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).

翻译：$L_2$范数（也称碰撞范数）是分布分析与概率算法中的核心量。Batu与Canonne（FOCS 2017）对$L_2$范数的算法特性及其与均匀性检验的关联进行了深入分析。然而在估计$L_2$范数本身时，其算法相较于实例特定的二阶矩界$O(1/(\varepsilon\|μ\|_2) + t_μ/\varepsilon^2)$（其中$t_μ= \|μ\|_3^3 / \|μ\|_2^4 - 1$）并非始终最优，这一结论由Batu（WoLA 2025，开放问题环节）指出。本文提出一种无偏$L_2$估计方法，其样本复杂度匹配实例特定的二阶矩分析。此外，我们证明$Ω(1/(\varepsilon \|μ\|_2) + t_μ/ \varepsilon^2)$确为通过采样估计分布$μ$范数时的逐实例下界（即便对于非无偏估计器亦然）。