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Instance-optimal estimation of L2-norm

Tomer Adar

TL;DR

An unbiased $L_2$-estimation algorithm whose sample complexity matches the instance-specific second-moment analysis and it is shown that $\Omega(1/(\varepsilon \|\mu\|_2))$ is indeed a per-instance lower bound for estimating the norm of a distribution $\mu$ by sampling (even for non-unbiased estimators).

Abstract

The $L_2$-norm, or collision norm, is a core entity in the analysis of distributions and probabilistic algorithms. Batu and Canonne (FOCS 2017) presented an extensive analysis of algorithmic aspects of the $L_2$-norm and its connection to uniformity testing. However, when it comes to estimating the $L_2$-norm itself, their algorithm is not always optimal compared to the instance-specific second-moment bounds, $O(1/(\varepsilon\|μ\|_2) + (\|μ\|_3^3 - \|μ\|_2^4) / (\varepsilon^2 \|μ\|_2^4))$, as stated by Batu (WoLA 2025, open problem session). In this paper, we present an unbiased $L_2$-estimation algorithm whose sample complexity matches the instance-specific second-moment analysis. Additionally, we show that $Ω(1/(\varepsilon \|μ\|_2))$ is indeed a per-instance lower bound for estimating the norm of a distribution $μ$ by sampling (even for non-unbiased estimators).

Instance-optimal estimation of L2-norm

TL;DR

An unbiased -estimation algorithm whose sample complexity matches the instance-specific second-moment analysis and it is shown that is indeed a per-instance lower bound for estimating the norm of a distribution by sampling (even for non-unbiased estimators).

Abstract

The -norm, or collision norm, is a core entity in the analysis of distributions and probabilistic algorithms. Batu and Canonne (FOCS 2017) presented an extensive analysis of algorithmic aspects of the -norm and its connection to uniformity testing. However, when it comes to estimating the -norm itself, their algorithm is not always optimal compared to the instance-specific second-moment bounds, , as stated by Batu (WoLA 2025, open problem session). In this paper, we present an unbiased -estimation algorithm whose sample complexity matches the instance-specific second-moment analysis. Additionally, we show that is indeed a per-instance lower bound for estimating the norm of a distribution by sampling (even for non-unbiased estimators).
Paper Structure (67 sections, 2 theorems, 155 equations, 2 tables)

This paper contains 67 sections, 2 theorems, 155 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard notations
  4. Common distributions
  5. Divergence measures
  6. Large-deviation bounds
  7. Estimation behaviors
  8. Non-technical overview
  9. Upper bound
  10. An $\Omega(1/\varepsilon\|\mu\|_2)$ lower bound
  11. Technical overview
  12. Elementary tools
  13. Amplification
  14. Other tools
  15. Reference $\|\mu\|_2^2$-estimators
  16. ...and 52 more sections

Key Result

theorem 1

For every $0 < \varepsilon \le 1$, there exists an unbiased estimator for $\|\mu\|_2^2$ whose input is a sampling access from a discrete distribution $\mu$ over an unknown domain such that:

Theorems & Definitions (63)

  • theorem 1: note=Short form of Lemma \ref{['lemma:estimate-L2-top-level']}
  • theorem 2: note=Short form of Lemma \ref{['lemma:lbnd-eps-mu2']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 53 more