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Fast and precise Lipschitz constant estimation of neural networks is an important task for deep learning. Researchers have recently found an intrinsic trade-off between the accuracy and smoothness of neural networks, so training a network…

Machine Learning · Computer Science 2022-10-12 Zi Wang , Gautam Prakriya , Somesh Jha

Maintaining numerical stability in machine learning models is crucial for their reliability and performance. One approach to maintain stability of a network layer is to integrate the condition number of the weight matrix as a regularizing…

Machine Learning · Computer Science 2024-10-02 Rossen Nenov , Daniel Haider , Peter Balazs

The Lipschitz constant is a key measure for certifying the robustness of neural networks to input perturbations. However, computing the exact constant is NP-hard, and standard approaches to estimate the Lipschitz constant involve solving a…

Machine Learning · Computer Science 2026-04-14 Yuezhu Xu , S. Sivaranjani

Due to their susceptibility to adversarial perturbations, neural networks (NNs) are hardly used in safety-critical applications. One measure of robustness to such perturbations in the input is the Lipschitz constant of the input-output map…

Machine Learning · Computer Science 2021-04-30 Patricia Pauli , Anne Koch , Julian Berberich , Paul Kohler , Frank Allgöwer

A widely believed explanation for the remarkable generalization capacities of overparameterized neural networks is that the optimization algorithms used for training induce an implicit bias towards benign solutions. To grasp this…

Machine Learning · Computer Science 2025-12-19 Maria Matveev , Vit Fojtik , Hung-Hsu Chou , Gitta Kutyniok , Johannes Maly

High sensitivity of neural networks against malicious perturbations on inputs causes security concerns. To take a steady step towards robust classifiers, we aim to create neural network models provably defended from perturbations. Prior…

Computer Vision and Pattern Recognition · Computer Science 2018-11-02 Yusuke Tsuzuku , Issei Sato , Masashi Sugiyama

We provide a theoretical explanation for the effectiveness of gradient clipping in training deep neural networks. The key ingredient is a new smoothness condition derived from practical neural network training examples. We observe that…

Optimization and Control · Mathematics 2020-02-12 Jingzhao Zhang , Tianxing He , Suvrit Sra , Ali Jadbabaie

Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain -- most notably,…

Machine Learning · Computer Science 2026-02-05 Róisín Luo

Real-life applications of deep neural networks are hindered by their unsteady predictions when faced with noisy inputs and adversarial attacks. The certified radius in this context is a crucial indicator of the robustness of models. However…

Machine Learning · Computer Science 2024-03-19 Blaise Delattre , Alexandre Araujo , Quentin Barthélemy , Alexandre Allauzen

We generalize the notion of average Lipschitz smoothness proposed by Ashlagi et al. (COLT 2021) by extending it to H\"older smoothness. This measure of the "effective smoothness" of a function is sensitive to the underlying distribution and…

Machine Learning · Computer Science 2023-10-31 Steve Hanneke , Aryeh Kontorovich , Guy Kornowski

Neural networks achieve outstanding accuracy in classification and regression tasks. However, understanding their behavior still remains an open challenge that requires questions to be addressed on the robustness, explainability and…

Machine Learning · Computer Science 2021-05-13 Anna-Kathrin Kopetzki , Stephan Günnemann

To foster trust in machine learning models, explanations must be faithful and stable for consistent insights. Existing relevant works rely on the $\ell_p$ distance for stability assessment, which diverges from human perception. Besides,…

Machine Learning · Computer Science 2024-12-30 Chao Chen , Chenghua Guo , Rufeng Chen , Guixiang Ma , Ming Zeng , Xiangwen Liao , Xi Zhang , Sihong Xie

In this paper, we analyze the properties of invertible neural networks, which provide a way of solving inverse problems. Our main focus lies on investigating and controlling the Lipschitz constants of the corresponding inverse networks.…

Machine Learning · Computer Science 2021-09-01 Paul Hagemann , Sebastian Neumayer

Several recent papers have discussed utilizing Lipschitz constants to limit the susceptibility of neural networks to adversarial examples. We analyze recently proposed methods for computing the Lipschitz constant. We show that the Lipschitz…

Machine Learning · Computer Science 2018-07-26 Todd Huster , Cho-Yu Jason Chiang , Ritu Chadha

Recent work has shown that state-of-the-art classifiers are quite brittle, in the sense that a small adversarial change of an originally with high confidence correctly classified input leads to a wrong classification again with high…

Machine Learning · Computer Science 2017-11-07 Matthias Hein , Maksym Andriushchenko

While the class of Polynomial Nets demonstrates comparable performance to neural networks (NN), it currently has neither theoretical generalization characterization nor robustness guarantees. To this end, we derive new complexity bounds for…

Machine Learning · Computer Science 2022-02-11 Zhenyu Zhu , Fabian Latorre , Grigorios G Chrysos , Volkan Cevher

A recent technique of randomized smoothing has shown that the worst-case (adversarial) $\ell_2$-robustness can be transformed into the average-case Gaussian-robustness by "smoothing" a classifier, i.e., by considering the averaged…

Machine Learning · Computer Science 2021-01-11 Jongheon Jeong , Jinwoo Shin

Attention is a powerful component of modern neural networks across a wide variety of domains. In this paper, we seek to quantify the regularity (i.e. the amount of smoothness) of the attention operation. To accomplish this goal, we propose…

Machine Learning · Statistics 2021-02-11 James Vuckovic , Aristide Baratin , Remi Tachet des Combes

Randomized smoothing has shown promising certified robustness against adversaries in classification tasks. Despite such success with only zeroth-order access to base models, randomized smoothing has not been extended to a general form of…

Machine Learning · Computer Science 2024-05-16 Aref Miri Rekavandi , Olga Ohrimenko , Benjamin I. P. Rubinstein

Recent work in imitation learning has shown that having an expert controller that is both suitably smooth and stable enables stronger guarantees on the performance of the learned controller. However, constructing such smoothed expert…

Systems and Control · Electrical Eng. & Systems 2024-10-02 Daniel Pfrommer , Swati Padmanabhan , Kwangjun Ahn , Jack Umenberger , Tobia Marcucci , Zakaria Mhammedi , Ali Jadbabaie