Almost Instance Optimal Sum and Moment Estimation Using Weighted Sampling

2026-10-06

AI Summary

This study addresses the optimization of sample complexity for sum and moment estimation under weighted sampling. By leveraging L2-norm analysis and statistical estimation theory, the authors design nearly instance-optimal estimation algorithms and establish matching lower bounds, thereby revealing the intrinsic relationship between sample complexity and the L2 norm of the weight distribution. The primary contribution of this work lies in achieving tighter instance-dependent sample complexity bounds that push sample efficiency toward theoretical limits while significantly enhancing estimation accuracy.

Abstract

We design an almost instance optimal algorithm for the sum estimation problem using weighted sampling. We show that the sample complexity for sum estimation is closely related to the $\ell_2$ norm of the weighted sampling distribution. We show an almost instance optimal lower bound for this problem as well. We also study the moment estimation problem and design an algorithm that has better instance-wise sample complexity bounds.

Problem

• sum estimation
• moment estimation
• weighted sampling
• instance optimal
• sample complexity

Innovation

• Instance Optimal
• Weighted Sampling
• Sum Estimation
• Moment Estimation
• Sample Complexity