Multi-Objective Weighted Sampling

Abstract

{\em Multi-objective samples} are powerful and versatile summaries of large data sets. For a set of keys $x\in X$ and associated values $f_x \geq 0$, a weighted sample taken with respect to $f$ allows us to approximate {\em segment-sum statistics} $\text{Sum}(f;H) = \text{sum}_{x\in H} f_x$, for any subset $H$ of the keys, with statistically-guaranteed quality that depends on sample size and the relative weight of $H$. When estimating $\text{Sum}(g;H)$ for $g\not=f$, however, quality guarantees are lost. A multi-objective sample with respect to a set of functions $F$ provides for each $f\in F$ the same statistical guarantees as a dedicated weighted sample while minimizing the summary size. We analyze properties of multi-objective samples and present sampling schemes and meta-algortithms for estimation and optimization while showcasing two important application domains. The first are key-value data sets, where different functions $f\in F$ applied to the values correspond to different statistics such as moments, thresholds, capping, and sum. A multi-objective sample allows us to approximate all statistics in $F$. The second is metric spaces, where keys are points, and each $f\in F$ is defined by a set of points $C$ with $f_x$ being the service cost of $x$ by $C$, and $\text{Sum}(f;X)$ models centrality or clustering cost of $C$. A multi-objective sample allows us to estimate costs for each $f\in F$. In these domains, multi-objective samples are often of small size, are efficiently to construct, and enable scalable estimation and optimization. We aim here to facilitate further applications of this powerful technique.