Duality for the Adversarial Total Variation
This work provides theoretical foundations for adversarial training by linking it to nonlocal total variation, offering new analytical tools for researchers in robust machine learning.
The paper establishes a dual representation and subdifferential characterization of the adversarial total variation, a regularizer arising from adversarial training of binary classifiers, using duality techniques in both continuous and bounded function spaces.
Adversarial training of binary classifiers can be reformulated as regularized risk minimization involving a nonlocal total variation. Building on this perspective, we establish a characterization of the subdifferential of this total variation using duality techniques. To achieve this, we derive a dual representation of the nonlocal total variation and a related integration of parts formula, involving a nonlocal gradient and divergence. We provide such duality statements both in the space of continuous functions vanishing at infinity on proper metric spaces and for the space of essentially bounded functions on Euclidean domains. Furthermore, under some additional conditions we provide characterizations of the subdifferential in these settings.