On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it

Abstract

Abstract We propose iterative algorithms to solve adversarial training problems in a variety of supervised learning settings of interest. Our algorithms, which can be interpreted as suitable ascent-descent dynamics in Wasserstein spaces, take the form of a system of interacting particles. These interacting particle dynamics are shown to converge toward appropriate mean-field limit equations in certain large number of particles regimes. In turn, we prove that, under certain regularity assumptions, these mean-field equations converge, in the large time limit, toward approximate Nash equilibria of the original adversarial learning problems. We present results for non-convex non-concave settings, as well as for non-convex concave ones. Numerical experiments illustrate our results.