Min-max optimization , with a nonconvex-nonconcave objective function f: R^d \times R^d \rightarrow R , arises in many areas, including optimization, economics, and deep learning. The nonconvexity-nonconcavity of f means that the problem of finding a global \varepsilon-min-max point cannot be solved in \mathrm{poly(d, 1{\varepsilon}) 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 pointfor 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^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star 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 f(x,y), and \nabla^2_y f(x,y), } that is polynomial in \frac{1{\varepsilon}}  , the dimension d, and the bounds on f and its first three derivatives. Our results do not require convexity, monotonicity, or special starting points .
Min-max optimization of an objective function f: R^d \times R^d \rightarrow R is an important model for robustness in an adversarial setting, with applications to many areas including optimization, economics, and deep learning. In many of these applications f may be nonconvex-nonconcave, and finding a global (d, 1{\varepsilon}) evaluations of f. Thus, most algorithms seek to obtain a certain notion of local } min-max point may be computationally intractable. There is a long line of work that seeks computationally tractable algorithms for alternatives to the min-max optimization model . However, many of the alternative models have solution points which are only guaranteed to exist under strong assumptions on f, such as convexity , monotonicity, or special properties of the starting point. In this paper, we propose an optimization model, the \varepsilon-greedy adversarial equilibrium, which can serve as a computationally tractable alternative to the min-max optimization model. Roughly , we say a point (x^\star, y^\star) is an \varepsilon-greedy ^d \times \mathbb{R}^d \rightarrow \mathbb{R} } adversarial equilibrium if y^\star is an \varepsilon-approximate local maximum for f(x^\star,\cdot) , and x^\star is an \varepsilon-approximate local minimum for a "greedy approximation" to the function \max _z f(x, z ) which can be efficiently estimated using second-order optimization algorithms. The existence follows from an algorithm that converges from any starting point to such a point in a number of evaluations to f, \nabla_{y f(x,y), and \nabla^2_y f(x,y), } that is polynomial in {\varepsilon}} 1/\varepsilon , the dimension d, and the bounds on f and its Lipschitz constant. In addition to existence, our model retains many desirable properties of the min-max model. For instance, it empowers the min-player to make updates that take into account the max-player's response, and in the case of strong convexity/concavity it corresponds to a global min-max solution with duality gap O(\epsilon^2) .
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