Variance-Optimal Offline and Streaming Stratified Random Sampling.

Stratified random sampling (SRS) is a fundamental sampling technique that provides accurate estimates for aggregate queries using a small size sample, and has been used widely for approximate query processing. A key question in SRS is how to partition a target sample size among different strata. While Neymanu0027s allocation provides a solution that minimizes the variance of an estimate using this sample, it works under the assumption that each stratum is abundant, i.e. has a large number of data points to choose from. This assumption may not hold in general: one or more strata may be bounded, and may not contain a large number of data points, even though the total data size may be large. first present VOILA, an offline method for allocating sample sizes to strata in a variance-optimal manner, even for the case when one or more strata may be bounded. We next consider SRS on streaming data that are continuously arriving. We show a lower bound, that any streaming algorithm for SRS must have (in the worst case) a variance that is $Omega(r)$ away from the optimal, where $r$ is the number of strata. We present S-VOILA, a practical streaming algorithm for SRS that is locally variance-optimal in its allocation of sample sizes to different strata. Both the offline and streaming algorithms are built on a method for reducing the size of a stratified random sample in a variance-optimal manner, which could be of independent interest. Our results from experiments on real and synthetic data show that that VOILA can have significantly smaller variance than Neymanu0027s allocation (VOILAu0027s variances are a factor of 1.4x-3000x smaller than that of Neyman allocation, with the same setting). The streaming algorithm S-VOIlA results in a variance that is typically close to VOILA, which was given the entire input beforehand.