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We propose iterative algorithms to solve adversarial problems in a variety ofsupervised learning settings of interest. Our algorithms, which can beinterpreted as suitable ascent-descent dynamics in Wasserstein spaces, take theform of a system of interacting particles. These interacting particle dynamicsare shown to converge toward appropriate mean-field limit equations in certainlarge number of particles regimes. In turn, we prove that, under certainregularity assumptions, these mean-field equations converge, in the large timelimit, toward approximate Nash equilibria of the original adversarial learningproblems. We present results for nonconvex-nonconcave settings, as well as fornonconvex-concave ones. Numerical experiments illustrate our results.

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