High Probability Frequency Moment Sketches

We consider the problem of sketching the p-th frequency moment of a vector, p>2, with multiplicative error at most 1± ε and with high confidence 1-δ. Despite the long sequence of work on this problem, tight bounds on this quantity are only known for constant δ. While one can obtain an upper bound with error probability δ by repeating a sketching algorithm with constant error probability O(log(1/δ)) times in parallel, and taking the median of the outputs, we show this is a suboptimal algorithm! Namely, we show optimal upper and lower bounds of Theta(n1-2/p log(1/δ) + n1-2/p log2/p (1/δ) log n) on the sketching dimension, for any constant approximation. Our result should be contrasted with results for estimating frequency moments for 1 ≤ p ≤ 2, for which we show the optimal algorithm for general δ is obtained by repeating the optimal algorithm for constant error probability O(log(1/δ)) times and taking the median output. We also obtain a matching lower bound for this problem, up to constant factors.