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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

We propose an algorithm to enhance certified robustness of a deep model ensemble by optimally weighting each base model. Unlike previous works on using ensembles to empirically improve robustness, our algorithm is based on optimizing a…

Machine Learning · Statistics 2019-11-01 Huan Zhang , Minhao Cheng , Cho-Jui Hsieh

Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not applicable. We will consider a way of introducing a derivative…

Machine Learning · Computer Science 2020-11-12 Julius Berner , Dennis Elbrächter , Philipp Grohs , Arnulf Jentzen

Lipschitz-based certification offers efficient, deterministic robustness guarantees but has struggled to scale in model size, training efficiency, and ImageNet performance. We introduce \emph{LipNeXt}, the first \emph{constraint-free} and…

Machine Learning · Computer Science 2026-01-27 Kai Hu , Haoqi Hu , Matt Fredrikson

Since the control of the Lipschitz constant has a great impact on the training stability, generalization, and robustness of neural networks, the estimation of this value is nowadays a real scientific challenge. In this paper we introduce a…

Machine Learning · Computer Science 2023-06-21 Blaise Delattre , Quentin Barthélemy , Alexandre Araujo , Alexandre Allauzen

Randomized smoothing has become a leading approach for certifying adversarial robustness in machine learning models. However, a persistent gap remains between theoretical certified robustness and empirical robustness accuracy. This paper…

Machine Learning · Computer Science 2025-04-10 Blaise Delattre , Paul Caillon , Quentin Barthélemy , Erwan Fagnou , Alexandre Allauzen

A globally robust deep neural network resists perturbations on all meaningful inputs. Current robustness certification methods emphasize local robustness, struggling to scale and generalize. This paper presents a systematic and efficient…

Machine Learning · Computer Science 2024-06-03 You Li , Guannan Zhao , Shuyu Kong , Yunqi He , Hai Zhou

We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bounds on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite…

Machine Learning · Computer Science 2020-04-21 Fabian Latorre , Paul Rolland , Volkan Cevher

Recent works have introduced input-convex neural networks (ICNNs) as learning models with advantageous training, inference, and generalization properties linked to their convex structure. In this paper, we propose a novel feature-convex…

Machine Learning · Computer Science 2023-10-11 Samuel Pfrommer , Brendon G. Anderson , Julien Piet , Somayeh Sojoudi

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

Robustness of neural networks is commonly quantified via local or global Lipschitz constants. However, Lipschitz continuity can be overly coarse or overly restrictive as global robustness measure, failing to capture nuanced, data-dependent…

Machine Learning · Statistics 2026-05-28 Jürgen Dölz , Michael Multerer , Michele Palma

Deployment of deep neural networks (DNNs) in safety- or security-critical systems requires provable guarantees on their correct behaviour. A common requirement is robustness to adversarial perturbations in a neighbourhood around an input.…

Machine Learning · Computer Science 2018-11-21 Wenjie Ruan , Min Wu , Youcheng Sun , Xiaowei Huang , Daniel Kroening , Marta Kwiatkowska

We propose a novel Branch-and-Bound method for reachability analysis of neural networks in both open-loop and closed-loop settings. Our idea is to first compute accurate bounds on the Lipschitz constant of the neural network in certain…

Systems and Control · Electrical Eng. & Systems 2023-04-20 Taha Entesari , Sina Sharifi , Mahyar Fazlyab

In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the…

Machine Learning · Computer Science 2019-09-13 Chris Finlay , Jeff Calder , Bilal Abbasi , Adam Oberman

Using weight decay to penalize the L2 norms of weights in neural networks has been a standard training practice to regularize the complexity of networks. In this paper, we show that a family of regularizers, including weight decay, is…

Machine Learning · Computer Science 2022-06-09 Ziquan Liu , Yufei Cui , Antoni B. Chan

A domain-theoretic framework is presented for validated robustness analysis of neural networks. First, global robustness of a general class of networks is analyzed. Then, using the fact that Edalat's domain-theoretic L-derivative coincides…

Machine Learning · Computer Science 2023-01-10 Can Zhou , Razin A. Shaikh , Yiran Li , Amin Farjudian

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

With deep neural networks providing state-of-the-art machine learning models for numerous machine learning tasks, quantifying the robustness of these models has become an important area of research. However, most of the research literature…

Machine Learning · Computer Science 2019-01-08 Tsui-Wei Weng , Pin-Yu Chen , Lam M. Nguyen , Mark S. Squillante , Ivan Oseledets , Luca Daniel

Training neural networks with verifiable robustness guarantees is challenging. Several existing approaches utilize linear relaxation based neural network output bounds under perturbation, but they can slow down training by a factor of…

Machine Learning · Computer Science 2019-11-28 Huan Zhang , Hongge Chen , Chaowei Xiao , Sven Gowal , Robert Stanforth , Bo Li , Duane Boning , Cho-Jui Hsieh

Neural networks (NNs) are now routinely implemented on systems that must operate in uncertain environments, but the tools for formally analyzing how this uncertainty propagates to NN outputs are not yet commonplace. Computing tight bounds…

Machine Learning · Computer Science 2020-12-08 Michael Everett , Golnaz Habibi , Jonathan P. How