Unbiased Insights: Optimal Streaming Algorithms for $\ell_p$ Sampling, the Forget Model, and Beyond
Honghao Lin, Hoai-An Nguyen, William Swartworth, David P. Woodruff
TL;DR
This paper advances frequency-moment estimation and ℓ_p sampling in insertion-only data streams under forget and non-linear update models. It develops nearly optimal one-pass ℓ_p samplers for p∈(0,2] and a near-optimal p=2 sampler, with continuous sampling capabilities and improved space bounds. Leveraging these samplers, the authors obtain nearly unbiased estimators for F_p in the α-RFDS Forget Model, resolving open problems, and extend the framework to prefix/suffix deletion and a broad class of contracting updates, including entropy estimation as a corollary. The approach blends heavy-hitter structures, adaptive sparse recovery, derandomization, and Taylor expansions to achieve tight space-accuracy trade-offs across p-regimes, supported by matching lower bounds. Overall, the work broadens the applicability of sublinear-space streaming sketches to non-linear, deletion, and time-windowed data scenarios with strong theoretical guarantees and practical implications for fast, memory-efficient data analysis.
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.