[Paper Review] Pan-private Algorithms: When Memory Does Not Help

Darakhshan Mir, S. Muthukrishnan+2 more|arXiv (Cornell University)|Sep 8, 2010

Privacy-Preserving Technologies in Data12 references3 citations

TL;DR

This paper presents improved pan-private streaming algorithms for fundamental counting tasks—distinct counts and heavy hitters—using noisy sketching techniques to ensure differential privacy even if internal memory is exposed. It establishes that unbounded memory does not significantly improve accuracy, proving tight lower bounds via noisy decoding, and achieves optimal additive error bounds for distinct count estimation and worst-case O(k) approximation for heavy hitters.

ABSTRACT

Consider updates arriving online in which the $t$th input is $(i_t,d_t)$, where $i_t$'s are thought of as IDs of users. Informally, a randomized function $f$ is {\em differentially private} with respect to the IDs if the probability distribution induced by $f$ is not much different from that induced by it on an input in which occurrences of an ID $j$ are replaced with some other ID $k$ Recently, this notion was extended to {\em pan-privacy} where the computation of $f$ retains differential privacy, even if the internal memory of the algorithm is exposed to the adversary (say by a malicious break-in or by fiat by the government). This is a strong notion of privacy, and surprisingly, for basic counting tasks such as distinct counts, heavy hitters and others, Dwork et al~\cite{dwork-pan} present pan-private algorithms with reasonable accuracy. The pan-private algorithms are nontrivial, and rely on sampling. We reexamine these basic counting tasks and show improved bounds. In particular, we estimate the distinct count $\Dt$ to within $(1\pm \eps)\Dt \pm O(\polylog m)$, where $m$ is the number of elements in the universe. This uses suitably noisy statistics on sketches known in the streaming literature. We also present the first known lower bounds for pan-privacy with respect to a single intrusion. Our lower bounds show that, even if allowed to work with unbounded memory, pan-private algorithms for distinct counts can not be significantly more accurate than our algorithms. Our lower bound uses noisy decoding. For heavy hitter counts, we present a pan private streaming algorithm that is accurate to within $O(k)$ in worst case; previously known bound for this problem is arbitrarily worse. An interesting aspect of our pan-private algorithms is that, they deliberately use very small (polylogarithmic) space and tend to be streaming algorithms, even though using more space is not forbidden.

Motivation & Objective

• To address the fundamental question of whether unbounded memory can improve accuracy in pan-private algorithms for basic counting tasks.
• To design pan-private streaming algorithms that maintain differential privacy even if internal state is exposed to an adversary.
• To establish tight lower bounds on the accuracy of pan-private algorithms for distinct count estimation, showing that memory does not help.
• To develop the first pan-private streaming algorithm with worst-case O(k) approximation for heavy hitter counts.
• To explore the limits of pan-privacy in streaming settings, particularly for problems involving both positive and negative updates.

Proposed method

• Uses noisy sketching techniques from the streaming literature to estimate distinct counts with (1±ε)D ± O(polylog m) additive error.
• Applies a randomized response mechanism on top of sketches to ensure pan-privacy, even if internal state is compromised.
• Employs a noisy decoding approach to prove lower bounds, showing that no pan-private algorithm can achieve significantly better accuracy than the proposed one, even with unbounded memory.
• Uses a randomized estimator for inner products and cropped frequency moments (T₂) via pairwise independent hashing and noise injection.
• Relies on Hoeffding’s inequality to bound estimation error in the noisy sketching process.
• Relaxes assumptions on known F₁ by running multiple parallel instances with exponentially increasing estimates of F₁, selecting the best-fitting one at query time.

Experimental results

Research questions

• RQ1 Can pan-private algorithms achieve significantly better accuracy for distinct count estimation if given unbounded memory?
• RQ2 What are the fundamental limits of accuracy for pan-private algorithms in estimating distinct counts, even with unlimited storage?
• RQ3 Can we design a pan-private streaming algorithm for heavy hitter counts with bounded worst-case error?
• RQ4 How does the use of noisy sketches affect the privacy and utility trade-off in pan-private computation?
• RQ5 To what extent can standard streaming techniques be adapted to satisfy pan-privacy under memory exposure?

Key findings

• The paper proves that pan-private algorithms for distinct count estimation cannot achieve better than O(polylog m) additive error, even with unbounded memory, using a noisy decoding argument.
• It presents a pan-private streaming algorithm that estimates distinct count D(t) within (1±ε)D(t) ± O(polylog m), matching the information-theoretic lower bound.
• For heavy hitter counts, the paper designs a pan-private algorithm with worst-case O(k) approximation error, improving upon previously known arbitrarily worse bounds.
• The proposed algorithms are inherently streaming, using only polylogarithmic space, despite the fact that larger memory is not forbidden by the model.
• The paper establishes that the use of noisy sketches and randomized response enables pan-privacy without requiring small memory, and that memory size does not improve accuracy.
• It shows that the cropped inner product estimator ˜(a·a′)(τ) is both pan-private and unbiased, with error bounded by Hoeffding’s inequality.