Lipschitz behavior of the robust regularization

TL;DR

This paper investigates the Lipschitz properties of epsilon-robust regularizations of functions, showing that for certain classes like semi-algebraic functions, the regularization is Lipschitz continuous around any point for small epsilon, even if the original function is not.

Contribution

It demonstrates that the epsilon-robust regularization is Lipschitz continuous for semi-algebraic functions, providing theoretical insights into the stability of robust regularization.

Findings

Robust regularization can be Lipschitz even if the original function is not.

Semi-algebraic functions have Lipschitz regularizations around any point for small epsilon.

The results apply to functions like spectral radius and convex quadratics.

Abstract

To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius of a matrix leads to pseudospectral computations. For favorable classes of functions, we show that the robust regularization is Lipschitz around any given point, for all small epsilon > 0, even if the original function is nonlipschitz (like the spectral radius). One such favorable class consists of the semi-algebraic functions. Such functions have graphs that are finite unions of sets defined by finitely-many polynomial inequalities, and are commonly encountered in applications.