Fingerprinting Codes and the Price of Approximate Differential Privacy
This work addresses the fundamental trade-off between privacy and accuracy in data analysis for researchers and practitioners in differential privacy, providing a foundational lower bound that is not incremental.
The paper tackles the problem of determining the sample complexity needed for differentially private algorithms to accurately answer large sets of counting queries, showing a lower bound of n ≥ Ω̃(√d log|Q|/(α²ε)), which is optimal up to poly-logarithmic factors and demonstrates that privacy requires asymptotically more samples than accuracy alone.
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 $$ n \geq \tildeΩ\Bigg(\frac{\sqrt{d} \log |\mathcal{Q}|}{α^2 \varepsilon} \Bigg). $$ 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.