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Many optimization algorithms converge to stationary points. When theunderlying problem is nonconvex, they may get trapped at local minimizers andoccasionally stagnate near saddle points. We propose the Run-and-InspectMethod, which adds an "inspect" phase to existing algorithms that helps escapefrom non-global stationary points. The inspection samples a set of points in aradius $R$ around the current point. When a sample point yields a sufficientdecrease in the objective, we move there and resume an existing algorithm. Ifno 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 beimplicitly decomposed into a smooth convex function plus a restricted functionthat is possibly nonconvex, nonsmooth. For high-dimensional problems, weintroduce blockwise inspections to overcome the curse of dimensionality whilestill maintaining optimality bounds up to a factor equal to the number ofblocks. Our method performs well on a set of artificial and realistic nonconvexproblems by coupling with gradient descent, coordinate descent, EM, andprox-linear algorithms.

Run-and-Inspect Method for Nonconvex Optimization and Global Optimality Bounds for R-Local Minimizers