Stream sampling for variance-optimal estimation of subset sums

TL;DR

This paper introduces $ ext{varopt}_k$, a reservoir sampling scheme that achieves variance-optimal unbiased estimation of subset sums from streaming data, outperforming previous methods in efficiency and accuracy.

Contribution

The paper presents a new reservoir sampling scheme, $ ext{varopt}_k$, that is variance-optimal for unbiased subset sum estimation, with improved bounds and efficiency.

Findings

Minimizes average variance over all subset sizes

Provides tighter worst-case variance bounds

Operates efficiently in $O( ext{log} k)$ time per item

Abstract

From a high volume stream of weighted items, we want to maintain a generic sample of a certain limited size $k$ that we can later use to estimate the total weight of arbitrary subsets. This is the classic context of on-line reservoir sampling, thinking of the generic sample as a reservoir. We present an efficient reservoir sampling scheme, $\varoptk$, that dominates all previous schemes in terms of estimation quality. $\varoptk$ provides {\em variance optimal unbiased estimation of subset sums}. More precisely, if we have seen $n$ items of the stream, then for {\em any} subset size $m$, our scheme based on $k$ samples minimizes the average variance over all subsets of size $m$. In fact, the optimality is against any off-line scheme with $k$ samples tailored for the concrete set of items seen. In addition to optimal average variance, our scheme provides tighter worst-case bounds on the variance of {\em particular} subsets than previously possible. It is efficient, handling each new item of the stream in $O(\log k)$ time. Finally, it is particularly well suited for combination of samples from different streams in a distributed setting.