Zero-sum games arise in a wide variety of problems, including robust optimization and adversarial learning. However, algorithms deployed for finding a local Nash equilibrium in these games often converge to non-Nash stationary points. This highlights a key challenge: for any algorithm, the stability properties of its underlying dynamical system can cause non-Nash points to be potential attractors. To overcome this challenge, algorithms must account for subtleties involving the curvatures of players' costs. To this end, we leverage dynamical system theory and develop a second-order algorithm for finding a local Nash equilibrium in the smooth, possibly nonconvex-nonconcave, zero-sum game setting. First, we prove that this novel method guarantees convergence to only local Nash equilibria with a local linear convergence rate. We then interpret a version of this method as a modified Gauss-Newton algorithm with local superlinear convergence to the neighborhood of a point that satisfies first-order local Nash equilibrium conditions. In comparison, current related state-of-the-art methods do not offer convergence rate guarantees. Furthermore, we show that this approach naturally generalizes to settings with convex and potentially coupled constraints while retaining earlier guarantees of convergence to only local (generalized) Nash equilibria.

翻译：零和博弈出现在多种问题中，包括鲁棒优化和对抗学习。然而，用于在这些博弈中寻找局部纳什均衡的算法通常收敛到非纳什的驻点。这凸显了一个关键挑战：对于任何算法，其底层动力系统的稳定性特性可能导致非纳什点成为潜在的吸引子。为克服这一挑战，算法必须考虑玩家成本曲率的细微差别。为此，我们利用动力系统理论，开发了一种二阶算法，用于在平滑、可能非凸-非凹的零和博弈场景中寻找局部纳什均衡。首先，我们证明这一新方法能保证仅收敛到局部纳什均衡，且具有局部线性收敛速度。随后，我们将该方法的一种变体解释为修正的高斯-牛顿算法，该算法在满足一阶局部纳什均衡条件的点附近具有局部超线性收敛性。相比之下，当前相关的最先进方法无法提供收敛速度保证。此外，我们表明该方法自然推广到具有凸且可能耦合约束的场景，同时保留仅收敛到局部（广义）纳什均衡的先前保证。