The Optimal Quantile Estimator for Compressed Counting

Abstract: Compressed Counting (CC) was recently proposed for very efficiently computing the (approximate) $\alpha$th frequency moments of data streams, where $0<\alpha <= 2$. Several estimators were reported including the geometric mean estimator, the harmonic mean estimator, the optimal power estimator, etc. The geometric mean estimator is particularly interesting for theoretical purposes. For example, when $\alpha -> 1$, the complexity of CC (using the geometric mean estimator) is $O(1/\epsilon)$, breaking the well-known large-deviation bound $O(1/\epsilon^2)$. The case $\alpha\approx 1$ has important applications, for example, computing entropy of data streams. For practical purposes, this study proposes the optimal quantile estimator. Compared with previous estimators, this estimator is computationally more efficient and is also more accurate when $\alpha> 1$.