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Wide machine learning tasks can be formulated as non-convex multi-playergames, where Nash equilibrium (NE) is an acceptable solution to all players,since no one can benefit from changing its strategy unilaterally. Attributed tothe non-convexity, obtaining the existence condition of global NE ischallenging, let alone designing theoretically guaranteed realizationalgorithms. This paper takes conjugate transformation to the formulation ofnon-convex multi-player games, and casts the complementary problem into avariational inequality (VI) problem with a continuous pseudo-gradient mapping.We then prove the existence condition of global NE: the solution to the VIproblem satisfies a duality relation. Based on this VI formulation, we design aconjugate-based ordinary differential equation (ODE) to approach global NE,which is proved to have an exponential convergence rate. To make the dynamicsmore implementable, we further derive a discretized algorithm. We apply ouralgorithm to two typical scenarios: multi-player generalized monotone game andmulti-player potential game. In the two settings, we prove that the step-sizesetting is required to be $\mathcal{O}(1/k)$ and $\mathcal{O}(1/\sqrt k)$ toyield the convergence rates of $\mathcal{O}(1/ k)$ and $\mathcal{O}(1/\sqrtk)$, respectively. Extensive experiments in robust neural network training andsensor localization are in full agreement with our theory.

Global Nash Equilibrium in Non-convex Multi-player Game: Theory and Algorithms