2013/11/13 by Mark Bun, Jonathan Ullman, Bun, Mark +3 · 5 citations
Computer Science · Social Sciences · #Cryptography and Data Security #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #Privacy, Security, and Data Protection #Privacy-Preserving Technologies in Data
paper · pdf · doi:10.48550/arxiv.1311.3158
openalex publication_date 2013/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show new lower bounds on the sample complexity of (ε, δ)-differentially private algorithms that accurately answer large sets of counting queries. A counting query on a database D ∈ (\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 Q of ≫ nd counting queries on D to within error ± α it is necessary that n ≥ Ω((√(d) log |Q|)/(α2 ε) ). 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 (ε, δ)-differential privacy is asymptotically larger than what is required merely for accuracy, which is O(log |Q| / α2). In addition, we show that our lower bound holds for the specific case of k-way marginal queries (where |Q| = 2k \binomdk) 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 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.