Neural Network Verification as Piecewise Linear Optimization: Formulations for the Composition of Staircase Functions

TL;DR

This paper develops a mixed-integer programming approach for neural network verification, introducing strong formulations and separation procedures for staircase functions, improving computational efficiency and accuracy in verification tasks.

Contribution

It introduces a novel strong MIP formulation and separation method for staircase activation functions, enhancing neural network verification efficiency and accuracy.

Findings

Reduced computational time in exact verification scenarios.

Improved false negative rates for inexact verifiers.

Applicable to various piecewise linear activation functions.

Abstract

We present a technique for neural network verification using mixed-integer programming (MIP) formulations. We derive a \emph{strong formulation} for each neuron in a network using piecewise linear activation functions. Additionally, as in general, these formulations may require an exponential number of inequalities, we also derive a separation procedure that runs in super-linear time in the input dimension. We first introduce and develop our technique on the class of \emph{staircase} functions, which generalizes the ReLU, binarized, and quantized activation functions. We then use results for staircase activation functions to obtain a separation method for general piecewise linear activation functions. Empirically, using our strong formulation and separation technique, we can reduce the computational time in exact verification settings based on MIP and improve the false negative rate for inexact verifiers relying on the relaxation of the MIP formulation.