# A Second-order Equilibrium in Nonconvex-Nonconcave Min-max Optimization: Existence and Algorithm

Min-max optimization, with a nonconvex-nonconcave objective function f: ℝ^d ×ℝ^d →ℝ, arises in many areas, including optimization, economics, and deep learning. The nonconvexity-nonconcavity of f means that the problem of finding a global ε-min-max point cannot be solved in poly(d, 1/ε) evaluations of f. Thus, most algorithms seek to obtain a certain notion of local min-max point where, roughly speaking, each player optimizes her payoff in a local sense. However, the classes of local min-max solutions which prior algorithms seek are only guaranteed to exist under very strong assumptions on f, such as convexity or monotonicity. We propose a notion of a greedy equilibrium point for min-max optimization and prove the existence of such a point for any function such that it and its first three derivatives are bounded. Informally, we say that a point (x^⋆, y^⋆) is an ε-greedy min-max equilibrium point of a function f: ℝ^d ×ℝ^d →ℝ if y^⋆ is a second-order local maximum for f(x^⋆,·) and, roughly, x^⋆ is a local minimum for a greedy optimization version of the function max_y f(x,y) which can be efficiently estimated using greedy algorithms. The existence follows from an algorithm that converges from any starting point to such a point in a number of gradient and function evaluations that is polynomial in 1/ε, the dimension d, and the bounds on f and its first three derivatives. Our results do not require convexity, monotonicity, or special starting points.