Publications by Year: 2015

Significant progress has been made recently on theoretical analysis of estimators for the stochastic block model (SBM). In this paper, we consider the multi-graph SBM, which serves as a foundation for many application settings including dynamic and multi-layer networks. We explore the asymptotic properties of two estimators for the multi-graph SBM, namely spectral clustering and the maximum-likelihood estimate (MLE), as the number of layers of the multi-graph increases. We derive sufficient conditions for consistency of both estimators and propose a variational approximation to the MLE that is computationally feasible for large networks. We verify the sufficient conditions via simulation and demonstrate that they are practical. In addition, we apply the model to two real data sets: a dynamic social network and a multi-layer social network with several types of relations.

The vast majority of social science research uses small (megabyte- or gigabyte-scale) datasets. These fixed-scale datasets are commonly downloaded to the researcher’s computer where the analysis is performed. The data can be shared, archived, and cited with well-established technologies, such as the Dataverse Project, to support the published results. The trend toward big data—including large-scale streaming data—is starting to transform research and has the potential to impact policymaking as well as our understanding of the social, economic, and political problems that affect human societies. However, big data research poses new challenges to the execution of the analysis, archiving and reuse of the data, and reproduction of the results. Downloading these datasets to a researcher’s computer is impractical, leading to analyses taking place in the cloud, and requiring unusual expertise, collaboration, and tool development. The increased amount of information in these large datasets is an advantage, but at the same time it poses an increased risk of revealing personally identifiable sensitive information. In this article, we discuss solutions to these new challenges so that the social sciences can realize the potential of big data.

This essay first appeared in the Internet Monitor project’s second annual report, Internet Monitor 2014: Reflections on the Digital World. The report, published by the Berkman Center for Internet & Society, is a collection of roughly three dozen short contributions that highlight and discuss some of the most compelling events and trends in the digitally networked environment over the past year.

We prove new upper and lower bounds on the sample complexity of (ϵ,δ) differentially private algorithms for releasing approximate answers to threshold functions. A threshold function cx over a totally ordered domain X evaluates to cx(y)=1 if y≤x, and evaluates to 0 otherwise. We give the first nontrivial lower bound for releasing thresholds with (ϵ,δ) differential privacy, showing that the task is impossible over an infinite domain X, and moreover requires sample complexity n≥Ω(log∗|X|), which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with n≤2(1+o(1))log∗|X| samples. This improves the previous best upper bound of 8(1+o(1))log∗|X| (Beimel et al., RANDOM '13).
Our sample complexity upper and lower bounds also apply to the tasks of learning distributions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with (ϵ,δ) differential privacy and learning without privacy. For properly learning thresholds in ℓ dimensions, this lower bound extends to n≥Ω(ℓ⋅log∗|X|).
To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database D of elements from X, the interior point problem asks for an element between the smallest and largest elements in D. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy.

In a search task, a group of agents compete to be the first to find the solution. Each agent has different private information to incorporate into its search. This problem is inspired by settings such as scientific research, Bitcoin hash inversion, or hunting for some buried treasure. A social planner such as a funding agency, mining pool, or pirate captain might like to convince the agents to collaborate, share their information, and greatly reduce the cost of searching. However, this cooperation is in tension with the individuals' competitive desire to each be the first to win the search. The planner's proposal should incentivize truthful information sharing, reduce the total cost of searching, and satisfy fairness properties that preserve the spirit of the competition. We design contract-based mechanisms for information sharing without money. The planner solicits the agents' information and assigns search locations to the agents, who may then search only within their assignments. Truthful reporting of information to the mechanism maximizes an agent's chance to win the search. Epsilon-voluntary participation is satisfied for large search spaces. In order to formalize the planner's goals of fairness and reduced search cost, we propose a simplified, simulated game as a benchmark and quantify fairness and search cost relative to this benchmark scenario. The game is also used to implement our mechanisms. Finally, we extend to the case where coalitions of agents may participate in the mechanism, forming larger coalitions recursively.

A new line of work, started with Dwork et al., studies the task of answering statistical queries using a sample and relates the problem to the concept of differential privacy. By the Hoeffding bound, a sample of size O(logk/α2) suffices to answer k non-adaptive queries within error α, where the answers are computed by evaluating the statistical queries on the sample. This argument fails when the queries are chosen adaptively (and can hence depend on the sample). Dwork et al. showed that if the answers are computed with (ϵ,δ)-differential privacy then O(ϵ) accuracy is guaranteed with probability 1−O(δϵ). Using the Private Multiplicative Weights mechanism, they concluded that the sample size can still grow polylogarithmically with the k.
Very recently, Bassily et al. presented an improved bound and showed that (a variant of) the private multiplicative weights algorithm can answer k adaptively chosen statistical queries using sample complexity that grows logarithmically in k. However, their results no longer hold for every differentially private algorithm, and require modifying the private multiplicative weights algorithm in order to obtain their high probability bounds.
We greatly simplify the results of Dwork et al. and improve on the bound by showing that differential privacy guarantees O(ϵ) accuracy with probability 1−O(δlog(1/ϵ)/ϵ). It would be tempting to guess that an (ϵ,δ)-differentially private computation should guarantee O(ϵ) accuracy with probability 1−O(δ). However, we show that this is not the case, and that our bound is tight (up to logarithmic factors).

We propose graph encryption schemes that efficiently support approximate shortest distance queries on large-scale encrypted graphs. Shortest distance queries are one of the most fundamental graph operations and have a wide range of applications. Using such graph encryption schemes, a client can outsource large-scale privacy-sensitive graphs to an untrusted server without losing the ability to query it. Other applications include encrypted graph databases and controlled disclosure systems. We propose GRECS (stands for GRaph EnCryption for approximate Shortest distance queries) which includes three schemes that are provably secure against any semi-honest server. Our first construction makes use of only symmetric-key operations, resulting in a computationally-efficient construction. Our second scheme, makes use of somewhat-homomorphic encryption and is less computationally-efficient but achieves optimal communication complexity (i.e., uses a minimal amount of bandwidth). Finally, our third scheme is both computationally-efficient and achieves optimal communication complexity at the cost of a small amount of additional leakage. We implemented and evaluated the efficiency of our constructions experimentally. The experiments demonstrate that our schemes are efficient and can be applied to graphs that scale up to 1.6 million nodes and 11 million edges.

We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to within constant error in a certain one-sided sense, then an OR of disjoint copies of that circuit cannot be pointwise approximated even with very high error. As our main application, we show that for every sequence of degrees d(n), there is an explicit depththree circuit F : {−1, 1} n → {−1, 1} of polynomial-size such that any degree-d polynomial cannot pointwise approximate F to error better than 1 − exp −Ω( ˜ nd−3/2 ) . As a consequence of our main result, we obtain an exp −Ω( ˜ n 2/5 ) upper bound on the the discrepancy of a function in AC0, and an exp Ω( ˜ n 2/5 ) lower bound on the threshold weight of AC0, improving over the previous best results of exp −Ω(n 1/3 ) and exp Ω(n 1/3 ) respectively. Our techniques also yield a new lower bound of Ω n 1/2/ log(d−2)/2 (n) on the approximate degree of the AND-OR tree of depth d, which is tight up to polylogarithmic factors for any constant d, as well as new bounds for read-once DNF formulas. In turn, these results imply new lower bounds on the communication and circuit complexity of these classes, and demonstrate strong limitations on existing PAC learning algorithms.

On September 24-25, 2013, the Privacy Tools for Sharing Research Data project at Harvard University held a workshop titled "Integrating Approaches to Privacy across the Research Data Lifecycle." Over forty leading experts in computer science, statistics, law, policy, and social science research convened to discuss the state of the art in data privacy research. The resulting conversations centered on the emerging tools and approaches from the participants’ various disciplines and how they should be integrated in the context of real-world use cases that involve the management of confidential research data.

Researchers are increasingly obtaining data from social networking websites, publicly-placed sensors, government records and other public sources. Much of this information appears public, at least to first impressions, and it is capable of being used in research for a wide variety of purposes with seemingly minimal legal restrictions. The insights about human behaviors we may gain from research that uses this data are promising. However, members of the research community are questioning the ethics of these practices, and at the heart of the matter are some difficult questions about the boundaries between public and private information. This workshop report, the second in a series, identifies selected questions and explores issues around the meaning of “public” in the context of using data about individuals for research purposes.

We show an essentially tight bound on the number of adaptively chosen statistical queries that a computationally efficient algorithm can answer accurately given n samples from an unknown distribution. A statistical query asks for the expectation of a predicate over the underlying distribution, and an answer to a statistical query is accurate if it is “close” to the correct expectation over the distribution. This question was recently studied by Dwork et al. (2015), who showed how to answer Ω( ˜ n 2 ) queries efficiently, and also by Hardt and Ullman (2014), who showed that answering O˜(n 3 ) queries is hard. We close the gap between the two bounds and 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 O(n 2 ) adaptively chosen statistical queries. An implication of our results is that computationally efficient algorithms for answering arbitrary, adaptively chosen statistical queries may as well be differentially private. We obtain our results using a new connection between the problem of answering adaptively chosen statistical queries and a combinatorial object called an interactive fingerprinting code Fiat and Tassa (2001). In order to optimize our hardness result, we give a new Fourier-analytic approach to analyzing fingerprinting codes that is simpler, more flexible, and yields better parameters than previous constructions.

A private learner is an algorithm that given a sample of labeled individual examples outputs a generalizing hypothesis while preserving the privacy of each individual. In 2008, Kasiviswanathan et al. (FOCS 2008) gave a generic construction of private learners, in which the sample complexity is (generally) higher than what is needed for non-private learners. This gap in the sample complexity was then further studied in several followup papers, showing that (at least in some cases) this gap is unavoidable. Moreover, those papers considered ways to overcome the gap, by relaxing either the privacy or the learning guarantees of the learner.
We suggest an alternative approach, inspired by the (non-private) models of semi-supervised learning and active-learning, where the focus is on the sample complexity of labeled examples whereas unlabeled examples are of a significantly lower cost. We consider private semi-supervised learners that operate on a random sample, where only a (hopefully small) portion of this sample is labeled. The learners have no control over which of the sample elements are labeled. Our main result is that the labeled sample complexity of private learners is characterized by the VC dimension.
We present two generic constructions of private semi-supervised learners. The first construction is of learners where the labeled sample complexity is proportional to the VC dimension of the concept class, however, the unlabeled sample complexity of the algorithm is as big as the representation length of domain elements. Our second construction presents a new technique for decreasing the labeled sample complexity of a given private learner, while only slightly increasing its unlabeled sample complexity. In addition, we show that in some settings the labeled sample complexity does not depend on the privacy parameters of the learner.