Truly Perfect Samplers for Data Streams and Sliding Windows

TL;DR

This paper introduces the concept of truly perfect data stream samplers that produce exact probability distributions, addressing limitations of previous approximate methods and exploring their complexity in streaming and sliding window models.

Contribution

It initiates the study of truly perfect samplers with exact distributions in data streams and analyzes their complexity in various streaming models.

Findings

First truly perfect samplers for data streams are proposed.

Analysis of the space complexity for perfect sampling algorithms.

Comparison with approximate sampling methods shows advantages in privacy and accuracy.

Abstract

In the $G$-sampling problem, the goal is to output an index $i$ of a vector $f \in\mathbb{R}^n$, such that for all coordinates $j \in [n]$, \[\textbf{Pr}[i=j] = (1 \pm \epsilon) \frac{G(f_j)}{\sum_{k\in[n]} G(f_k)} + \gamma,\] where $G:\mathbb{R} \to \mathbb{R}_{\geq 0}$ is some non-negative function. If $\epsilon = 0$ and $\gamma = 1/\text{poly}(n)$, the sampler is called perfect. In the data stream model, $f$ is defined implicitly by a sequence of updates to its coordinates, and the goal is to design such a sampler in small space. Jayaram and Woodruff (FOCS 2018) gave the first perfect $L_p$ samplers in turnstile streams, where $G(x)=|x|^p$, using $\text{polylog}(n)$ space for $p\in(0,2]$. However, to date all known sampling algorithms are not truly perfect, since their output distribution is only point-wise $\gamma = 1/\text{poly}(n)$ close to the true distribution. This small error can be significant when samplers are run many times on successive portions of a stream, and leak potentially sensitive information about the data stream. In this work, we initiate the study of truly perfect samplers, with $\epsilon = \gamma = 0$, and comprehensively investigate their complexity in the data stream and sliding window models. Abstract truncated due to arXiv limits; please see paper for full abstract.