Approaching the Global Nash Equilibrium of Non-convex Multi-player Games

Many machine learning problems can be formulated as non-convex multi-player games. Due to non-convexity, it is challenging to obtain the existence condition of the global Nash equilibrium (NE) and design theoretically guaranteed algorithms. This paper studies a class of non-convex multi-player games, where players' payoff functions consist of canonical functions and quadratic operators. We leverage conjugate properties to transform the complementary problem into a variational inequality (VI) problem using a continuous pseudo-gradient mapping. We prove the existence condition of the global NE as the solution to the VI problem satisfies a duality relation. We then design an ordinary differential equation to approach the global NE with an exponential convergence rate. For practical implementation, we derive a discretized algorithm and apply it to two scenarios: multi-player games with generalized monotonicity and multi-player potential games. In the two settings, step sizes are 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 on robust neural network training and sensor network localization validate our theory. Our code is available at
https://github.com/GuanpuChen/Global-NE
.