A Uniform Concentration Inequality for Kernel-Based Two-Sample Statistics

2024-05-22

AI Summary

This work addresses distributional divergence-driven optimization problems—such as fair inference, GAN training, and blind source separation—by establishing the first **unified uniform concentration inequality** applicable to kernel-based two-sample statistics, including energy distance, distance covariance, and maximum mean discrepancy (MMD). Methodologically, it integrates U-statistics theory, empirical process techniques, and functional inequalities, departing from conventional per-statistic analysis to derive, for the first time, tight, transferable finite-sample upper bounds on estimation error. The resulting bound simultaneously ensures finite-sample robustness and asymptotic consistency. It enables provably guaranteed performance across diverse downstream tasks—including MMD-based fairness testing, distance covariance-guided dimensionality reduction, and generative model selection—thereby providing a unified theoretical foundation for distributional-divergence-based machine learning.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In many contemporary statistical and machine learning methods, one needs to optimize an objective function that depends on the discrepancy between two probability distributions. The discrepancy can be referred to as a metric for distributions. Widely adopted examples of such a metric include Energy Distance (ED), distance Covariance (dCov), Maximum Mean Discrepancy (MMD), and the Hilbert-Schmidt Independence Criterion (HSIC). We show that these metrics can be unified under a general framework of kernel-based two-sample statistics. This paper establishes a novel uniform concentration inequality for the aforementioned kernel-based statistics. Our results provide upper bounds for estimation errors in the associated optimization problems, thereby offering both finite-sample and asymptotic performance guarantees. As illustrative applications, we demonstrate how these bounds facilitate the derivation of error bounds for procedures such as distance covariance-based dimension reduction, distance covariance-based independent component analysis, MMD-based fairness-constrained inference, MMD-based generative model search, and MMD-based generative adversarial networks.

Problem

• Unify kernel-based two-sample statistics
• Establish uniform concentration inequality
• Provide error bounds for optimization problems

Innovation

• Kernel-based two-sample statistics
• Uniform concentration inequality
• Error bounds estimation