Estimating the Effective Support Size in Constant Query Complexity

Estimating the support size of a distribution is a well-studied problem in statistics. Motivated by the fact that this problem is highly non-robust (as small perturbations in the distributions can drastically affect the support size) and thus hard to estimate, Goldreich [ECCC 2019] studied the query complexity of estimating the $ε$-\emph{effective support size} $\text{Ess}_ε$ of a distribution ${P}$, which is equal to the smallest support size of a distribution that is $ε$-far in total variation distance from ${P}$. In his paper, he shows an algorithm in the dual access setting (where we may both receive random samples and query the sampling probability $p(x)$ for any $x$) for a bicriteria approximation, giving an answer in $[\text{Ess}_{(1+β)ε},(1+γ) \text{Ess}_ε]$ for some values $β, γ> 0$. However, his algorithm has either super-constant query complexity in the support size or super-constant approximation ratio $1+γ= ω(1)$. He then asked if this is necessary, or if it is possible to get a constant-factor approximation in a number of queries independent of the support size. We answer his question by showing that not only is complexity independent of $n$ possible for $γ>0$, but also for $γ=0$, that is, that the bicriteria relaxation is not necessary. Specifically, we show an algorithm with query complexity $O(\frac{1}{β^3 ε^3})$. That is, for any $0 < ε, β< 1$, we output in this complexity a number $\tilde{n} \in [\text{Ess}_{(1+β)ε},\text{Ess}_ε]$. We also show that it is possible to solve the approximate version with approximation ratio $1+γ$ in complexity $O\left(\frac{1}{β^2 ε} + \frac{1}{βεγ^2}\right)$. Our algorithm is very simple, and has $4$ short lines of pseudocode.