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 nonconcave 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 provide conditions under which vanilla GDA provably converges not merely to 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 a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence results are non-local despite working in the setting of non-convex non-concave games. Critically, under proper assumptions we 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 framework ranging from generative adversarial networks to evolutionary biology.

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 nonconcave 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 provide conditions under which vanilla GDA provably converges not merely to 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 a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence results are non-local despite working in the setting of non-convex non-concave games. Critically, under proper assumptions we 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 framework ranging from generative adversarial networks to evolutionary biology.

Reaching consensus and convergence to equilibrium are two major challenges of multi-agent systems. Although each has attracted significant attention, relatively few studies address both challenges at the same time. This paper examines the connection between the notions of consensus and equilibrium in a multi-agent system where multiple interacting sub-populations coexist. We argue that consensus can be seen as an intricate component of intra-population stability, whereas equilibrium can be seen as encoding inter-population stability. We show that smooth fictitious play, a well-known learning model in game theory, can achieve both consensus and convergence to equilibrium in diverse multi-agent settings. Moreover, we show that the consensus formation process plays a crucial role in the seminal thorny problem of equilibrium selection in multi-agent learning.

Neural Information Processing Systems http://nips.cc/

An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint.

Notef = andg = are functions. Theorem 4.4 withrespecttok k1-norm; see Example A.3 in Section A.2.1.

Neural Information Processing Systems http://nips.cc/

This phenomenon, that we coin "fast and furious" learning in games, sets a new benchmark about what is possible both in max-min optimization as well as in multi-agent systems.