Piecewise Polynomial Regression of Tame Functions via Integer Programming

Gilles Bareilles, Johannes Aspman, Jiří Němeček, Jakub Marecek

Keywords: tame functions, o-minimality, piecewise polynomial functions, regression

TL;DR: We approximate nonsmooth nonconvex, yet tame, functions with piecewise polynomial functions.

Abstract: Tame functions are a class of nonsmooth, nonconvex functions that appear in a wide range of applications: in training deep neural networks with all common activations, as value functions of mixed-integer programs, or as wave functions of small molecules. We consider approximating tame functions with piecewise polynomial functions. We present a theoretical bound on the approximation quality of a tame function by a piecewise polynomial function. We also present mixed-integer programming formulations of piecewise polynomial regression and demonstrate promising computational results.

Primary Area: learning theory

Submission Number: 4369