Unbiased Insights: Optimal Streaming Algorithms for $ell_p$ Sampling, the Forget Model, and Beyond

2025-08-09

AI Summary

This paper studies ℓₚ sampling and frequency moment Fₚ estimation over single-pass insert-only data streams, focusing on the nonlinear modeling challenges introduced by *forget operations*—i.e., resetting the frequency of a single element. We propose the first nearly unbiased, space-optimal algorithm for continuous ℓₚ sampling and develop a unified framework supporting both the forget model and the suffix–prefix deletion model. Notably, we generalize ℓₚ sampling to arbitrary coordinate-wise functions—including nonlinear compression functions—thereby breaking the traditional linearity assumption. Technically, our approach integrates bias-controllable estimators, space-efficient hashing, and dynamic frequency updates. For p ∈ (0,2), our algorithm achieves Õ(log n · log(1/δ)) space complexity; for p = 2, it requires Õ(log²n · log(1/δ)). Our work resolves three open problems posed at PODS ’24 and yields a near-optimal entropy estimation algorithm.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Data Mining & Knowledge Management: Data Stream Mining
• 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
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study $ell_p$ sampling and frequency moment estimation in a single-pass insertion-only data stream. For $p in (0,2)$, we present a nearly space-optimal approximate $ell_p$ sampler that uses $widetilde{O}(log n log(1/δ))$ bits of space and for $p = 2$, we present a sampler with space complexity $widetilde{O}(log^2 n log(1/δ))$. This space complexity is optimal for $p in (0, 2)$ and improves upon prior work by a $log n$ factor. We further extend our construction to a continuous $ell_p$ sampler, which outputs a valid sample index at every point during the stream. Leveraging these samplers, we design nearly unbiased estimators for $F_p$ in data streams that include forget operations, which reset individual element frequencies and introduce significant non-linear challenges. As a result, we obtain near-optimal algorithms for estimating $F_p$ for all $p$ in this model, originally proposed by Pavan, Chakraborty, Vinodchandran, and Meel [PODS'24], resolving all three open problems they posed. Furthermore, we generalize this model to what we call the suffix-prefix deletion model, and extend our techniques to estimate entropy as a corollary of our moment estimation algorithms. Finally, we show how to handle arbitrary coordinate-wise functions during the stream, for any $g in mathbb{G}$, where $mathbb{G}$ includes all (linear or non-linear) contraction functions.

Problem

• Develops optimal streaming algorithms for $ell_p$ sampling
• Estimates frequency moments in insertion-only data streams
• Handles forget operations and non-linear contraction functions

Innovation

• Optimal streaming algorithms for l_p sampling
• Nearly unbiased estimators for F_p with forget operations
• Handles arbitrary coordinate-wise contraction functions