Deep Neural Networks as 0-1 Mixed Integer Linear Programs: A Feasibility Study

TL;DR

This paper explores modeling deep neural networks with ReLU activations as 0-1 mixed integer linear programs, discussing solution techniques, applications in feature visualization and adversarial example generation, and preliminary computational results.

Contribution

It introduces a bound-tightening technique for 0-1 MILP models of DNNs and investigates their application in feature visualization and adversarial example construction.

Findings

Effective bound-tightening improves MILP solution efficiency.

Preliminary results on small DNNs show promising computational performance.

Applications demonstrated include feature visualization and adversarial example generation.

Abstract

Deep Neural Networks (DNNs) are very popular these days, and are the subject of a very intense investigation. A DNN is made by layers of internal units (or neurons), each of which computes an affine combination of the output of the units in the previous layer, applies a nonlinear operator, and outputs the corresponding value (also known as activation). A commonly-used nonlinear operator is the so-called rectified linear unit (ReLU), whose output is just the maximum between its input value and zero. In this (and other similar cases like max pooling, where the max operation involves more than one input value), one can model the DNN as a 0-1 Mixed Integer Linear Program (0-1 MILP) where the continuous variables correspond to the output values of each unit, and a binary variable is associated with each ReLU to model its yes/no nature. In this paper we discuss the peculiarity of this kind of 0-1 MILP models, and describe an effective bound-tightening technique intended to ease its solution. We also present possible applications of the 0-1 MILP model arising in feature visualization and in the construction of adversarial examples. Preliminary computational results are reported, aimed at investigating (on small DNNs) the computational performance of a state-of-the-art MILP solver when applied to a known test case, namely, hand-written digit recognition.