The Complexity of Constrained Min-Max Optimization

Despite its important applications in Machine Learning, min-max optimization of nonconvex-nonconcave objectives remains elusive. Not only are there no known first-order methods converging even to approximate local min-max points, but the computational complexity of identifying them is also poorly understood. In this paper, we provide a characterization of the computational complexity of the problem, as well as of the limitations of first-order methods in constrained min-max optimization problems with nonconvex-nonconcave objectives and linear constraints. As a warm-up, we show that, even when the objective is a Lipschitz and smooth differentiable function, deciding whether a min-max point exists, in fact even deciding whether an approximate min-max point exists, is NP-hard. More importantly, we show that an approximate local min-max point of large enough approximation is guaranteed to exist, but finding one such point is PPAD-complete. The same is true of computing an approximate fixed point of Gradient Descent/Ascent. An important byproduct of our proof is to establish an unconditional hardness result in the Nemirovsky-Yudin model. We show that, given oracle access to some function f : P → [-1, 1] and its gradient nabla f, where P subseteq [0, 1]d is a known convex polytope, every algorithm that finds a varepsilon-approximate local min-max point needs to make a number of queries that is exponential in at least one of 1/varepsilon, L, G, or d, where L and G are respectively the smoothness and Lipschitzness of f and d is the dimension. This comes in sharp contrast to minimization problems, where finding approximate local minima in the same setting can be done with Projected Gradient Descent using O(L/varepsilon) many queries. Our result is the first to show an exponential separation between these two fundamental optimization problems.