A theory on the absence of spurious solutions for nonconvex and nonsmooth optimization

TL;DR

This paper introduces the concept of global functions, a class of continuous functions with no spurious local optima, providing theoretical guarantees for nonconvex nonsmooth optimization problems, including tensor decomposition and robust $ ext{l}_1$ norm methods.

Contribution

It defines global functions and proves their properties, establishing the first theoretical results for nonconvex nonsmooth optimization problems.

Findings

Global functions have properties similar to continuous functions in classical analysis.

Nonconvex nonsmooth problems in tensor decomposition are shown to be global functions.

The $ ext{l}_1$ norm effectively avoids outliers in nonconvex optimization.

Abstract

We study the set of continuous functions that admit no spurious local optima (i.e. local minima that are not global minima) which we term \textit{global functions}. They satisfy various powerful properties for analyzing nonconvex and nonsmooth optimization problems. For instance, they satisfy a theorem akin to the fundamental uniform limit theorem in the analysis regarding continuous functions. Global functions are also endowed with useful properties regarding the composition of functions and change of variables. Using these new results, we show that a class of nonconvex and nonsmooth optimization problems arising in tensor decomposition applications are global functions. This is the first result concerning nonconvex methods for nonsmooth objective functions. Our result provides a theoretical guarantee for the widely-used $\ell_1$ norm to avoid outliers in nonconvex optimization.