Probabilistic Verification of ReLU Neural Networks via Characteristic Functions

TL;DR

This paper introduces a probabilistic verification method for ReLU neural networks using characteristic functions, enabling distribution propagation without requiring moments, and providing guarantees on network performance.

Contribution

It presents a novel approach leveraging characteristic functions for probabilistic verification of neural networks, avoiding the need for well-defined moments.

Findings

Effective distribution propagation through neural networks

No requirement for moments or moment generating functions

Demonstrated on two examples with performance comparison

Abstract

Verifying the input-output relationships of a neural network so as to achieve some desired performance specification is a difficult, yet important, problem due to the growing ubiquity of neural nets in many engineering applications. We use ideas from probability theory in the frequency domain to provide probabilistic verification guarantees for ReLU neural networks. Specifically, we interpret a (deep) feedforward neural network as a discrete dynamical system over a finite horizon that shapes distributions of initial states, and use characteristic functions to propagate the distribution of the input data through the network. Using the inverse Fourier transform, we obtain the corresponding cumulative distribution function of the output set, which can be used to check if the network is performing as expected given any random point from the input set. The proposed approach does not require distributions to have well-defined moments or moment generating functions. We demonstrate our proposed approach on two examples, and compare its performance to related approaches.