Theoretical Convergence of SMOTE-Generated Samples

2026-01-05

arXiv.org

AI Summary

This study addresses the lack of theoretical convergence guarantees for SMOTE in handling class-imbalanced data by providing the first rigorous theoretical analysis of its behavior. Leveraging tools from probability theory and statistics, the authors prove that samples generated by SMOTE converge in probability to the true underlying data distribution and achieve mean convergence under compact support conditions. Furthermore, the work quantitatively characterizes the relationship between the nearest-neighbor parameter \(k\) and the convergence rate, demonstrating that smaller values of \(k\) lead to faster convergence. These theoretical findings are corroborated through numerical experiments on both synthetic and real-world datasets, thereby establishing a solid theoretical foundation for SMOTE-based data augmentation techniques.

Technology Category

• Machine Learning: Learning Theory
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Imbalanced data affects a wide range of machine learning applications, from healthcare to network security. As SMOTE is one of the most popular approaches to addressing this issue, it is imperative to validate it not only empirically but also theoretically. In this paper, we provide a rigorous theoretical analysis of SMOTE's convergence properties. Concretely, we prove that the synthetic random variable Z converges in probability to the underlying random variable X. We further prove a stronger convergence in mean when X is compact. Finally, we show that lower values of the nearest neighbor rank lead to faster convergence offering actionable guidance to practitioners. The theoretical results are supported by numerical experiments using both real-life and synthetic data. Our work provides a foundational understanding that enhances data augmentation techniques beyond imbalanced data scenarios.

Problem

• imbalanced data
• SMOTE
• theoretical convergence
• synthetic samples
• data augmentation

Innovation

• SMOTE
• theoretical convergence
• imbalanced data
• data augmentation
• nearest neighbor rank