First Order Methods take Exponential Time to Converge to Global Minimizers of Non-Convex Functions
This work addresses the fundamental computational limits of optimization in machine learning, showing that global minimization for non-convex functions is inherently hard, which is foundational for algorithm design.
The authors tackled the problem of global optimization for non-convex functions, proving that first-order methods can require exponential time to converge to a global minimizer by linking it to parameter estimation hardness.
Machine learning algorithms typically perform optimization over a class of non-convex functions. In this work, we provide bounds on the fundamental hardness of identifying the global minimizer of a non convex function. Specifically, we design a family of parametrized non-convex functions and employ statistical lower bounds for parameter estimation. We show that the parameter estimation problem is equivalent to the problem of function identification in the given family. We then claim that non convex optimization is at least as hard as function identification. Jointly, we prove that any first order method can take exponential time to converge to a global minimizer.