Piecewise Polynomial Regression of Tame Functions via Integer Programming

TL;DR

This paper introduces a novel mixed-integer programming approach for piecewise polynomial regression to approximate tame functions, which are prevalent in various complex applications, providing theoretical bounds and promising computational results.

Contribution

It presents the first mixed-integer programming formulation for piecewise polynomial regression of tame functions, along with approximation bounds.

Findings

Bounded approximation quality for tame functions by piecewise polynomials.

First MIP formulation for piecewise polynomial regression.

Promising computational results demonstrating practical applicability.

Abstract

Tame functions are a class of nonsmooth, nonconvex functions, which feature in a wide range of applications: functions encountered in the training of deep neural networks with all common activations, value functions of mixed-integer programs, or wave functions of small molecules. We consider approximating tame functions with piecewise polynomial functions. We bound the quality of approximation of a tame function by a piecewise polynomial function with a given number of segments on any full-dimensional cube. We also present the first mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.