A Domain-Theoretic Framework for Robustness Analysis of Neural Networks

TL;DR

This paper introduces a domain-theoretic framework for validated robustness analysis of neural networks, enabling both global and local assessments, and includes a validated algorithm for estimating Lipschitz constants with proven correctness.

Contribution

It develops a novel domain-theoretic approach that unifies analysis of differentiable and non-differentiable networks, and provides a validated algorithm for Lipschitz constant estimation.

Findings

Validated algorithm for Lipschitz constant estimation with proven completeness.

Framework handles both differentiable and non-differentiable networks uniformly.

Experimental results demonstrate the effectiveness of the validated software implementation.

Abstract

A domain-theoretic framework is presented for validated robustness analysis of neural networks. First, global robustness of a general class of networks is analyzed. Then, using the fact that Edalat's domain-theoretic L-derivative coincides with Clarke's generalized gradient, the framework is extended for attack-agnostic local robustness analysis. The proposed framework is ideal for designing algorithms which are correct by construction. This claim is exemplified by developing a validated algorithm for estimation of Lipschitz constant of feedforward regressors. The completeness of the algorithm is proved over differentiable networks, and also over general position ReLU networks. Computability results are obtained within the framework of effectively given domains. Using the proposed domain model, differentiable and non-differentiable networks can be analyzed uniformly. The validated algorithm is implemented using arbitrary-precision interval arithmetic, and the results of some experiments are presented. The software implementation is truly validated, as it handles floating-point errors as well.