The Complexity of Min-Max Optimization for Quadratic Polynomials

The paper establishes hardness for approximate solutions, not just exact ones. This is the sharper result: even relaxed convergence targets remain computationally intractable for broad quadratic problem classes, which closes off the typical escape hatch practitioners use when exact solutions are infeasible.

This hardness result sits in tension with recent work on learning under geometric and structural constraints. The shape space analysis paper from the same day emphasizes that preserving geometric invariance improves robustness, yet this min-max finding suggests that even when you know the geometry (quadratic structure), finding equilibria remains hard. Similarly, the gradient stability work on residual connections formalizes why deep networks suffer training pathology, but this result implies that even well-behaved architectures face fundamental computational barriers when cast as adversarial optimization problems. The practical implication: architectural fixes and geometric priors help, but they don't resolve the underlying complexity ceiling for competitive multi-agent settings.

If practitioners report that heuristic min-max solvers (like alternating gradient descent) achieve acceptable convergence on real robotics or game-playing tasks despite this hardness result, that signals the hardness applies to worst-case instances that rarely appear in practice. Conversely, if new papers cite this result to explain why adversarial training plateaus on specific benchmarks, the theory is making contact with observed behavior.