arXiv research

Recently, neural networks in machine learning use rectified linear units (ReLUs) in early processing layers for better performance. Training these structures sometimes results in "dying ReLU units" with near-zero outputs. We first explore this condition via simulation using the CIFAR-10 dataset and variants of two popu…

A new method for optimizing regression problems with ReLU units converges.

The paper investigates how target normalization and momentum affect dying ReLUs in neural networks.

We study layered neural networks of rectified linear units (ReLU) in a modelling framework for stochastic training processes. The comparison with sigmoidal activation functions is in the center of interest. We compute typical learning curves for shallow networks with K hidden units in matching student teacher scenarios…

In this paper we investigate the performance of different types of rectified activation functions in convolutional neural network: standard rectified linear unit (ReLU), leaky rectified linear unit (Leaky ReLU), parametric rectified linear unit (PReLU) and a new randomized leaky rectified linear units (RReLU). We evalu…

This paper presents a basic property of region dividing of ReLU (rectified linear unit) deep learning when new layers are successively added, by which two new perspectives of interpreting deep learning are given. The first is related to decision trees and forests; we construct a deep learning structure equivalent to a …

Neural networks approximate unit spheres as polytopes.

PHP connects to ReLU neural networks for scalable Bayesian inference.

Study approximates nonlinear functionals using deep ReLU networks.

Study shows shallow ReLU networks struggle with high-dimensional Lipschitz functions.

SPLASH units improve deep neural network accuracy and adversarial robustness.

Algorithm learns two-layer residual units using ReLU activations from samples.

In adversarial attacks to machine-learning classifiers, small perturbations are added to input that is correctly classified. The perturbations yield adversarial examples, which are virtually indistinguishable from the unperturbed input, and yet are misclassified. In standard neural networks used for deep learning, atta…

Two networks are equivalent if they produce the same output for any given input. In this paper, we study the possibility of transforming a deep neural network to another network with a different number of units or layers, which can be either equivalent, a local exact approximation, or a global linear approximation of t…

The paper calculates bounds on the local Lipschitz constants of neural network layers.

Quantile regression with ReLU networks achieves minimax rates for various function types.

Activation functions play a key role in providing remarkable performance in deep neural networks, and the rectified linear unit (ReLU) is one of the most widely used activation functions. Various new activation functions and improvements on ReLU have been proposed, but each carry performance drawbacks. In this paper, w…

In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give an algorithm to train a ReLU DNN with one hidden layer to *global optimality* with runtime polynomial in the data size albeit exponential in the input dimension. Further, we impro…

Study on optimal ReLU networks with weight decay for interpolation.

New activations improve deep network reproducibility without sacrificing accuracy.

The Rectified Linear Unit (ReLU) is a foundational activation function in artficial neural networks. Recent literature frequently misattributes its origin to the 2018 (initial) version of this paper, which exclusively investigated ReLU at the classification layer. This paper formally corrects the citation record by tra…

We introduce exact macroscopic on-line learning dynamics of two-layer neural networks with ReLU units in the form of a system of differential equations, using techniques borrowed from statistical physics. For the first experiments, numerical solutions reveal similar behavior compared to sigmoidal activation researched …

Adversarial attacks add perturbations to the input features with the intent of changing the classification produced by a machine learning system. Small perturbations can yield adversarial examples which are misclassified despite being virtually indistinguishable from the unperturbed input. Classifiers trained with stan…

This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width $w_{\text{min}}(d)$ (and arbitrary depth) can approximate any continuous functio…

Examines learning efficiency in neural networks and related models.

This informal technical report details the geometric illustration of decision boundaries for ReLU units in a three layer fully connected neural network. The network is designed and trained to predict pixel intensity from an (x, y) input location. The Geometric Illustration of Neural Networks (GINN) tool was built to vi…

We present a novel neural network algorithm, the Tensor Switching (TS) network, which generalizes the Rectified Linear Unit (ReLU) nonlinearity to tensor-valued hidden units. The TS network copies its entire input vector to different locations in an expanded representation, with the location determined by its hidden un…

Deep ReLU networks need Ω(N) parameters to interpolate at irregularly spaced points.

We give the first dimension-efficient algorithms for learning Rectified Linear Units (ReLUs), which are functions of the form $\mathbf{x} \mapsto \max(0, \mathbf{w} \cdot \mathbf{x})$ with $\mathbf{w} \in \mathbb{S}^{n-1}$. Our algorithm works in the challenging Reliable Agnostic learning model of Kalai, Kanade, and Ma…

New maximum score estimators using ReLU functions and deep neural networks.

Paper improves confidence intervals and variance estimation for deep learning models.

Study on ReLU regression with Massart noise, achieving exact parameter recovery.

Deep gated networks help understand training and generalization in deep learning.

`Biologically inspired' activation functions, such as the logistic sigmoid, have been instrumental in the historical advancement of machine learning. However in the field of deep learning, they have been largely displaced by rectified linear units (ReLU) or similar functions, such as its exponential linear unit (ELU) v…

Dropout training improves neural networks' performance.

We establish $L^{\infty}$ and $L^2$ error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with $\ell^1$ and $\ell^0$ controls on their inner and outer parameters. With the squared ReLU ridge function, we show th…

We investigate deep Bayesian neural networks with Gaussian weight priors and a class of ReLU-like nonlinearities. Bayesian neural networks with Gaussian priors are well known to induce an L2, "weight decay", regularization. Our results characterize a more intricate regularization effect at the level of the unit activat…

New algorithms accelerate solving nonlinear matrix decomposition with ReLU.

The paper reveals how clustering in ReLU networks enhances interpretability.

ReLU networks don't exponentially distort curve lengths as previously thought.

This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be approximated by deep ReLU networks of width $\mathcal{O}(N…

Rectified Linear Units (ReLU) have become the main model for the neural units in current deep learning systems. This choice has been originally suggested as a way to compensate for the so called vanishing gradient problem which can undercut stochastic gradient descent (SGD) learning in networks composed of multiple lay…

We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze the performance of gradient descent for training such kind of n…

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 underlying data distribution being e.g. Gaussian, or require the network size an…

Activation functions are essential for deep learning methods to learn and perform complex tasks such as image classification. Rectified Linear Unit (ReLU) has been widely used and become the default activation function across the deep learning community since 2012. Although ReLU has been popular, however, the hard zero…

Paper reveals hidden convexities in deep learning models using sparse signal processing.

Develops exact convex optimization formulations for neural networks.

Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available m…