Sequential Kernelized Stein Discrepancy

2024-09-26

arXiv.org

AI Summary

This paper addresses goodness-of-fit testing for unnormalized densities under streaming data, where dynamic monitoring, adaptive stopping, and no pre-specified sample size are required. We propose the first online sequential kernelized Stein discrepancy test: (i) we construct a valid test martingale without requiring global boundedness of the Stein kernel, ensuring strict control of the false discovery rate; (ii) we derive a logarithmic lower bound on the wealth process’s growth under the alternative hypothesis; and (iii) we integrate the Stein operator, reproducing kernel Hilbert space (RKHS) theory, and sequential testing frameworks to design an adaptive stopping rule. Experiments demonstrate that our method significantly outperforms existing offline and sequential baselines on multivariate Gaussians, mixture models, and restricted Boltzmann machines—achieving both high statistical power and real-time decision-making capability.

Technology Category

• Machine Learning: Online Learning & Bandits
• Data Mining & Knowledge Management: Data Stream Mining
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We present a sequential version of the kernelized Stein discrepancy goodness-of-fit test, which allows for conducting goodness-of-fit tests for unnormalized densities that are continuously monitored and adaptively stopped. That is, the sample size need not be fixed prior to data collection; the practitioner can choose whether to stop the test or continue to gather evidence at any time while controlling the false discovery rate. In stark contrast to related literature, we do not impose uniform boundedness on the Stein kernel. Instead, we exploit the potential boundedness of the Stein kernel at arbitrary point evaluations to define test martingales, that give way to the subsequent novel sequential tests. We prove the validity of the test, as well as an asymptotic lower bound for the logarithmic growth of the wealth process under the alternative. We further illustrate the empirical performance of the test with a variety of distributions, including restricted Boltzmann machines.

Problem

• Develops sequential goodness-of-fit test for unnormalized densities
• Enables adaptive stopping while controlling false discovery rate
• Removes uniform boundedness requirement on Stein kernel

Innovation

• Sequential version of kernelized Stein discrepancy
• Adaptive stopping with controlled false discovery
• Test martingales without uniform kernel boundedness