNets) to their joint infinite depth L, hidden width M, and embedding dimension D limit. Specifically, we consider ResNets with two-layer perceptron blocks in…
This paper considers the following question: how well can depth-two ReLU networks with randomly initialized bottom-level weights represent smooth functions? We give near-matching upper- and lower-bounds for $L_2$-approximation in terms of…
Deep neural networks are highly expressive machine learning models with the ability to interpolate arbitrary datasets. Deep nets are typically optimized via first-order methods and the optimization process crucially depends on the…
Deep neural networks have achieved remarkable success in practice, yet a mechanistic understanding of how features evolve during training remains incomplete, especially in the large-depth limit. For ResNets under depth-$\mu$P scaling, prior…
Rectified Linear Units (ReLUs) have been shown to ameliorate the vanishing gradient problem, allow for efficient backpropagation, and empirically promote sparsity in the learned parameters. They have led to state-of-the-art results in a…
We prove that for an $L$-layer fully-connected linear neural network, if the width of every hidden layer is $\tilde\Omega (L \cdot r \cdot d_{\mathrm{out}} \cdot \kappa^3 )$, where $r$ and $\kappa$ are the rank and the condition number of…
This theoretical paper is devoted to developing a rigorous theory for demystifying the global convergence phenomenon in a challenging scenario: learning over-parameterized Rectified Linear Unit (ReLU) nets for very high dimensional dataset…
Neural networks with REctified Linear Unit (ReLU) activation functions (a.k.a. ReLU networks) have achieved great empirical success in various domains. Nonetheless, existing results for learning ReLU networks either pose assumptions on the…
The paper briefy reviews several recent results on hierarchical architectures for learning from examples, that may formally explain the conditions under which Deep Convolutional Neural Networks perform much better in function approximation…
We propose and analyze a new family of algorithms for training neural networks with ReLU activations. Our algorithms are based on the technique of alternating minimization: estimating the activation patterns of each ReLU for all given…
Motivated by the resurgence of neural networks in being able to solve complex learning tasks we undertake a study of high depth networks using ReLU gates which implement the function $x \mapsto \max\{0,x\}$. We try to understand the role of…
We investigate the local spectral statistics of the loss surface Hessians of artificial neural networks, where we discover excellent agreement with Gaussian Orthogonal Ensemble statistics across several network architectures and datasets.…
We explore the phase diagram of approximation rates for deep neural networks and prove several new theoretical results. In particular, we generalize the existing result on the existence of deep discontinuous phase in ReLU networks to…
Quantile regression is the task of estimating a specified percentile response, such as the median, from a collection of known covariates. We study quantile regression with rectified linear unit (ReLU) neural networks as the chosen model…
This article considers fully connected neural networks with Gaussian random weights and biases as well as $L$ hidden layers, each of width proportional to a large parameter $n$. For polynomially bounded non-linearities we give sharp…
Overparameterized ML models, including neural networks, typically induce underdetermined training objectives with multiple global minima. The implicit bias refers to the limiting global minimum that is attained by a common optimization…
We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are…
We study the loss landscape of both shallow and deep, mildly overparameterized ReLU neural networks on a generic finite input dataset for the squared error loss. We show both by count and volume that most activation patterns correspond to…
In this article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…
We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on $n$ data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to…