Journal Article

PUBDB-2025-03598

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Adversarial flows: A gradient flow characterization of adversarial attacks

Weigand, L.DESY* ; Roith, T. (Corresponding author)Extern*DESY* ; Burger, M.DESY*

2025
Cambridge Univ. Press Cambridge

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Please use a persistent id in citations: doi:10.1017/S0956792525100120  doi:10.3204/PUBDB-2025-03598

Report No.: arXiv:2406.05376

Abstract: A popular method to perform adversarial attacks on neuronal networks is the so-called fast gradient sign method and its iterative variant. In this paper, we interpret this method as an explicit Euler discretization of a differential inclusion, where we also show convergence of the discretization to the associated gradient flow. To do so, we consider the concept of p-curves of maximal slope in the case $p=\infty$. We prove existence of $\infty$-curves of maximum slope and derive an alternative characterization via differential inclusions. Furthermore, we also consider Wasserstein gradient flows for potential energies, where we show that curves in the Wasserstein space can be characterized by a representing measure on the space of curves in the underlying Banach space, which fulfill the differential inclusion. The application of our theory to the finite-dimensional setting is twofold: On the one hand, we show that a whole class of normalized gradient descent methods (in particular signed gradient descent) converge, up to subsequences, to the flow, when sending the step size to zero. On the other hand, in the distributional setting, we show that the inner optimization task of adversarial training objective can be characterized via $\infty$-curves of maximum slope on an appropriate optimal transport space.

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Contributing Institute(s):

1. Computational Imaging (FS-CI)

Research Program(s):

1. 623 - Data Management and Analysis (POF4-623) (POF4-623)

Experiment(s):

1. No specific instrument

Appears in the scientific report 2025

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