# Do Machine Learning Models Learn Statistical Rules Inferred from Data?

arXiv: https://arxiv.org/abs/2303.01433 · HTML: https://mlsanity.com/paper/2303.01433/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 42 references that are in the map.
- First posted on arXiv 2 Mar 2023.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [The Cityscapes Dataset for Semantic Urban Scene Understanding](https://mlsanity.com/paper/1604.01685/) (2016)
- [Revisiting Deep Learning Models for Tabular Data](https://mlsanity.com/paper/2106.11959/) (2021)
- [Mitigating Unwanted Biases with Adversarial Learning](https://mlsanity.com/paper/1801.07593/) (2018)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [FinBERT: Financial Sentiment Analysis with Pre-trained Language Models](https://mlsanity.com/paper/1908.10063/) (2019)
- [MEMO: Test Time Robustness via Adaptation and Augmentation](https://mlsanity.com/paper/2110.09506/) (2021)
- [Snorkel: Rapid Training Data Creation with Weak Supervision](https://mlsanity.com/paper/1711.10160/) (2017)
- [Good Debt or Bad Debt: Detecting Semantic Orientations in Economic Texts](https://mlsanity.com/paper/1307.5336/) (2013)
- [Contrastive Test-Time Adaptation](https://mlsanity.com/paper/2204.10377/) (2022)
- [Data Programming: Creating Large Training Sets, Quickly](https://mlsanity.com/paper/1605.07723/) (2016)
- [SAITS: Self-Attention-based Imputation for Time Series](https://mlsanity.com/paper/2202.08516/) (2022)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [EfficientPS: Efficient Panoptic Segmentation](https://mlsanity.com/paper/2004.02307/) (2020)
- [Test-Time Adaptation via Conjugate Pseudo-labels](https://mlsanity.com/paper/2207.09640/) (2022)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Rain rendering for evaluating and improving robustness to bad weather](https://mlsanity.com/paper/2009.03683/) (2020)
- [Divide and Contrast: Source-free Domain Adaptation via Adaptive Contrastive Learning](https://mlsanity.com/paper/2211.06612/) (2022)
- [ImageNet-X: Understanding Model Mistakes with Factor of Variation Annotations](https://mlsanity.com/paper/2211.01866/) (2022)
- [Learn to Explain Efficiently via Neural Logic Inductive Learning](https://mlsanity.com/paper/1910.02481/) (2019)
- [If your data distribution shifts, use self-learning](https://mlsanity.com/paper/2104.12928/) (2021)
- [The ILASP system for Inductive Learning of Answer Set Programs](https://mlsanity.com/paper/2005.00904/) (2020)
- [Learning to Generalize across Domains on Single Test Samples](https://mlsanity.com/paper/2202.08045/) (2022)
- [sigmoidF1: A Smooth F1 Score Surrogate Loss for Multilabel Classification](https://mlsanity.com/paper/2108.10566/) (2021)
- [Twitter Topic Classification](https://mlsanity.com/paper/2209.09824/) (2022)
- [Controlling Neural Networks with Rule Representations](https://mlsanity.com/paper/2106.07804/) (2021)

## Built on by (most cited first, 3 of 3)

- [Robustness and Cybersecurity in the EU Artificial Intelligence Act](https://mlsanity.com/paper/2502.16184/) (2025)
- [On the Promise for Assurance of Differentiable Neurosymbolic Reasoning Paradigms](https://mlsanity.com/paper/2502.08932/) (2025)
- [DISCRET: Synthesizing Faithful Explanations For Treatment Effect Estimation](https://mlsanity.com/paper/2406.00611/) (2024)

## Abstract (arXiv)

Machine learning models can make critical errors that are easily hidden within vast amounts of data. Such errors often run counter to rules based on human intuition. However, rules based on human knowledge are challenging to scale or to even formalize. We thereby seek to infer statistical rules from the data and quantify the extent to which a model has learned them. We propose a framework SQRL that integrates logic-based methods with statistical inference to derive these rules from a model's training data without supervision. We further show how to adapt models at test time to reduce rule violations and produce more coherent predictions. SQRL generates up to 300K rules over datasets from vision, tabular, and language settings. We uncover up to 158K violations of those rules by state-of-the-art models for classification, object detection, and data imputation. Test-time adaptation reduces these violations by up to 68.7% with relative performance improvement up to 32%. SQRL is available at https://github.com/DebugML/sqrl.