  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.
