Generalised Lipschitz Regularisation Equals Distributional Robustness
This provides a foundational theoretical framework connecting adversarial robustness and distributional robustness, applicable to exotic function classes like universal approximators.
The paper establishes a theoretical equivalence between generalized Lipschitz regularization and distributional robustness with transportation cost uncertainty sets, enabling tight robustness certification for Lipschitz-regularized models under mild assumptions.
The problem of adversarial examples has highlighted the need for a theory of regularisation that is general enough to apply to exotic function classes, such as universal approximators. In response, we give a very general equality result regarding the relationship between distributional robustness and regularisation, as defined with a transportation cost uncertainty set. The theory allows us to (tightly) certify the robustness properties of a Lipschitz-regularised model with very mild assumptions. As a theoretical application we show a new result explicating the connection between adversarial learning and distributional robustness. We then give new results for how to achieve Lipschitz regularisation of kernel classifiers, which are demonstrated experimentally.