# Improving Certified Robustness via Statistical Learning with Logical Reasoning

arXiv: https://arxiv.org/abs/2003.00120 · HTML: https://mlsanity.com/paper/2003.00120/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2020.

## Built on by (most cited first, 12 of 12)

- [R2-Guard: Robust Reasoning Enabled LLM Guardrail via Knowledge-Enhanced Logical Reasoning](https://mlsanity.com/paper/2407.05557/) (2024)
- [LOT: Layer-wise Orthogonal Training on Improving ℓ2 Certified Robustness](https://mlsanity.com/paper/2210.11620/) (2022)
- [SafeAuto: Knowledge-Enhanced Safe Autonomous Driving with Multimodal Foundation Models](https://mlsanity.com/paper/2503.00211/) (2025)
- [Efficient Query-Based Attack against ML-Based Android Malware Detection under Zero Knowledge Setting](https://mlsanity.com/paper/2309.01866/) (2023)
- [COLEP: Certifiably Robust Learning-Reasoning Conformal Prediction via Probabilistic Circuits](https://mlsanity.com/paper/2403.11348/) (2024)
- [CAUSAL3D: A Comprehensive Benchmark for Causal Learning from Visual Data](https://mlsanity.com/paper/2503.04852/) (2025)
- [Neural Probabilistic Circuits: Enabling Compositional and Interpretable Predictions through Logical Reasoning](https://mlsanity.com/paper/2501.07021/) (2025)
- [Improving deep learning with prior knowledge and cognitive models: A survey on enhancing explainability, adversarial robustness and zero-shot learning](https://mlsanity.com/paper/2403.07078/) (2024)
- [Factor Graph-based Interpretable Neural Networks](https://mlsanity.com/paper/2502.14572/) (2025)
- [Understanding and Improving Adversarial Robustness of Neural Probabilistic Circuits](https://mlsanity.com/paper/2509.20549/) (2025)
- [VISAT: Benchmarking Adversarial and Distribution Shift Robustness in Traffic Sign Recognition with Visual Attributes](https://mlsanity.com/paper/2510.26833/) (2025)
- [LogiC-Diff: Embedding Security Properties Into AI-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2609.38381/) (2026)

## Abstract (arXiv)

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.