Probabilistic Verification and Reachability Analysis of Neural Networks via Semidefinite Programming

TL;DR

This paper introduces a probabilistic verification method for neural networks using semidefinite programming, enabling safety analysis and confidence ellipsoid estimation under input uncertainties with known moments.

Contribution

It proposes a novel approach to probabilistic safety verification and confidence ellipsoid estimation for neural networks by abstracting nonlinear activations with affine and quadratic constraints.

Findings

Effective semidefinite programming-based analysis.

Successful numerical experiments demonstrating approach.

Improved safety bounds under probabilistic input uncertainties.

Abstract

Quantifying the robustness of neural networks or verifying their safety properties against input uncertainties or adversarial attacks have become an important research area in learning-enabled systems. Most results concentrate around the worst-case scenario where the input of the neural network is perturbed within a norm-bounded uncertainty set. In this paper, we consider a probabilistic setting in which the uncertainty is random with known first two moments. In this context, we discuss two relevant problems: (i) probabilistic safety verification, in which the goal is to find an upper bound on the probability of violating a safety specification; and (ii) confidence ellipsoid estimation, in which given a confidence ellipsoid for the input of the neural network, our goal is to compute a confidence ellipsoid for the output. Due to the presence of nonlinear activation functions, these two problems are very difficult to solve exactly. To simplify the analysis, our main idea is to abstract the nonlinear activation functions by a combination of affine and quadratic constraints they impose on their input-output pairs. We then show that the safety of the abstracted network, which is sufficient for the safety of the original network, can be analyzed using semidefinite programming. We illustrate the performance of our approach with numerical experiments.