# CARE: Certifiably Robust Learning with Reasoning via Variational Inference

arXiv: https://arxiv.org/abs/2209.05055 · HTML: https://mlsanity.com/paper/2209.05055/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 80 references that are in the map.
- First posted on arXiv 12 Sep 2022.

## Builds on (most cited first, 30 of 48)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Evasion Attacks against Machine Learning at Test Time](https://mlsanity.com/paper/1708.06131/) (2017)
- [Adversarial Examples Are Not Easily Detected: Bypassing Ten Detection Methods](https://mlsanity.com/paper/1705.07263/) (2017)
- [Zero-Shot Learning -- A Comprehensive Evaluation of the Good, the Bad and the Ugly](https://mlsanity.com/paper/1707.00600/) (2017)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Defense-GAN: Protecting Classifiers Against Adversarial Attacks Using Generative Models](https://mlsanity.com/paper/1805.06605/) (2018)
- [Provably Robust Deep Learning via Adversarially Trained Smoothed Classifiers](https://mlsanity.com/paper/1906.04584/) (2019)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Certified Defenses against Adversarial Examples](https://mlsanity.com/paper/1801.09344/) (2018)
- [Generating Adversarial Examples with Adversarial Networks](https://mlsanity.com/paper/1801.02610/) (2018)
- [Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks](https://mlsanity.com/paper/1705.01320/) (2017)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Towards Stable and Efficient Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1906.06316/) (2019)
- [Spatially Transformed Adversarial Examples](https://mlsanity.com/paper/1801.02612/) (2018)
- [Semidefinite relaxations for certifying robustness to adversarial examples](https://mlsanity.com/paper/1811.01057/) (2018)
- [Randomized Smoothing of All Shapes and Sizes](https://mlsanity.com/paper/2002.08118/) (2020)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [(Certified!!) Adversarial Robustness for Free!](https://mlsanity.com/paper/2206.10550/) (2022)
- [An approach to reachability analysis for feed-forward ReLU neural networks](https://mlsanity.com/paper/1706.07351/) (2017)
- [A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks](https://mlsanity.com/paper/1902.08722/) (2019)
- [MACER: Attack-free and Scalable Robust Training via Maximizing Certified Radius](https://mlsanity.com/paper/2001.02378/) (2020)
- [SoK: Certified Robustness for Deep Neural Networks](https://mlsanity.com/paper/2009.04131/) (2020)

## Built on by (most cited first, 10 of 10)

- [R2-Guard: Robust Reasoning Enabled LLM Guardrail via Knowledge-Enhanced Logical Reasoning](https://mlsanity.com/paper/2407.05557/) (2024)
- [SafeAuto: Knowledge-Enhanced Safe Autonomous Driving with Multimodal Foundation Models](https://mlsanity.com/paper/2503.00211/) (2025)
- [COLEP: Certifiably Robust Learning-Reasoning Conformal Prediction via Probabilistic Circuits](https://mlsanity.com/paper/2403.11348/) (2024)
- [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)
- [KnowGraph: Knowledge-Enabled Anomaly Detection via Logical Reasoning on Graph Data](https://mlsanity.com/paper/2410.08390/) (2024)
- [Pixel-wise Smoothing for Certified Robustness against Camera Motion Perturbations](https://mlsanity.com/paper/2309.13150/) (2023)
- [Understanding and Improving Adversarial Robustness of Neural Probabilistic Circuits](https://mlsanity.com/paper/2509.20549/) (2025)
- [KoALA: KL-L0 Adversarial Detector via Label Agreement](https://mlsanity.com/paper/2510.12752/) (2025)
- [VISAT: Benchmarking Adversarial and Distribution Shift Robustness in Traffic Sign Recognition with Visual Attributes](https://mlsanity.com/paper/2510.26833/) (2025)

## Abstract (arXiv)

Despite great recent advances achieved by deep neural networks (DNNs), they are often vulnerable to adversarial attacks. Intensive research efforts have been made to improve the robustness of DNNs; however, most empirical defenses can be adaptively attacked again, and the theoretically certified robustness is limited, especially on large-scale datasets. One potential root cause of such vulnerabilities for DNNs is that although they have demonstrated powerful expressiveness, they lack the reasoning ability to make robust and reliable predictions. In this paper, we aim to integrate domain knowledge to enable robust learning with the reasoning paradigm. In particular, we propose a certifiably robust learning with reasoning pipeline (CARE), which consists of a learning component and a reasoning component. Concretely, we use a set of standard DNNs to serve as the learning component to make semantic predictions, and we leverage the probabilistic graphical models, such as Markov logic networks (MLN), to serve as the reasoning component to enable knowledge/logic reasoning. However, it is known that the exact inference of MLN (reasoning) is #P-complete, which limits the scalability of the pipeline. To this end, we propose to approximate the MLN inference via variational inference based on an efficient expectation maximization algorithm. In particular, we leverage graph convolutional networks (GCNs) to encode the posterior distribution during variational inference and update the parameters of GCNs (E-step) and the weights of knowledge rules in MLN (M-step) iteratively. We conduct extensive experiments on different datasets and show that CARE achieves significantly higher certified robustness compared with the state-of-the-art baselines. We additionally conducted different ablation studies to demonstrate the empirical robustness of CARE and the effectiveness of different knowledge integration.