Stream Sampling Framework and Application for Frequency Cap Statistics

Abstract

Unaggregated data, in a streamed or distributed form, are prevalent and come from diverse sources such as interactions of users with web services and IP traffic. Data elements have keys (cookies, users, queries), and elements with different keys interleave. Analytics on such data typically utilizes statistics expressed as a sum over keys in a specified segment of a function f applied to the frequency (the total number of occurrences) of the key. In particular, Distinct is the number of active keys in the segment, Sum is the sum of their frequencies, and both are special cases of frequency cap statistics, which cap the frequency by a parameter T . Random samples can be very effective for quick and efficient estimation of statistics at query time. Ideally, to estimate statistics for a given function f , our sample would include a key with frequency w with probability roughly proportional to f ( w ). The challenge is that while such “gold-standard” samples can be easily computed after aggregating the data (computing the set of key-frequency pairs), this aggregation is costly: It requires structure of size that is proportional to the number of active keys, which can be very large. We present a sampling framework for unaggregated data that uses a single pass (for streams) or two passes (for distributed data) and structure size proportional to the desired sample size. Our design unifies classic solutions for Distinct and Sum. Specifically, our ℓ-capped samples provide nonnegative unbiased estimates of any monotone non-decreasing frequency statistics and statistical guarantees on quality that are close to gold standard for cap statistics with T =Θ (ℓ). Furthermore, our multi-objective samples provide these statistical guarantees on quality for all concave sub-linear statistics (the nonnegative span of cap functions) while incurring only a logarithmic overhead on sample size.