Differentially Private Fractional Frequency Moments Estimation with Polylogarithmic Space

We prove that Fp sketch, a well-celebrated streaming algorithm for frequency moments estimation, is differentially private as is when p ∈ (0, 1]. Fp sketch uses only polylogarithmic space, exponentially better than existing DP baselines and only worse than the optimal non-private baseline by a logarithmic factor. The evaluation shows that Fp sketch can achieve reasonable accuracy with differential privacy guarantee. The evaluation code is included in the supplementary material

Sketching and streaming algorithms

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 136-145).A sketch of a dataset is a small-space data structure supporting some prespecified set of queries (and possibly updates) while consuming space substantially sublinear in the space required to actually store all the data. Furthermore, it is often desirable, or required by the application, that the sketch itself be computable by a small-space algorithm given just one pass over the data, a so-called streaming algorithm. Sketching and streaming have found numerous applications in network traffic monitoring, data mining, trend detection, sensor networks, and databases. In this thesis, I describe several new contributions in the area of sketching and streaming algorithms. The first space-optimal streaming algorithm for the distinct elements problem. Our algorithm also achieves 0(1) update and reporting times. A streaming algorithm for Hamming norm estimation in the turnstile model which achieves the best known space complexity. The first space-optimal algorithm for pth moment estimation in turnstile streams for 0 < p < 2, with matching lower bounds, and another space-optimal algorithm which also has a fast O(log²(1/[epsilon]) log log(1[epsilon])) update time for (1+/-[epsilon])- approximation. A general reduction from empirical entropy estimation in turnstile streams to moment estimation, providing the only known near-optimal space-complexity upper bound for this problem. A proof of the Johnson-Lindenstrauss lemma where every matrix in the support of the embedding distribution is much sparser than previous known constructions. In particular, to achieve distortion (1+/-[epsilon]) with probability 1-[delta], we embed into optimal dimension 0([epsilon]-²log(1/[delta])) and such that every matrix in the support of the distribution has 0([epsilon]-¹ log(1/[delta])) non-zero entries per column.by Jelani Nelson.Ph.D

A Near-Optimal Algorithm for L1-Difference

We give the first L1-sketching algorithm for integer vectors which produces nearly optimal sized sketches in nearly linear time. This answers the first open problem in the list of open problems from the 2006 IITK Workshop on Algorithms for Data Streams. Specifically, suppose Alice receives a vector x ∈ {−M, . . . , M}n and Bob receives y ∈ {−M, . . . , M}n, and the two parties share randomness. Each party must output a short sketch of their vector such that a third party can later quickly recover a (1 ± ε)-approximation to ||x − y||1 with 2/3 probability given only the sketches. We give a sketching algorithm which produces O(ε−2log(1/ε) log(nM))-bit sketches in O(n log2(nM)) time, independent of ε. The previous best known sketching algorithm for L1 is due to [Feigenbaum et al., SICOMP 2002], which achieved the optimal sketch length of O(ε−2log(nM)) bits but had a running time of O(n log(nM)/ε2). Notice that our running time is near-linear for every ε, whereas for sufficiently small values of ε, the running time of the previous algorithm can be as large as quadratic. Like their algorithm, our sketching procedure also yields a small-space, one-pass streaming algorithm which works even if the entries of x, y are given in arbitrary order.Engineering and Applied Science