[Paper Review] Fingerprinting Codes and the Price of Approximate Differential Privacy

Mark Bun, Jonathan Ullman+1 more|arXiv (Cornell University)|Nov 13, 2013

Privacy-Preserving Technologies in Data23 references4 citations

TL;DR

This paper establishes tight information-theoretic lower bounds on the sample complexity required for $(\varepsilon,\delta)$-differentially private algorithms to accurately answer large sets of counting queries. It demonstrates that the price of approximate differential privacy is asymptotically larger than pure accuracy requirements, proving a lower bound of $\tilde{\Omega}\left(\frac{\sqrt{d}\log|\mathcal{Q}|}{\alpha^2\varepsilon}\right)$, which matches the best-known upper bounds up to logarithmic factors.

ABSTRACT

We show new lower bounds on the sample complexity of $(\varepsilon, δ)$-differentially private algorithms that accurately answer large sets of counting queries. A counting query on a database $D \in (\{0,1\}^d)^n$ has the form "What fraction of the individual records in the database satisfy the property $q$?" We show that in order to answer an arbitrary set $\mathcal{Q}$ of $\gg nd$ counting queries on $D$ to within error $\pm α$ it is necessary that $$ n \geq ildeΩ\Bigg(\frac{\sqrt{d} \log |\mathcal{Q}|}{α^2 \varepsilon} \Bigg). $$ This bound is optimal up to poly-logarithmic factors, as demonstrated by the Private Multiplicative Weights algorithm (Hardt and Rothblum, FOCS'10). In particular, our lower bound is the first to show that the sample complexity required for accuracy and $(\varepsilon, δ)$-differential privacy is asymptotically larger than what is required merely for accuracy, which is $O(\log |\mathcal{Q}| / α^2)$. In addition, we show that our lower bound holds for the specific case of $k$-way marginal queries (where $|\mathcal{Q}| = 2^k \binom{d}{k}$) when $α$ is not too small compared to $d$ (e.g. when $α$ is any fixed constant). Our results rely on the existence of short \emph{fingerprinting codes} (Boneh and Shaw, CRYPTO'95, Tardos, STOC'03), which we show are closely connected to the sample complexity of differentially private data release. We also give a new method for combining certain types of sample complexity lower bounds into stronger lower bounds.

Motivation & Objective

• To determine the minimum sample complexity required for $(\varepsilon,\delta)$-differentially private algorithms to accurately answer large families of counting queries.
• To establish that the price of differential privacy is asymptotically larger than the sample complexity required for accuracy alone.
• To show that fingerprinting codes are fundamentally connected to the sample complexity of private data release.
• To prove a tight lower bound for $k$-way marginal queries and general counting query families under approximate differential privacy.

Proposed method

• Reduces the problem of private query release to the existence of short fingerprinting codes with specific robustness properties.
• Uses a novel composition theorem to combine lower bounds from different query families into a stronger overall bound.
• Applies weakly robust fingerprinting codes (specifically Tardos codes) and proves their strong robustness via probabilistic and concentration arguments.
• Employs a reduction from full robustness to weak robustness in fingerprinting codes to establish sample complexity lower bounds.
• Uses Markov's inequality and tail bounds on binomial and trigonometric integrals to control error probabilities in the analysis.
• Leverages the connection between fingerprinting codes and the VC-dimension of query families to derive information-theoretic limits.

Experimental results

Research questions

• RQ1 What is the minimum sample size required for an $(\varepsilon,\delta)$-differentially private algorithm to accurately answer $|\mathcal{Q}|$ counting queries with error $\pm\alpha$?
• RQ2 Does the requirement of $(\varepsilon,\delta)$-differential privacy incur an asymptotic cost beyond the sample complexity needed for accurate statistical estimation?
• RQ3 How are fingerprinting codes related to the sample complexity of differentially private data release?
• RQ4 Can the sample complexity lower bound be improved or tightened using structural properties of query families such as $k$-way marginals?
• RQ5 Is the Private Multiplicative Weights algorithm's sample complexity optimal up to logarithmic factors?

Key findings

• The sample complexity for $(\varepsilon,\delta)$-differentially private release of $|\mathcal{Q}|$ counting queries is at least $\tilde{\Omega}\left(\frac{\sqrt{d}\log|\mathcal{Q}|}{\alpha^2\varepsilon}\right)$, which is tight up to poly-logarithmic factors.
• This lower bound is strictly larger than the $O(\log|\mathcal{Q}|/\alpha^2)$ sample complexity required for accurate estimation without privacy, establishing a non-trivial price of differential privacy.
• For $k$-way marginal queries with $|\mathcal{Q}| = 2^k \binom{d}{k}$, the lower bound is $\Omega(\sqrt{d}/\alpha^2\varepsilon)$ when $\alpha$ is a constant, and $\Omega(k/\varepsilon)$ when $\alpha$ is small.
• Tardos' fingerprinting code is shown to be weakly robust, and this property is used to derive strong lower bounds via a reduction to full robustness.
• The analysis proves that with high probability, a randomly generated Tardos codebook contains $\Omega(n^{3/2}\log(n/\xi))$ marked columns, which is essential for the lower bound construction.
• The paper establishes that the Private Multiplicative Weights algorithm achieves the optimal sample complexity up to logarithmic factors, confirming the tightness of the lower bound.