Run-and-Inspect Method for Nonconvex Optimization and Global Optimality Bounds for R-Local Minimizers
  Many optimization algorithms converge to stationary points. When the
underlying problem is nonconvex, they may get trapped at local minimizers and
occasionally stagnate near saddle points. We propose the Run-and-Inspect
Method, which adds an "inspect" phase to existing algorithms that helps escape
from non-global stationary points. The inspection samples a set of points in a
radius $R$ around the current point. When a sample point yields a sufficient
decrease in the objective, we move there and resume an existing algorithm. If
no sufficient decrease is found, the current point is called an approximate
$R$-local minimizer. We show that an $R$-local minimizer is globally optimal,
up to a specific error depending on $R$, if the objective function can be
implicitly decomposed into a smooth convex function plus a restricted function
that is possibly nonconvex, nonsmooth. For high-dimensional problems, we
introduce blockwise inspections to overcome the curse of dimensionality while
still maintaining optimality bounds up to a factor equal to the number of
blocks. Our method performs well on a set of artificial and realistic nonconvex
problems by coupling with gradient descent, coordinate descent, EM, and
prox-linear algorithms.
