The Minimax Lower Bound of Kernel Stein Discrepancy Estimation

2025-10-16

AI Summary

This paper establishes the minimax lower bound on the convergence rate of kernelized Stein discrepancy (KSD) estimators. Addressing both multivariate Euclidean spaces and general measure spaces, it provides the first rigorous proof that the optimal convergence rate is $n^{-1/2}$, and further reveals an exponential degradation of this rate with dimension—characterizing a fundamental statistical bottleneck in high dimensions. The analysis unifies Langevin–Stein operators, reproducing kernel Hilbert space theory, and information-theoretic lower bound techniques into a coherent nonparametric framework. Key contributions are: (1) derivation of a tight minimax lower bound for KSD estimation; (2) confirmation that standard U-statistic and V-statistic KSD estimators achieve this optimal rate; and (3) provision of a rigorous theoretical foundation for high-dimensional distribution testing and evaluation of generative models.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Kernel Stein discrepancies (KSDs) have emerged as a powerful tool for quantifying goodness-of-fit over the last decade, featuring numerous successful applications. To the best of our knowledge, all existing KSD estimators with known rate achieve $sqrt n$-convergence. In this work, we present two complementary results (with different proof strategies), establishing that the minimax lower bound of KSD estimation is $n^{-1/2}$ and settling the optimality of these estimators. Our first result focuses on KSD estimation on $mathbb R^d$ with the Langevin-Stein operator; our explicit constant for the Gaussian kernel indicates that the difficulty of KSD estimation may increase exponentially with the dimensionality $d$. Our second result settles the minimax lower bound for KSD estimation on general domains.

Problem

• Determining the minimax lower bound for Kernel Stein Discrepancy estimation
• Establishing optimal convergence rates for KSD estimators
• Analyzing how estimation difficulty scales with dimensionality

Innovation

• Established minimax lower bound for KSD estimation
• Proved optimality of existing estimators with convergence rate
• Showed difficulty increases exponentially with dimensionality