Instance-optimal estimation of L2-norm

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, $O(1/(\varepsilon\|μ\|_2) + t_μ/\varepsilon^2)$, for $t_μ= \|μ\|_3^3 / \|μ\|_2^4 - 1$, 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 $Ω(1/(\varepsilon \|μ\|_2) + t_μ/ \varepsilon^2)$ is indeed the per-instance lower bound for estimating the norm of a distribution $μ$ by sampling (even for non-unbiased estimators).