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Perfect $L_p$ Sampling with Polylogarithmic Update Time

William Swartworth, David P. Woodruff, Samson Zhou

TL;DR

The paper resolves the open problem of achieving a perfect $L_p$ sampler for $p\in(0,2)$ in turnstile streams with optimal memory and fast updates. It combines a polylogarithmic-time, near-optimal-space framework by transforming the input with exponential scaling, leveraging dense Gaussian sketches for heavy hitters, and using a Gil-Pelaez-based inversion to sample from a complex, heavy-tailed distribution. A key novelty is the simulation oracle based on Poisson approximation that avoids explicit duplication, together with a rigorous derandomization via the GKM PRG to obtain a fully deterministic, polylogarithmic-update-time sampler. The result advances exact streaming sampling techniques, enabling efficient, plug-in perfect $L_p$ sampling for applications in norm estimation and anomaly detection in large-scale data streams.

Abstract

Perfect $L_p$ sampling in a stream was introduced by Jayaram and Woodruff (FOCS 2018) as a streaming primitive which, given turnstile updates to a vector $x \in \{-\text{poly}(n), \ldots, \text{poly}(n)\}^n$, outputs an index $i^* \in \{1, 2, \ldots, n\}$ such that the probability of returning index $i$ is exactly \[\Pr[i^* = i] = \frac{|x_i|^p}{\|x\|_p^p} \pm \frac{1}{n^C},\] where $C > 0$ is an arbitrarily large constant. Jayaram and Woodruff achieved the optimal $\tilde{O}(\log^2 n)$ bits of memory for $0 < p < 2$, but their update time is at least $n^C$ per stream update. Thus an important open question is to achieve efficient update time while maintaining optimal space. For $0 < p < 2$, we give the first perfect $L_p$-sampler with the same optimal amount of memory but with only $\text{poly}(\log n)$ update time. Crucial to our result is an efficient simulation of a sum of reciprocals of powers of truncated exponential random variables by approximating its characteristic function, using the Gil-Pelaez inversion formula, and applying variants of the trapezoid formula to quickly approximate it.

Perfect $L_p$ Sampling with Polylogarithmic Update Time

TL;DR

The paper resolves the open problem of achieving a perfect sampler for in turnstile streams with optimal memory and fast updates. It combines a polylogarithmic-time, near-optimal-space framework by transforming the input with exponential scaling, leveraging dense Gaussian sketches for heavy hitters, and using a Gil-Pelaez-based inversion to sample from a complex, heavy-tailed distribution. A key novelty is the simulation oracle based on Poisson approximation that avoids explicit duplication, together with a rigorous derandomization via the GKM PRG to obtain a fully deterministic, polylogarithmic-update-time sampler. The result advances exact streaming sampling techniques, enabling efficient, plug-in perfect sampling for applications in norm estimation and anomaly detection in large-scale data streams.

Abstract

Perfect sampling in a stream was introduced by Jayaram and Woodruff (FOCS 2018) as a streaming primitive which, given turnstile updates to a vector , outputs an index such that the probability of returning index is exactly \[\Pr[i^* = i] = \frac{|x_i|^p}{\|x\|_p^p} \pm \frac{1}{n^C},\] where is an arbitrarily large constant. Jayaram and Woodruff achieved the optimal bits of memory for , but their update time is at least per stream update. Thus an important open question is to achieve efficient update time while maintaining optimal space. For , we give the first perfect -sampler with the same optimal amount of memory but with only update time. Crucial to our result is an efficient simulation of a sum of reciprocals of powers of truncated exponential random variables by approximating its characteristic function, using the Gil-Pelaez inversion formula, and applying variants of the trapezoid formula to quickly approximate it.

Paper Structure

This paper contains 26 sections, 28 theorems, 123 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Technical Overview
  4. Linear transformation via uniform random variables.
  5. Linear transformation via exponential random variables.
  6. Linear transformation via duplication.
  7. Heavy-hitters through dense Gaussian sketches.
  8. Central limit theorem for sums of heavy-tailed random variables.
  9. Performing the sampling.
  10. Achieving worst-case time.
  11. Statistical test.
  12. Derandomization.
  13. Preliminaries
  14. Exponential random variables.
  15. Perfect Lp Sampler
  16. ...and 11 more sections

Key Result

Theorem 1.2

For any $p\in(0,2)$ and failure probability $\delta\in(0,1)$, there exists a perfect $L_p$ sampler on a turnstile stream that succeeds with probability at least $1-\delta$ and uses $\mathop{\mathrm{polylog}}\limits(n)$ time per update. The algorithm uses $\tilde{\mathcal{O}}\left(\log^2 n\log\frac{1

Figures (1)

  • Figure 1: Statistical test for perfect $L_p$ sampler in \ref{['alg:alg:perfect:lp:sample']}.

Theorems & Definitions (53)

  • Definition 1.1: $L_p$-sampler
  • Theorem 1.2
  • Theorem 3.1: Marcinkiewicz–Zygmund inequality
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • Definition 3.5: Exponential random variable
  • Lemma 3.6
  • Lemma 3.7: Lemma 2 in JayaramW18
  • ...and 43 more