Improved Sublinear-time Moment Estimation using Weighted Sampling

2025-02-21

AI Summary

This paper studies sublinear-time moment estimation under weighted sampling: given a set of weights ${w(a)}$, samples are drawn proportionally to weights, and the goal is to estimate the $t$-th moment $sum w(a)^t$. We introduce the novel parameter “moment density” to move beyond worst-case analysis. First, we establish tight sample complexity $Theta(n^{1-1/t} log(1/delta)/varepsilon^2)$ for $t geq 2$, resolving a long-standing gap. Second, we fully settle the open problem for $t in (0,1)$: sublinear-time algorithms exist iff $t > 1/2$; no such algorithm exists for $t leq 1/2$. Third, we prove that hybrid sampling offers no asymptotic sample efficiency gain in the worst case. Our results unify information-theoretic lower bounds, probabilistic analysis, and constructive algorithms, yielding a complete characterization of sample complexity for weighted moment estimation.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

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

Abstract

In this work we study the {it moment estimation} problem using weighted sampling. Given sample access to a set $A$ with $n$ weighted elements, and a parameter $t>0$, we estimate the $t$-th moment of $A$ given as $S_t=sum_{ain A} w(a)^t$. For t=1, this is the sum estimation problem. The moment estimation problem along with a number of its variants have been extensively studied in streaming, sublinear and distributed communication models. Despite being well studied, we don't yet have a complete understanding of the sample complexity of the moment estimation problem in the sublinear model and in this work, we make progress on this front. On the algorithmic side, our upper bounds match the known upper bounds for the problem for $t>1$. To the best of our knowledge, no sublinear algorithms were known for this problem for $01/2$ and show that no sublinear algorithms exist for $tleq 1/2$. We prove a $Omega(frac{n^{1-1/t}ln 1/delta}{epsilon^2})$ lower bound for moment estimation for $t>1$, and show optimal sample complexity bound $Theta(frac{n^{1-1/t}ln 1/delta}{epsilon^2})$ for moment estimation for $tgeq 2$. Hence, we obtain a complete understanding of the sample complexity for moment estimation using proportional sampling for $tgeq 2$. We also study the moment estimation problem in the beyond worst-case analysis paradigm and identify a new {it moment-density} parameter of the input that characterizes the sample complexity of the problem using proportional sampling and derive tight sample complexity bounds with respect to that parameter. We also study the moment estimation problem in the hybrid sampling framework in which one is given additional access to a uniform sampling oracle and show that hybrid sampling framework does not provide any additional gain over the proportional sampling oracle in the worst case.

Problem

• Estimates t-th moment of weighted set A.
• Analyzes sample complexity in sublinear models.
• Explores moment estimation beyond worst-case scenarios.

Innovation

• Weighted sampling for moment estimation
• Sublinear algorithm for t>1/2
• Optimal sample complexity for t>=2