April 20, 2014

Theory Lunch: Jon Ullman – October 11, 2013

Title: Fingerprinting Codes and the Price of Approximate Differential Privacy
Speaker: Jon Ullman, Harvard University

[Complete Video]

Abstract: 

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 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 of >> nd linear queries, Q, on D to within error +/- α it is necessary that n > Ω~(d^{1/2} log|Q| / α2 ε).  This bound is optimal up to poly-logarithmic factors, as demonstrated by the Private Multiplicative Weights algorithm of 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 marginals (where |Q|~(2d)k) when α is a constant.
 
Our results rely on the existence of short fingerprinting codes (Boneh-Shaw, CRYPTO'95), 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.
 
Joint work with Mark Bun and Salil Vadhan.