Fingerprinting codes and the price of approximate differential privacy

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 [EQUATION] This bound is optimal up to poly-logarithmic factors, as demonstrated by the Private Multiplicative Weights algorithm (Hardt and Rothblum, FOCS'10). It is also the first to show that the sample complexity required for (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(d/k)) when α is a 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.Engineering and Applied Science

Private Multiplicative Weights Beyond Linear Queries

A wide variety of fundamental data analyses in machine learning, such as linear and logistic regression, require minimizing a convex function defined by the data. Since the data may contain sensitive information about individuals, and these analyses can leak that sensitive information, it is important to be able to solve convex minimization in a privacy-preserving way. A series of recent results show how to accurately solve a single convex minimization problem in a differentially private manner. However, the same data is often analyzed repeatedly, and little is known about solving multiple convex minimization problems with differential privacy. For simpler data analyses, such as linear queries, there are remarkable differentially private algorithms such as the private multiplicative weights mechanism (Hardt and Rothblum, FOCS 2010) that accurately answer exponentially many distinct queries. In this work, we extend these results to the case of convex minimization and show how to give accurate and differentially private solutions to *exponentially many* convex minimization problems on a sensitive dataset

Tight Lower Bounds for Differentially Private Selection

A pervasive task in the differential privacy literature is to select the $k$ items of "highest quality" out of a set of $d$ items, where the quality of each item depends on a sensitive dataset that must be protected. Variants of this task arise naturally in fundamental problems like feature selection and hypothesis testing, and also as subroutines for many sophisticated differentially private algorithms. The standard approaches to these tasks---repeated use of the exponential mechanism or the sparse vector technique---approximately solve this problem given a dataset of $n = O(\sqrt{k}\log d)$ samples. We provide a tight lower bound for some very simple variants of the private selection problem. Our lower bound shows that a sample of size $n = \Omega(\sqrt{k} \log d)$ is required even to achieve a very minimal accuracy guarantee. Our results are based on an extension of the fingerprinting method to sparse selection problems. Previously, the fingerprinting method has been used to provide tight lower bounds for answering an entire set of $d$ queries, but often only some much smaller set of $k$ queries are relevant. Our extension allows us to prove lower bounds that depend on both the number of relevant queries and the total number of queries

Preventing False Discovery in Interactive Data Analysis is Hard

We show that, under a standard hardness assumption, there is no computationally efficient algorithm that given $n$ samples from an unknown distribution can give valid answers to $n^{3+o(1)}$ adaptively chosen statistical queries. A statistical query asks for the expectation of a predicate over the underlying distribution, and an answer to a statistical query is valid if it is "close" to the correct expectation over the distribution. Our result stands in stark contrast to the well known fact that exponentially many statistical queries can be answered validly and efficiently if the queries are chosen non-adaptively (no query may depend on the answers to previous queries). Moreover, a recent work by Dwork et al. shows how to accurately answer exponentially many adaptively chosen statistical queries via a computationally inefficient algorithm; and how to answer a quadratic number of adaptive queries via a computationally efficient algorithm. The latter result implies that our result is tight up to a linear factor in $n.$ Conceptually, our result demonstrates that achieving statistical validity alone can be a source of computational intractability in adaptive settings. For example, in the modern large collaborative research environment, data analysts typically choose a particular approach based on previous findings. False discovery occurs if a research finding is supported by the data but not by the underlying distribution. While the study of preventing false discovery in Statistics is decades old, to the best of our knowledge our result is the first to demonstrate a computational barrier. In particular, our result suggests that the perceived difficulty of preventing false discovery in today's collaborative research environment may be inherent

Order-Revealing Encryption and the Hardness of Private Learning

An order-revealing encryption scheme gives a public procedure by which two ciphertexts can be compared to reveal the ordering of their underlying plaintexts. We show how to use order-revealing encryption to separate computationally efficient PAC learning from efficient $(\epsilon, \delta)$-differentially private PAC learning. That is, we construct a concept class that is efficiently PAC learnable, but for which every efficient learner fails to be differentially private. This answers a question of Kasiviswanathan et al. (FOCS '08, SIAM J. Comput. '11). To prove our result, we give a generic transformation from an order-revealing encryption scheme into one with strongly correct comparison, which enables the consistent comparison of ciphertexts that are not obtained as the valid encryption of any message. We believe this construction may be of independent interest.Comment: 28 page