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