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).

翻译：$L_2$范数，或称碰撞范数，是分布分析与概率算法分析中的一个核心实体。Batu与Canonne（FOCS 2017）对$L_2$范数的算法层面及其与均匀性检验的联系进行了广泛分析。然而，在估计$L_2$范数本身时，相较于实例特定的二阶矩界$O(1/(\varepsilon\|μ\|_2) + (\|μ\|_3^3 - \|μ\|_2^4) / (\varepsilon^2 \|μ\|_2^4))$（如Batu在WoLA 2025开放问题环节所述），他们的算法并非总是最优的。本文提出了一种无偏的$L_2$范数估计算法，其样本复杂度与实例特定的二阶矩分析相匹配。此外，我们证明$Ω(1/(\varepsilon \|μ\|_2))$确实是通过抽样估计分布$μ$范数的一个逐实例下界（即使对于非无偏估计器亦然）。