Run-and-Inspect Method for Nonconvex Optimization and Global Optimality Bounds for R-Local Minimizers
This addresses the challenge of global optimization in nonconvex settings for researchers and practitioners, though it is incremental as it builds on existing algorithms.
The paper tackles the problem of optimization algorithms getting trapped in local minima or saddle points in nonconvex optimization by proposing the Run-and-Inspect Method, which adds an inspection phase to escape non-global stationary points and shows that R-local minimizers are globally optimal up to a specific error depending on R, with performance demonstrated on artificial and realistic problems.
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.