arXiv subjects

Salil Vadhan

Publications and source records attributed to Salil Vadhan.

At least 55 records · Page 3Linked to original sources

arXiv2019

Differential Privacy on Finite Computers

We consider the problem of designing and analyzing differentially private algorithms that can be implemented on {\em discrete} models of computation in {\em strict} polynomial time, motivated by known attacks on floating point implementations of real-arithmetic differentially private algorithms (Mironov, CCS 2012) and the potential for timing attacks on expected polynomial-time algorithms. As a case study, we examine the basic problem of approximating the histogram of a categorical dataset over a possibly large data universe $\mathcal{X}$. The classic Laplace Mechanism (Dwork, McSherry, Nissim, Smith, TCC 2006 and J. Privacy \& Confidentiality 2017) does not satisfy our requirements, as it is based on real arithmetic, and natural discrete analogues, such as the Geometric Mechanism (Ghosh, Roughgarden, Sundarajan, STOC 2009 and SICOMP 2012), take time at least linear in $|\mathcal{X}|$, which can be exponential in the bit length of the input. In this paper, we provide strict polynomial-time discrete algorithms for approximate histograms whose simultaneous accuracy (the maximum error over all bins) matches that of the Laplace Mechanism up to constant factors, while retaining the same (pure) differential privacy guarantee. One of our algorithms produces a sparse histogram as output. Its "per-bin accuracy" (the error on individual bins) is worse than that of the Laplace Mechanism by a factor of $\log|\mathcal{X}|$, but we prove a lower bound showing that this is necessary for any algorithm that produces a sparse histogram. A second algorithm avoids this lower bound, and matches the per-bin accuracy of the Laplace Mechanism, by producing a compact and efficiently computable representation of a dense histogram, it is based on an $(n+1)$-wise independent implementation of an appropriately clamped version of the Discrete Geometric Mechanism.

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arXiv2018

Fingerprinting Codes and the Price of Approximate Differential Privacy

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.

cs.CR↗

arXiv2018

Usable Differential Privacy: A Case Study with PSI

Differential privacy is a promising framework for addressing the privacy concerns in sharing sensitive datasets for others to analyze. However differential privacy is a highly technical area and current deployments often require experts to write code, tune parameters, and optimize the trade-off between the privacy and accuracy of statistical releases. For differential privacy to achieve its potential for wide impact, it is important to design usable systems that enable differential privacy to be used by ordinary data owners and analysts. PSI is a tool that was designed for this purpose, allowing researchers to release useful differentially private statistical information about their datasets without being experts in computer science, statistics, or privacy. We conducted a thorough usability study of PSI to test whether it accomplishes its goal of usability by non-experts. The usability test illuminated which features of PSI are most user-friendly and prompted us to improve aspects of the tool that caused confusion. The test also highlighted some general principles and lessons for designing usable systems for differential privacy, which we discuss in depth.

cs.HC↗

arXiv2018

PSI (Ψ): a Private data Sharing Interface

We provide an overview of PSI ("a Private data Sharing Interface"), a system we are developing to enable researchers in the social sciences and other fields to share and explore privacy-sensitive datasets with the strong privacy protections of differential privacy.

cs.CR↗

arXiv2017

Finite Sample Differentially Private Confidence Intervals

We study the problem of estimating finite sample confidence intervals of the mean of a normal population under the constraint of differential privacy. We consider both the known and unknown variance cases and construct differentially private algorithms to estimate confidence intervals. Crucially, our algorithms guarantee a finite sample coverage, as opposed to an asymptotic coverage. Unlike most previous differentially private algorithms, we do not require the domain of the samples to be bounded. We also prove lower bounds on the expected size of any differentially private confidence set showing that our the parameters are optimal up to polylogarithmic factors.

cs.CR↗

arXiv2017

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space

We give a deterministic $\tilde{O}(\log n)$-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using $O(\log n)$ space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using $O(\log^{3/2} n)$ space (Saks and Zhou, FOCS 1995 and JCSS 1999). Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC `04; Peng and Spielman, STOC `14) with ideas used to show that Undirected S-T Connectivity is in deterministic logspace (Reingold, STOC `05 and JACM `08; Rozenman and Vadhan, RANDOM `05).

cs.CC↗

arXiv2017

Locating a Small Cluster Privately

We present a new algorithm for locating a small cluster of points with differential privacy [Dwork, McSherry, Nissim, and Smith, 2006]. Our algorithm has implications to private data exploration, clustering, and removal of outliers. Furthermore, we use it to significantly relax the requirements of the sample and aggregate technique [Nissim, Raskhodnikova, and Smith, 2007], which allows compiling of "off the shelf" (non-private) analyses into analyses that preserve differential privacy.

cs.DS↗

arXiv2016

Differentially Private Chi-Squared Hypothesis Testing: Goodness of Fit and Independence Testing

Hypothesis testing is a useful statistical tool in determining whether a given model should be rejected based on a sample from the population. Sample data may contain sensitive information about individuals, such as medical information. Thus it is important to design statistical tests that guarantee the privacy of subjects in the data. In this work, we study hypothesis testing subject to differential privacy, specifically chi-squared tests for goodness of fit for multinomial data and independence between two categorical variables. We propose new tests for goodness of fit and independence testing that like the classical versions can be used to determine whether a given model should be rejected or not, and that additionally can ensure differential privacy. We give both Monte Carlo based hypothesis tests as well as hypothesis tests that more closely follow the classical chi-squared goodness of fit test and the Pearson chi-squared test for independence. Crucially, our tests account for the distribution of the noise that is injected to ensure privacy in determining significance. We show that these tests can be used to achieve desired significance levels, in sharp contrast to direct applications of classical tests to differentially private contingency tables which can result in wildly varying significance levels. Moreover, we study the statistical power of these tests. We empirically show that to achieve the same level of power as the classical non-private tests our new tests need only a relatively modest increase in sample size.

math.ST↗

arXiv2016

The Complexity of Computing the Optimal Composition of Differential Privacy

In the study of differential privacy, composition theorems (starting with the original paper of Dwork, McSherry, Nissim, and Smith (TCC'06)) bound the degradation of privacy when composing several differentially private algorithms. Kairouz, Oh, and Viswanath (ICML'15) showed how to compute the optimal bound for composing $k$ arbitrary $(ε,δ)$-differentially private algorithms. We characterize the optimal composition for the more general case of $k$ arbitrary $(ε_{1},δ_{1}),\ldots,(ε_{k},δ_{k})$-differentially private algorithms where the privacy parameters may differ for each algorithm in the composition. We show that computing the optimal composition in general is $\#$P-complete. Since computing optimal composition exactly is infeasible (unless FP=$\#$P), we give an approximation algorithm that computes the composition to arbitrary accuracy in polynomial time. The algorithm is a modification of Dyer's dynamic programming approach to approximately counting solutions to knapsack problems (STOC'03).

cs.CC↗

arXiv2016

Towards a Privacy Research Roadmap for the Computing Community

Great advances in computing and communication technology are bringing many benefits to society, with transformative changes and financial opportunities being created in health care, transportation, education, law enforcement, national security, commerce, and social interactions. Many of these benefits, however, involve the use of sensitive personal data, and thereby raise concerns about privacy. Failure to address these concerns can lead to a loss of trust in the private and public institutions that handle personal data, and can stifle the independent thought and expression that is needed for our democracy to flourish. This report, sponsored by the Computing Community Consortium (CCC), suggests a roadmap for privacy research over the next decade, aimed at enabling society to appropriately control threats to privacy while enjoying the benefits of information technology and data science. We hope that it will be useful to the agencies of the Federal Networking and Information Technology Research and Development (NITRD) Program as they develop a joint National Privacy Research Strategy over the coming months. The report synthesizes input drawn from the privacy and computing communities submitted to both the CCC and NITRD, as well as past reports on the topic.

cs.CY↗

arXiv2015

Pseudorandomness for Read-Once, Constant-Depth Circuits

For Boolean functions computed by read-once, depth-$D$ circuits with unbounded fan-in over the de Morgan basis, we present an explicit pseudorandom generator with seed length $\tilde{O}(\log^{D+1} n)$. The previous best seed length known for this model was $\tilde{O}(\log^{D+4} n)$, obtained by Trevisan and Xue (CCC `13) for all of $AC^0$ (not just read-once). Our work makes use of Fourier analytic techniques for pseudorandomness introduced by Reingold, Steinke, and Vadhan (RANDOM `13) to show that the generator of Gopalan et al. (FOCS `12) fools read-once $AC^0$. To this end, we prove a new Fourier growth bound for read-once circuits, namely that for every $F: \{0,1\}^n\to\{0,1\}$ computed by a read-once, depth-$D$ circuit, \begin{equation*}\sum_{s\subseteq[n], |s|=k}|\hat{F}[s]|\le O(\log^{D-1}n)^k,\end{equation*} where $\hat{F}$ denotes the Fourier transform of $F$ over $\mathbb{Z}^n_2$.

cs.CC↗

arXiv2014

Privacy Games

The problem of analyzing the effect of privacy concerns on the behavior of selfish utility-maximizing agents has received much attention lately. Privacy concerns are often modeled by altering the utility functions of agents to consider also their privacy loss. Such privacy aware agents prefer to take a randomized strategy even in very simple games in which non-privacy aware agents play pure strategies. In some cases, the behavior of privacy aware agents follows the framework of Randomized Response, a well-known mechanism that preserves differential privacy. Our work is aimed at better understanding the behavior of agents in settings where their privacy concerns are explicitly given. We consider a toy setting where agent A, in an attempt to discover the secret type of agent B, offers B a gift that one type of B agent likes and the other type dislikes. As opposed to previous works, B's incentive to keep her type a secret isn't the result of "hardwiring" B's utility function to consider privacy, but rather takes the form of a payment between B and A. We investigate three different types of payment functions and analyze B's behavior in each of the resulting games. As we show, under some payments, B's behavior is very different than the behavior of agents with hardwired privacy concerns and might even be deterministic. Under a different payment we show that B's BNE strategy does fall into the framework of Randomized Response.

cs.GT↗

arXiv2014

Pseudorandomness and Fourier Growth Bounds for Width 3 Branching Programs

We present an explicit pseudorandom generator for oblivious, read-once, width-$3$ branching programs, which can read their input bits in any order. The generator has seed length $\tilde{O}( \log^3 n ).$ The previously best known seed length for this model is $n^{1/2+o(1)}$ due to Impagliazzo, Meka, and Zuckerman (FOCS '12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM '13) for \textit{permutation} branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any $f:\{0,1\}^n\rightarrow \{0,1\}$ computed by such a branching program, and $k\in [n],$ $$\sum_{s\subseteq [n]: |s|=k} \left| \hat{f}[s] \right| \leq n^2 \cdot (O(\log n))^k,$$ where $\widehat{f}[s] = \mathbb{E}\left[f[U] \cdot (-1)^{s \cdot U}\right]$ is the standard Fourier transform over $\mathbb{Z}_2^n$. The base $O(\log n)$ of the Fourier growth is tight up to a factor of $\log \log n$.

cs.CC↗

arXiv2014

Faster Algorithms for Privately Releasing Marginals

We study the problem of releasing $k$-way marginals of a database $D \in (\{0,1\}^d)^n$, while preserving differential privacy. The answer to a $k$-way marginal query is the fraction of $D$'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 of a dataset, and designing efficient algorithms for privately releasing marginal queries has been identified as an important open problem in private data analysis (cf. Barak et. al., PODS '07). We give an algorithm that runs in time $d^{O(\sqrt{k})}$ and releases a private summary capable of answering any $k$-way marginal query with at most $\pm .01$ error on every query as long as $n \geq d^{O(\sqrt{k})}$. To our knowledge, ours is the first algorithm capable of privately releasing marginal queries with non-trivial worst-case accuracy guarantees in time substantially smaller than the number of $k$-way marginal queries, which is $d^{Θ(k)}$ (for $k \ll d$).

cs.DS↗

arXiv2014

Redrawing the Boundaries on Purchasing Data from Privacy-Sensitive Individuals

We prove new positive and negative results concerning the existence of truthful and individually rational mechanisms for purchasing private data from individuals with unbounded and sensitive privacy preferences. We strengthen the impossibility results of Ghosh and Roth (EC 2011) by extending it to a much wider class of privacy valuations. In particular, these include privacy valuations that are based on (ε, δ)-differentially private mechanisms for non-zero δ, ones where the privacy costs are measured in a per-database manner (rather than taking the worst case), and ones that do not depend on the payments made to players (which might not be observable to an adversary). To bypass this impossibility result, we study a natural special setting where individuals have mono- tonic privacy valuations, which captures common contexts where certain values for private data are expected to lead to higher valuations for privacy (e.g. having a particular disease). We give new mech- anisms that are individually rational for all players with monotonic privacy valuations, truthful for all players whose privacy valuations are not too large, and accurate if there are not too many players with too-large privacy valuations. We also prove matching lower bounds showing that in some respects our mechanism cannot be improved significantly.

cs.GT↗

arXiv2013

Locally Testable Codes and Cayley Graphs

We give two new characterizations of ($\F_2$-linear) locally testable error-correcting codes in terms of Cayley graphs over $\F_2^h$: \begin{enumerate} \item A locally testable code is equivalent to a Cayley graph over $\F_2^h$ whose set of generators is significantly larger than $h$ and has no short linear dependencies, but yields a shortest-path metric that embeds into $\ell_1$ with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into $\ell_1$. \item A locally testable code is equivalent to a Cayley graph over $\F_2^h$ that has significantly more than $h$ eigenvalues near 1, which have no short linear dependencies among them and which "explain" all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues. \end{enumerate}

cs.CC↗

arXiv2013

Pseudorandomness for Regular Branching Programs via Fourier Analysis

We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is $O(\log^2 n)$, where $n$ is the length of the branching program. The previous best seed length known for this model was $n^{1/2+o(1)}$, which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of $s^{1/2+o(1)}$ for arbitrary branching programs of size $s$). Our techniques also give seed length $n^{1/2+o(1)}$ for general oblivious, read-once branching programs of width $2^{n^{o(1)}}$, which is incomparable to the results of Impagliazzo et al.Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width $w$ has Fourier mass at most $(2w^2)^k$ at level $k$, independent of the length of the program.

cs.CC↗

arXiv2012

Truthful Mechanisms for Agents that Value Privacy

Recent work has constructed economic mechanisms that are both truthful and differentially private. In these mechanisms, privacy is treated separately from the truthfulness; it is not incorporated in players' utility functions (and doing so has been shown to lead to non-truthfulness in some cases). In this work, we propose a new, general way of modelling privacy in players' utility functions. Specifically, we only assume that if an outcome $o$ has the property that any report of player $i$ would have led to $o$ with approximately the same probability, then $o$ has small privacy cost to player $i$. We give three mechanisms that are truthful with respect to our modelling of privacy: for an election between two candidates, for a discrete version of the facility location problem, and for a general social choice problem with discrete utilities (via a VCG-like mechanism). As the number $n$ of players increases, the social welfare achieved by our mechanisms approaches optimal (as a fraction of $n$).

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