Optimal Pattern Detection Tree for Symbolic Rule-Based Classification

Young-Chae Hong, Yangho Chen

Abstract: Pattern discovery in data plays a crucial role across diverse domains, including healthcare, risk assessment, and machinery maintenance. In contrast to black-box deep learning models, symbolic rule discovery emerges as a key data mining task, generating human-interpretable rules that offer both transparency and intuitive explainability. This paper introduces the Optimal Pattern Detection Tree (OPDT), a rule-based machine learning model based on novel mixed-integer programming to discover a single optimal pattern in data through binary classification. To incorporate prior knowledge and compliance requirements, we further introduce the Branching Structure Constraints (BSC) framework, which enables decision makers to encode domain knowledge and constraints directly into the model. This optimization-based approach discovers a hidden underlying pattern in datasets, when it exists, by identifying an optimal rule that maximizes coverage while minimizing the false positive rate due to misclassification. Our computational experiments show that OPDT discovers a pattern with optimality guarantees on moderately sized datasets within reasonable runtime.

Submission Type: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

Revision 01 (2026-02-20)
- We have revised the Metrics subsection in Section 3.2 to clarify the motivation of Volume-Impurity Index and add the definition of a rule in OPDT.
- We have updated Section 3.1 to clarify an ambiguous sentence (e.g., mutually exclusive impacts on classification) about feature groups.
- We have updated Figure 1 and added explanation in the caption.
- We have updated Figure 2 and added explanation in the caption.
- We have added the following sentence to the Approach subsection in Section 3.2:
- *While SCDT allows different feature groups at each branching node as shown in Figure 1, OPDT enforces the same feature group at each level of the decision tree to systematically explore all negation combinations (i.e., $\geq$ or $<$ for each feature within a feature group) across the chain of rule conditions.*
- We have updated Tables 1 and 2 and Equations (1)-(29).
- We have updated the Branching Structure Constraints subsection in Section 3.2.
- We have updated Algorithm 1 in Section 3.2.
- We have moved Table 7 from Appendix to main body.

Revision 02 (2026-02-28)
- We have updated the abstract to mention the BSC framework for prior knowledge.
- We have fixed a typo in the third paragraph in Section 2.1.
- We have corrected the wrong citation at the end of the second paragraph in Section 2.1.
- We have added more description for $\epsilon_p$, $\epsilon_{\max}$, and $M$ in the MIP formulation.
- We have removed the duplicated Constraint (15).
- We have fixed the typo "Frist" $\to$ "First".
- We have added model size complexity on page 9.
- We have added a paragraph specifying which particular constraints are additionally introduced for readability in the MIP formulation.

Revision 03 (2026-03-02)
- We have added Figure 3 to illustrate how OPDT enhances the VI value through the MIP solver as a visual example.

Revision 04 (2026-03-04)
- Fixed the wrong dataset names assigned to each row between "Australian", "German", "Blood Transfusion", and "Breast Cancer" in Table 8. (No change to the numbers, which remain consistent with Table 9.) No change is needed for the paragraph, as no sentence references a specific dataset name.
- We have added additional description for $\epsilon_p$ and $\epsilon_{\max}$.
- We have added "single" to the abstract to emphasize the single-rule scope.
- We have updated the Introduction section to emphasize a single pattern.

Revision 05 (2026-03-08)
- We have fixed a typo in the German dataset: the BSCCART value has been corrected from $1.80$ to $-1.80$.
- We have updated the captions of Tables 6 and 8 from "OPDT Performance at Various Methods ..." to "OPDT Performance Compared to Benchmark Methods ...".
- We have added Appendix B: Feature Grouping Using ML-Based Feature Importance.
- We have renamed the section heading of Table 10 from "Appendix" to "OPDT and Benchmark Methods Performance," since there are multiple sections in the Appendix.
- We have enhanced the paragraph on page 15 with additional descriptions for the weak cases on test data.
- We have added Table 9 for the sensitivity analysis of parameter $w$.
- We have added Figure 4 and Table 11 to demonstrate feature grouping using ML-based feature importance.

Revision 06 (2026-03-09)
- We have specified the default value $w = 10$ in Section 4.3 and added a sentence for $w$ to Section 4 before presenting the experimental results.
- We have added the future work items to Section 5 (Conclusion and Future Work) based on the reviewer's feedback and comments.

Revision 07 (2026-04-11)
- fix minor typos and grammatical errors throughout the manuscript.
- add individual dataset citations in Section 4.1 after "we focus on healthcare and finance datasets"

Assigned Action Editor: ~Satoshi_Hara1

Submission Number: 6854