On the Variance of Subset Sum Estimation

For high volume data streams and large data warehouses, sampling is used for efficient approximate answers to aggregate queries over selected subsets. Mathematically, we are dealing with a set of weighted items and want to support queries to arbitrary subset sums. With unit weights, we can compute subset sizes which together with the previous sums provide the subset averages. The question addressed here is which sampling scheme we should use to get the most accurate subset sum estimates. We present a simple theorem on the variance of subset sum estimation and use it to prove variance optimality and near-optimality of subset sum estimation with different known sampling schemes. This variance is measured as the average over all subsets of any given size. By optimal we mean there is no set of input weights for which any sampling scheme can have a better average variance. Such powerful results can never be established experimentally. The results of this paper are derived mathematically. For example, we show that appropriately weighted systematic sampling is simultaneously optimal for all subset sizes. More standard schemes such as uniform sampling and probability-proportional-to-size sampling with replacement can be arbitrarily bad. Knowing the variance optimality of different sampling schemes can help deciding which sampling scheme to apply in a given context.