arXiv subjects

Jonathan Ullman

Publications and source records attributed to Jonathan Ullman.

At least 73 records · Page 4Linked to original sources

arXiv2016

Make Up Your Mind: The Price of Online Queries in Differential Privacy

We consider the problem of answering queries about a sensitive dataset subject to differential privacy. The queries may be chosen adversarially from a larger set Q of allowable queries in one of three ways, which we list in order from easiest to hardest to answer: Offline: The queries are chosen all at once and the differentially private mechanism answers the queries in a single batch. Online: The queries are chosen all at once, but the mechanism only receives the queries in a streaming fashion and must answer each query before seeing the next query. Adaptive: The queries are chosen one at a time and the mechanism must answer each query before the next query is chosen. In particular, each query may depend on the answers given to previous queries. Many differentially private mechanisms are just as efficient in the adaptive model as they are in the offline model. Meanwhile, most lower bounds for differential privacy hold in the offline setting. This suggests that the three models may be equivalent. We prove that these models are all, in fact, distinct. Specifically, we show that there is a family of statistical queries such that exponentially more queries from this family can be answered in the offline model than in the online model. We also exhibit a family of search queries such that exponentially more queries from this family can be answered in the online model than in the adaptive model. We also investigate whether such separations might hold for simple queries like threshold queries over the real line.

cs.CR↗

arXiv2016

Space Lower Bounds for Itemset Frequency Sketches

Given a database, computing the fraction of rows that contain a query itemset or determining whether this fraction is above some threshold are fundamental operations in data mining. A uniform sample of rows is a good sketch of the database in the sense that all sufficiently frequent itemsets and their approximate frequencies are recoverable from the sample, and the sketch size is independent of the number of rows in the original database. For many seemingly similar problems there are better sketching algorithms than uniform sampling. In this paper we show that for itemset frequency sketching this is not the case. That is, we prove that there exist classes of databases for which uniform sampling is a space optimal sketch for approximate itemset frequency analysis, up to constant or iterated-logarithmic factors.

cs.DS↗

arXiv2016

Some Pairs Problems

A common form of MapReduce application involves discovering relationships between certain pairs of inputs. Similarity joins serve as a good example of this type of problem, which we call a "some-pairs" problem. In the framework of Afrati et al. (VLDB 2013), algorithms are measured by the tradeoff between reducer size (maximum number of inputs a reducer can handle) and the replication rate (average number of reducers to which an input must be sent. There are two obvious approaches to solving some-pairs problems in general. We show that no general-purpose MapReduce algorithm can beat both of these two algorithms in the worst case. We then explore a recursive algorithm for solving some-pairs problems and heuristics for beating the lower bound on common instances of the some-pairs class of problems.

cs.DB↗

arXiv2015

Robust Mediators in Large Games

A mediator is a mechanism that can only suggest actions to players, as a function of all agents' reported types, in a given game of incomplete information. We study what is achievable by two kinds of mediators, "strong" and "weak." Players can choose to opt-out of using a strong mediator but cannot misrepresent their type if they opt-in. Such a mediator is "strong" because we can view it as having the ability to verify player types. Weak mediators lack this ability--- players are free to misrepresent their type to a weak mediator. We show a striking result---in a prior-free setting, assuming only that the game is large and players have private types, strong mediators can implement approximate equilibria of the complete-information game. If the game is a congestion game, then the same result holds using only weak mediators. Our result follows from a novel application of differential privacy, in particular, a variant we propose called joint differential privacy.

cs.GT↗

arXiv2015

Mechanism Design in Large Games: Incentives and Privacy

We study the problem of implementing equilibria of complete information games in settings of incomplete information, and address this problem using "recommender mechanisms." A recommender mechanism is one that does not have the power to enforce outcomes or to force participation, rather it only has the power to suggestion outcomes on the basis of voluntary participation. We show that despite these restrictions, recommender mechanisms can implement equilibria of complete information games in settings of incomplete information under the condition that the game is large---i.e. that there are a large number of players, and any player's action affects any other's payoff by at most a small amount. Our result follows from a novel application of differential privacy. We show that any algorithm that computes a correlated equilibrium of a complete information game while satisfying a variant of differential privacy---which we call joint differential privacy---can be used as a recommender mechanism while satisfying our desired incentive properties. Our main technical result is an algorithm for computing a correlated equilibrium of a large game while satisfying joint differential privacy. Although our recommender mechanisms are designed to satisfy game-theoretic properties, our solution ends up satisfying a strong privacy property as well. No group of players can learn "much" about the type of any player outside the group from the recommendations of the mechanism, even if these players collude in an arbitrary way. As such, our algorithm is able to implement equilibria of complete information games, without revealing information about the realized types.

cs.GT↗

arXiv2015

Watch and Learn: Optimizing from Revealed Preferences Feedback

A Stackelberg game is played between a leader and a follower. The leader first chooses an action, then the follower plays his best response. The goal of the leader is to pick the action that will maximize his payoff given the follower's best response. In this paper we present an approach to solving for the leader's optimal strategy in certain Stackelberg games where the follower's utility function (and thus the subsequent best response of the follower) is unknown. Stackelberg games capture, for example, the following interaction between a producer and a consumer. The producer chooses the prices of the goods he produces, and then a consumer chooses to buy a utility maximizing bundle of goods. The goal of the seller here is to set prices to maximize his profit---his revenue, minus the production cost of the purchased bundle. It is quite natural that the seller in this example should not know the buyer's utility function. However, he does have access to revealed preference feedback---he can set prices, and then observe the purchased bundle and his own profit. We give algorithms for efficiently solving, in terms of both computational and query complexity, a broad class of Stackelberg games in which the follower's utility function is unknown, using only "revealed preference" access to it. This class includes in particular the profit maximization problem, as well as the optimal tolling problem in nonatomic congestion games, when the latency functions are unknown. Surprisingly, we are able to solve these problems even though the optimization problems are non-convex in the leader's actions.

cs.DS↗

arXiv2015

More General Queries and Less Generalization Error in Adaptive Data Analysis

Adaptivity is an important feature of data analysis---typically the choice of questions asked about a dataset depends on previous interactions with the same dataset. However, generalization error is typically bounded in a non-adaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC '15) and Hardt and Ullman (FOCS '14) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution $\mathcal{P}$ and a set of $n$ independent samples $x$ is drawn from $\mathcal{P}$. We seek an algorithm that, given $x$ as input, "accurately" answers a sequence of adaptively chosen "queries" about the unknown distribution $\mathcal{P}$. How many samples $n$ must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions towards resolving this question: *We give upper bounds on the number of samples $n$ that are needed to answer statistical queries that improve over the bounds of Dwork et al. *We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and the important class of convex risk minimization queries. As in Dwork et al., our algorithms are based on a connection between differential privacy and generalization error, but we feel that our analysis is simpler and more modular, which may be useful for studying these questions in the future.

cs.LG↗

arXiv2015

Algorithmic Stability for Adaptive Data Analysis

Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution $\mathbf{P}$ and a set of $n$ independent samples $\mathbf{x}$ is drawn from $\mathbf{P}$. We seek an algorithm that, given $\mathbf{x}$ as input, accurately answers a sequence of adaptively chosen queries about the unknown distribution $\mathbf{P}$. How many samples $n$ must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions: (i) We give upper bounds on the number of samples $n$ that are needed to answer statistical queries. The bounds improve and simplify the work of Dwork et al. (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015, NIPS, 2015). (ii) We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries. As in Dwork et al., our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stability implies low generalization error. We also study weaker stability guarantees such as bounded KL divergence and total variation distance.

cs.LG↗

arXiv2015

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the $n$-stage repeated version of the game exist if and only if both players have $Ω(n)$ random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the $n$-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on "random-like" sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then $Ω(n)$ random bits remain necessary for equilibria to exist.

cs.GT↗

arXiv2015

Inducing Approximately Optimal Flow Using Truthful Mediators

We revisit a classic coordination problem from the perspective of mechanism design: how can we coordinate a social welfare maximizing flow in a network congestion game with selfish players? The classical approach, which computes tolls as a function of known demands, fails when the demands are unknown to the mechanism designer, and naively eliciting them does not necessarily yield a truthful mechanism. Instead, we introduce a weak mediator that can provide suggested routes to players and set tolls as a function of reported demands. However, players can choose to ignore or misreport their type to this mediator. Using techniques from differential privacy, we show how to design a weak mediator such that it is an asymptotic ex-post Nash equilibrium for all players to truthfully report their types to the mediator and faithfully follow its suggestion, and that when they do, they end up playing a nearly optimal flow. Notably, our solution works in settings of incomplete information even in the absence of a prior distribution on player types. Along the way, we develop new techniques for privately solving convex programs which may be of independent interest.

cs.GT↗

arXiv2015

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.

cs.DS↗

arXiv2015

Interactive Fingerprinting Codes and the Hardness of Preventing False Discovery

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., who showed how to answer $\tildeΩ(n^2)$ queries efficiently, and also by Hardt and Ullman, who showed that answering $\tilde{O}(n^3)$ 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. 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.

cs.CR↗

arXiv2015

Between Pure and Approximate Differential Privacy

We show a new lower bound on the sample complexity of $(\varepsilon, δ)$-differentially private algorithms that accurately answer statistical queries on high-dimensional databases. The novelty of our bound is that it depends optimally on the parameter $δ$, which loosely corresponds to the probability that the algorithm fails to be private, and is the first to smoothly interpolate between approximate differential privacy ($δ> 0$) and pure differential privacy ($δ= 0$). Specifically, we consider a database $D \in \{\pm1\}^{n \times d}$ and its \emph{one-way marginals}, which are the $d$ queries of the form "What fraction of individual records have the $i$-th bit set to $+1$?" We show that in order to answer all of these queries to within error $\pm α$ (on average) while satisfying $(\varepsilon, δ)$-differential privacy, it is necessary that $$ n \geq Ω\left( \frac{\sqrt{d \log(1/δ)}}{α\varepsilon} \right), $$ which is optimal up to constant factors. To prove our lower bound, we build on the connection between \emph{fingerprinting codes} and lower bounds in differential privacy (Bun, Ullman, and Vadhan, STOC'14). In addition to our lower bound, we give new purely and approximately differentially private algorithms for answering arbitrary statistical queries that improve on the sample complexity of the standard Laplace and Gaussian mechanisms for achieving worst-case accuracy guarantees by a logarithmic factor.

cs.DS↗

arXiv2014

An Anti-Folk Theorem for Large Repeated Games with Imperfect Monitoring

We study infinitely repeated games in settings of imperfect monitoring. We first prove a family of theorems that show that when the signals observed by the players satisfy a condition known as $(ε, γ)$-differential privacy, that the folk theorem has little bite: for values of $ε$ and $γ$ sufficiently small, for a fixed discount factor, any equilibrium of the repeated game involve players playing approximate equilibria of the stage game in every period. Next, we argue that in large games ($n$ player games in which unilateral deviations by single players have only a small impact on the utility of other players), many monitoring settings naturally lead to signals that satisfy $(ε,γ)$-differential privacy, for $ε$ and $γ$ tending to zero as the number of players $n$ grows large. We conclude that in such settings, the set of equilibria of the repeated game collapse to the set of equilibria of the stage game.

cs.GT↗

arXiv2014

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.

cs.LG↗

arXiv2014

Privately Solving Linear Programs

In this paper, we initiate the systematic study of solving linear programs under differential privacy. The first step is simply to define the problem: to this end, we introduce several natural classes of private linear programs that capture different ways sensitive data can be incorporated into a linear program. For each class of linear programs we give an efficient, differentially private solver based on the multiplicative weights framework, or we give an impossibility result.

cs.DS↗

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↗

arXiv2013

Answering n^{2+o(1)} Counting Queries with Differential Privacy is Hard

A central problem in differentially private data analysis is how to design efficient algorithms capable of answering large numbers of counting queries on a sensitive database. Counting queries of the form "What fraction of individual records in the database satisfy the property q?" We prove that if one-way functions exist, then there is no algorithm that takes as input a database D in ({0,1}^d)^n, and k = n^{2+o(1)} arbitrary efficiently computable counting queries, runs in time poly(d, n), and returns an approximate answer to each query, while satisfying differential privacy. We also consider the complexity of answering "simple" counting queries, and make some progress in this direction by showing that the above result holds even when we require that the queries are computable by constant depth (AC-0) circuits. Our result is almost tight in the sense that nearly n^2 counting queries can be answered efficiently while satisfying differential privacy. Moreover, super-polynomially many queries can be answered in exponential time. We prove our results by extending the connection between differentially private counting query release and cryptographic traitor-tracing schemes to the setting where the queries are given to the sanitizer as input, and by constructing a traitor-tracing scheme that is secure in this setting.

cs.CR↗