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 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 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^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: R^d \times R^d \rightarrow R if y^\star is a second-order local maximum for f(x^\star,\cdot) in a sense of Nesterov 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 that is polynomial in \frac{1{\epsilon}} {\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, 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 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 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^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: R^d \times R^d \rightarrow 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 that is polynomial in {\epsilon}} \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.
D
in a sense of Nesterov
D
\frac{1
A
\frac{1