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

TL;DR

The paper introduces the Run-and-Inspect Method, enhancing existing optimization algorithms with an inspection phase to escape local minima and provide global optimality bounds for nonconvex problems, especially in high dimensions.

Contribution

It proposes a novel inspection technique that improves nonconvex optimization algorithms and establishes bounds for R-local minimizers, including blockwise methods for high-dimensional problems.

Findings

Method effectively escapes local minima in nonconvex problems.

Provides global optimality bounds for R-local minimizers.

Performs well with various algorithms on artificial and real-world problems.

Abstract

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.