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Deep Neural Networks (DNNs) are very popular these days, and are the subjectof a very intense investigation. A DNN is made by layers of internal units (orneurons), each of which computes an affine combination of the output of theunits in the previous layer, applies a nonlinear operator, and outputs thecorresponding value (also known as activation). A commonly-used nonlinearoperator is the so-called rectified linear unit (ReLU), whose output is justthe maximum between its input value and zero. In this (and other similar caseslike 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) wherethe continuous variables correspond to the output values of each unit, and abinary variable is associated with each ReLU to model its yes/no nature. Inthis paper we discuss the peculiarity of this kind of 0-1 MILP models, anddescribe an effective bound-tightening technique intended to ease its solution.We also present possible applications of the 0-1 MILP model arising in featurevisualization and in the construction of adversarial examples. Preliminarycomputational results are reported, aimed at investigating (on small DNNs) thecomputational performance of a state-of-the-art MILP solver when applied to aknown test case, namely, hand-written digit recognition.

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