Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent
This provides theoretical guarantees for optimization in AI architectures like GANs, though it is incremental as it builds on existing GDA methods.
The paper tackles the problem of understanding gradient descent ascent (GDA) dynamics in non-convex non-concave zero-sum games with hidden structure, proving that vanilla GDA converges to the von-Neumann solution under strict convex-concave conditions and that regularization ensures convergence in other cases.
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-convex non-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 general zero-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 prove that if the hidden game is strictly convex-concave then vanilla GDA converges not merely to local Nash, but typically to the von-Neumann solution. If the game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to a von-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. Finally, we discuss connections of our framework with generative adversarial networks.