Improving Certified Robustness via Statistical Learning with Logical Reasoning

TL;DR

This paper enhances certified robustness in machine learning by integrating logical reasoning with statistical models via Markov logic networks, providing new robustness bounds and demonstrating superior performance on diverse datasets.

Contribution

It introduces a novel approach combining logical reasoning with statistical ML models to improve certified robustness and derives the first robustness bounds for Markov logic networks.

Findings

Robustness certification for MLN is #P-hard.

The proposed method outperforms state-of-the-art robustness techniques.

Experiments on five datasets confirm significant robustness improvements.

Abstract

Intensive algorithmic efforts have been made to enable the rapid improvements of certificated robustness for complex ML models recently. However, current robustness certification methods are only able to certify under a limited perturbation radius. Given that existing pure data-driven statistical approaches have reached a bottleneck, in this paper, we propose to integrate statistical ML models with knowledge (expressed as logical rules) as a reasoning component using Markov logic networks (MLN, so as to further improve the overall certified robustness. This opens new research questions about certifying the robustness of such a paradigm, especially the reasoning component (e.g., MLN). As the first step towards understanding these questions, we first prove that the computational complexity of certifying the robustness of MLN is #P-hard. Guided by this hardness result, we then derive the first certified robustness bound for MLN by carefully analyzing different model regimes. Finally, we conduct extensive experiments on five datasets including both high-dimensional images and natural language texts, and we show that the certified robustness with knowledge-based logical reasoning indeed significantly outperforms that of the state-of-the-arts.