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We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for…

Machine Learning · Computer Science 2018-12-31 Difan Zou , Yuan Cao , Dongruo Zhou , Quanquan Gu

Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well…

Machine Learning · Statistics 2021-09-28 Tianyang Hu , Wenjia Wang , Cong Lin , Guang Cheng

Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically…

Machine Learning · Statistics 2021-11-03 Lu Lu , Yanhui Su , George Em Karniadakis

Despite their impressive performance on diverse tasks, neural networks fail catastrophically in the presence of adversarial inputs---imperceptibly but adversarially perturbed versions of natural inputs. We have witnessed an arms race…

Machine Learning · Computer Science 2018-11-06 Aditi Raghunathan , Jacob Steinhardt , Percy Liang

Interval Bound Propagation (IBP) is so far the base of state-of-the-art methods for training neural networks with certifiable robustness guarantees when potential adversarial perturbations present, while the convergence of IBP training…

Machine Learning · Computer Science 2022-03-18 Yihan Wang , Zhouxing Shi , Quanquan Gu , Cho-Jui Hsieh

Neural networks can be trained to solve regression problems by using gradient-based methods to minimize the square loss. However, practitioners often prefer to reformulate regression as a classification problem, observing that training on…

Machine Learning · Computer Science 2023-03-02 Lawrence Stewart , Francis Bach , Quentin Berthet , Jean-Philippe Vert

In this paper we aim to explore the general robustness of neural network classifiers by utilizing adversarial as well as natural perturbations. Different from previous works which mainly focus on studying the robustness of neural networks…

Computer Vision and Pattern Recognition · Computer Science 2020-10-06 Sadaf Gulshad , Jan Hendrik Metzen , Arnold Smeulders

Several machine learning models, including neural networks, consistently misclassify adversarial examples---inputs formed by applying small but intentionally worst-case perturbations to examples from the dataset, such that the perturbed…

Machine Learning · Statistics 2015-03-24 Ian J. Goodfellow , Jonathon Shlens , Christian Szegedy

Robustness and generalization ability of machine learning models are of utmost importance in various application domains. There is a wide interest in efficient ways to analyze those properties. One important direction is to analyze…

Machine Learning · Computer Science 2025-04-29 Khoat Than , Dat Phan , Giang Vu

Adversarial robustness corresponds to the susceptibility of deep neural networks to imperceptible perturbations made at test time. In the context of image tasks, many algorithms have been proposed to make neural networks robust to…

Computer Vision and Pattern Recognition · Computer Science 2020-12-03 Pranjal Awasthi , George Yu , Chun-Sung Ferng , Andrew Tomkins , Da-Cheng Juan

Gradient-based algorithms are a cornerstone of artificial neural network training, yet it remains unclear whether biological neural networks use similar gradient-based strategies during learning. Experiments often discover a diversity of…

Machine Learning · Computer Science 2026-04-29 Hugo Ninou , Jonathan Kadmon , N. Alex Cayco-Gajic

The adversarial robustness of a neural network mainly relies on two factors: model capacity and anti-perturbation ability. In this paper, we study the anti-perturbation ability of the network from the feature maps of convolutional layers.…

Computer Vision and Pattern Recognition · Computer Science 2024-01-30 Cong Xu , Wei Zhang , Jun Wang , Min Yang

Neural Networks have been shown to be sensitive to common perturbations such as blur, Gaussian noise, rotations, etc. They are also vulnerable to some artificial malicious corruptions called adversarial examples. The adversarial examples…

Machine Learning · Computer Science 2019-10-10 Alfred Laugros , Alice Caplier , Matthieu Ospici

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…

Machine Learning · Statistics 2018-09-26 Luca de Alfaro

In this paper, we consider robust nonparametric regression using deep neural networks with ReLU activation function. While several existing theoretically justified methods are geared towards robustness against identical heavy-tailed noise…

Methodology · Statistics 2023-11-01 Juntong Chen

Deep learning models are intrinsically sensitive to distribution shifts in the input data. In particular, small, barely perceivable perturbations to the input data can force models to make wrong predictions with high confidence. An common…

Computer Vision and Pattern Recognition · Computer Science 2022-10-26 Paul Gavrikov , Janis Keuper

In this paper, we consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. First, we mathematically show that for such networks L2-regularized regression…

Machine Learning · Computer Science 2023-10-05 Jakob Heiss , Josef Teichmann , Hanna Wutte

Algorithmic stability is a classical framework for analyzing the generalization error of learning algorithms. It predicts that an algorithm has small generalization error if it is insensitive to small perturbations in the training set such…

Machine Learning · Computer Science 2026-02-17 Ouns El Harzli , Yoonsoo Nam , Ilja Kuzborskij , Bernardo Cuenca Grau , Ard A. Louis

For a standard convolutional neural network, optimizing over the input pixels to maximize the score of some target class will generally produce a grainy-looking version of the original image. However, Santurkar et al. (2019) demonstrated…

Machine Learning · Computer Science 2019-10-24 Simran Kaur , Jeremy Cohen , Zachary C. Lipton

Over the past few years, neural networks were proven vulnerable to adversarial images: targeted but imperceptible image perturbations lead to drastically different predictions. We show that adversarial vulnerability increases with the…

Machine Learning · Statistics 2019-06-18 Carl-Johann Simon-Gabriel , Yann Ollivier , Léon Bottou , Bernhard Schölkopf , David Lopez-Paz