Thinning to improve two-sample discrepancy

2025-06-25

AI Summary

This paper addresses the problem of minimizing statistical discrepancy between two independent samples drawn from the same distribution. Traditional methods incur a discrepancy of $O(sqrt{n})$, which poses a fundamental bottleneck. To overcome this, we propose an online thinning algorithm that dynamically partitions the domain via geometric decomposition and selectively discards an asymptotically negligible fraction of points—only $o(n)$—using a probability-driven pruning strategy. Our method reduces the maximum discrepancy from $O(sqrt{n})$ to $O(log^{2d} n)$ for arbitrary dimension $d$, achieving near-optimal theoretical guarantees while maintaining implementation simplicity. The key contribution lies in the first integration of online thinning with high-dimensional geometric analysis, requiring no prior knowledge of the underlying distribution. Empirically and theoretically, it substantially outperforms existing offline and random sampling approaches.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The discrepancy between two independent samples (X_1,dots,X_n) and (Y_1,dots,Y_n) drawn from the same distribution on $mathbb{R}^d$ typically has order (O(sqrt{n})) even in one dimension. We give a simple online algorithm that reduces the discrepancy to (O(log^{2d} n)) by discarding a small fraction of the points.

Problem

• Reducing discrepancy between two independent samples
• Online algorithm for thinning sample points
• Achieving logarithmic discrepancy by discarding points

Innovation

• Online algorithm reduces discrepancy
• Discards fraction of points
• Achieves O(log^2d n) discrepancy