arXiv subjects

Jonathan Ullman

Publications and source records attributed to Jonathan Ullman.

77 records · Page 5Linked to original sources

arXiv2013

Faster Private Release of Marginals on Small Databases

We study the problem of answering \emph{$k$-way marginal} queries on a database $D \in (\{0,1\}^d)^n$, while preserving differential privacy. The answer to a $k$-way marginal query is the fraction of the database's records $x \in \{0,1\}^d$ with a given value in each of a given set of up to $k$ columns. Marginal queries enable a rich class of statistical analyses on a dataset, and designing efficient algorithms for privately answering marginal queries has been identified as an important open problem in private data analysis. For any $k$, we give a differentially private online algorithm that runs in time $$ \min{\exp(d^{1-Ω(1/\sqrt{k})}), \exp(d / \log^{.99} d)\} $$ per query and answers any (possibly superpolynomially long and adaptively chosen) sequence of $k$-way marginal queries up to error at most $\pm .01$ on every query, provided $n \gtrsim d^{.51} $. To the best of our knowledge, this is the first algorithm capable of privately answering marginal queries with a non-trivial worst-case accuracy guarantee on a database of size $\poly(d, k)$ in time $\exp(o(d))$. Our algorithms are a variant of the private multiplicative weights algorithm (Hardt and Rothblum, FOCS '10), but using a different low-weight representation of the database. We derive our low-weight representation using approximations to the OR function by low-degree polynomials with coefficients of bounded $L_1$-norm. We also prove a strong limitation on our approach that is of independent approximation-theoretic interest. Specifically, we show that for any $k = o(\log d)$, any polynomial with coefficients of $L_1$-norm $poly(d)$ that pointwise approximates the $d$-variate OR function on all inputs of Hamming weight at most $k$ must have degree $d^{1-O(1/\sqrt{k})}$.

cs.DS↗

arXiv2013

Differential Privacy for the Analyst via Private Equilibrium Computation

We give new mechanisms for answering exponentially many queries from multiple analysts on a private database, while protecting differential privacy both for the individuals in the database and for the analysts. That is, our mechanism's answer to each query is nearly insensitive to changes in the queries asked by other analysts. Our mechanism is the first to offer differential privacy on the joint distribution over analysts' answers, providing privacy for data analysts even if the other data analysts collude or register multiple accounts. In some settings, we are able to achieve nearly optimal error rates (even compared to mechanisms which do not offer analyst privacy), and we are able to extend our techniques to handle non-linear queries. Our analysis is based on a novel view of the private query-release problem as a two-player zero-sum game, which may be of independent interest.

cs.DS↗

arXiv2011

Privately Releasing Conjunctions and the Statistical Query Barrier

Suppose we would like to know all answers to a set of statistical queries C on a data set up to small error, but we can only access the data itself using statistical queries. A trivial solution is to exhaustively ask all queries in C. Can we do any better? + We show that the number of statistical queries necessary and sufficient for this task is---up to polynomial factors---equal to the agnostic learning complexity of C in Kearns' statistical query (SQ) model. This gives a complete answer to the question when running time is not a concern. + We then show that the problem can be solved efficiently (allowing arbitrary error on a small fraction of queries) whenever the answers to C can be described by a submodular function. This includes many natural concept classes, such as graph cuts and Boolean disjunctions and conjunctions. While interesting from a learning theoretic point of view, our main applications are in privacy-preserving data analysis: Here, our second result leads to the first algorithm that efficiently releases differentially private answers to of all Boolean conjunctions with 1% average error. This presents significant progress on a key open problem in privacy-preserving data analysis. Our first result on the other hand gives unconditional lower bounds on any differentially private algorithm that admits a (potentially non-privacy-preserving) implementation using only statistical queries. Not only our algorithms, but also most known private algorithms can be implemented using only statistical queries, and hence are constrained by these lower bounds. Our result therefore isolates the complexity of agnostic learning in the SQ-model as a new barrier in the design of differentially private algorithms.

cs.DS↗

arXiv2011

Iterative Constructions and Private Data Release

In this paper we study the problem of approximately releasing the cut function of a graph while preserving differential privacy, and give new algorithms (and new analyses of existing algorithms) in both the interactive and non-interactive settings. Our algorithms in the interactive setting are achieved by revisiting the problem of releasing differentially private, approximate answers to a large number of queries on a database. We show that several algorithms for this problem fall into the same basic framework, and are based on the existence of objects which we call iterative database construction algorithms. We give a new generic framework in which new (efficient) IDC algorithms give rise to new (efficient) interactive private query release mechanisms. Our modular analysis simplifies and tightens the analysis of previous algorithms, leading to improved bounds. We then give a new IDC algorithm (and therefore a new private, interactive query release mechanism) based on the Frieze/Kannan low-rank matrix decomposition. This new release mechanism gives an improvement on prior work in a range of parameters where the size of the database is comparable to the size of the data universe (such as releasing all cut queries on dense graphs). We also give a non-interactive algorithm for efficiently releasing private synthetic data for graph cuts with error O(|V|^{1.5}). Our algorithm is based on randomized response and a non-private implementation of the SDP-based, constant-factor approximation algorithm for cut-norm due to Alon and Naor. Finally, we give a reduction based on the IDC framework showing that an efficient, private algorithm for computing sufficiently accurate rank-1 matrix approximations would lead to an improved efficient algorithm for releasing private synthetic data for graph cuts. We leave finding such an algorithm as our main open problem.

cs.DS↗

arXiv2011

On the Zero-Error Capacity Threshold for Deletion Channels

We consider the zero-error capacity of deletion channels. Specifically, we consider the setting where we choose a codebook ${\cal C}$ consisting of strings of $n$ bits, and our model of the channel corresponds to an adversary who may delete up to $pn$ of these bits for a constant $p$. Our goal is to decode correctly without error regardless of the actions of the adversary. We consider what values of $p$ allow non-zero capacity in this setting. We suggest multiple approaches, one of which makes use of the natural connection between this problem and the problem of finding the expected length of the longest common subsequence of two random sequences.

cs.IT↗