Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent

Dec 6, 2021



Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convexnon-concave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike generalzero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the “hidden" convex-concave game. We provide conditions under which vanilla GDA provably converges not merelyto local Nash, but the actual von-Neumann solution. If the hidden game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to avon-Neumann solution of a slightly perturbed zero-sum game. Our convergence guarantees are non-local, which as far as we know is a first-of-its-kind type of result in non-convex non-concave games. Critically, under proper assumptionswe combine the Center-Stable Manifold Theorem along with novel type of initialization dependent Lyapunov functions to prove that almost all initial conditions converge to the solution. Finally, we discuss diverse applications of our frameworkranging from generative adversarial networks to evolutionary biology.


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