Using Bifurcations for Diversity in Differentiable Games

Abstract

Ridge Rider (RR) is an algorithm for finding diverse solutions to optimization problems by following eigenvectors of the Hessian ("ridges''). RR is designed for conservative gradient systems (i.e. settings involving a single loss function), where it branches at saddles -- the only relevant bifurcation points. We generalize this idea to non-conservative, multi-agent gradient systems by identifying new types of bifurcation points and proposing a method to follow eigenvectors with complex eigenvalues. We give theoretical motivation for our method -- denoted Game Ridge Rider (GRR) -- by leveraging machinery from the field of dynamical systems. Finally, we empirically motivate our method by constructing novel toy problems where we can visualize new phenomena and by finding diverse solutions in the iterated prisoners' dilemma, where existing methods often converge to a single solution mode.