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This theoretical study establishes rigorous lower bounds on the local robustness of deep ReLU networks by leveraging high-dimensional geometry and concentration of measure principles. The analysis reveals that decision-boundary neighborhoods containing adversarial examples shrink rapidly as input dimension increases, providing critical mathematical insight into the inherent vulnerability scaling of neural architectures. These findings offer a foundational understanding of how network width and depth geometrically constrain resistance to random perturbations, independent of specific training data or architecture choices.
Theoretical Lower Bounds on the Robustness of Deep ReLU Networks
arXiv:2602.18674v2 Announce Type: replace Abstract: We present a theoretical study of the robustness of parameterized neural networks to random input perturbations. Specifically, we analyze local robustness by quantifying the probability that a random L_2-perturbation of a given input results in a correct classification. For deep ReLU networks, we derive lower bounds on local robustness by combining tools from high-dimensional geometry, in particular concentration of measure, with a new characterization of the geometric structure induced by their input-output functions. We prove that each convex polyhedral region in the partition of the input space induced by a ReLU network has at most as many faces as there are network units, regardless of the network depth or architecture. This geometric property serves as the key ingredient in our robustness analysis. Finally, we analyze how local robustness scales with input dimension and characterize the sets of inputs whose neighborhoods are most likely to contain adversarial examples. We show that the width of decision-boundary neighborhoods containing vulnerable points shrinks rapidly as dimension increases and grows only logarithmically with the number of network units. We also discuss the volume of a set of vulnerable points in terms of approximately space-filling shapes of decision boundaries.
Abstract:We present a theoretical study of the robustness of parameterized neural networks to random input perturbations. Specifically, we analyze local robustness by quantifying the probability that a random L_2-perturbation of a given input results in a correct classification. For deep ReLU networks, we derive lower bounds on local robustness by combining tools from high-dimensional geometry, in particular concentration of measure, with a new characterization of the geometric structure induced by their input-output functions. We prove that each convex polyhedral region in the partition of the input space induced by a ReLU network has at most as many faces as there are network units, regardless of the network depth or architecture. This geometric property serves as the key ingredient in our robustness analysis. Finally, we analyze how local robustness scales with input dimension and characterize the sets of inputs whose neighborhoods are most likely to contain adversarial examples. We show that the width of decision-boundary neighborhoods containing vulnerable points shrinks rapidly as dimension increases and grows only logarithmically with the number of network units. We also discuss the volume of a set of vulnerable points in terms of approximately space-filling shapes of decision boundaries.
Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE)
arXiv:2602.18674 [cs.LG]
(or arXiv:2602.18674v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2602.18674 Focus to learn more arXiv-issued DOI via DataCite
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