In Proximity of ReLU DNN, PWA Function, and Explicit MPC

TL;DR

This paper explores the relationship between ReLU deep neural networks, piecewise affine functions, and explicit model predictive control, providing theoretical insights and methods for representation and reconstruction.

Contribution

It introduces a method to identify PWA functions from ReLU DNNs and studies inverse mp-LP and mp-QP problems for reconstructing control policies.

Findings

ReLU DNNs can be approximated by PWA functions over polyhedral regions.

The paper provides theorems on the complexity and architecture of ReLU DNNs.

An inverse problem approach for reconstructing control constraints and costs from PWA functions.

Abstract

Rectifier (ReLU) deep neural networks (DNN) and their connection with piecewise affine (PWA) functions is analyzed. The paper is an effort to find and study the possibility of representing explicit state feedback policy of model predictive control (MPC) as a ReLU DNN, and vice versa. The complexity and architecture of DNN has been examined through some theorems and discussions. An approximate method has been developed for identification of input-space in ReLU net which results a PWA function over polyhedral regions. Also, inverse multiparametric linear or quadratic programs (mp-LP or mp-QP) has been studied which deals with reconstruction of constraints and cost function given a PWA function.