Differential privacy is a notion that has emerged in the community of statistical databases, as a response to the problem of protecting the privacy of the database's participants when performing statistical queries. The idea is that a randomized query satisfies differential privacy if the likelihood of obtaining a certain answer for a database x is not too different from the likelihood of obtaining the same answer on adjacent databases, i.e. databases which differ from x for only one individual. Information flow is an area of Security concerned with the problem of controlling the leakage of confidential information in programs and protocols. Nowadays, one of the most established approaches to quantify and to reason about leakage is based on the Renyi min entropy version of information theory. In this paper, we analyze critically the notion of differential privacy in light of the conceptual framework provided by the Renyi min information theory. We show that there is a close relation between differential privacy and leakage, due to the graph symmetries induced by the adjacency relation. Furthermore, we consider the utility of the randomized answer, which measures its expected degree of accuracy. We focus on certain kinds of utility functions called "binary", which have a close correspondence with the Renyi min mutual information. Again, it turns out that there can be a tight correspondence between differential privacy and utility, depending on the symmetries induced by the adjacency relation and by the query. Depending on these symmetries we can also build an optimal-utility randomization mechanism while preserving the required level of differential privacy. Our main contribution is a study of the kind of structures that can be induced by the adjacency relation and the query, and how to use them to derive bounds on the leakage and achieve the optimal utility.
In @cite_3 the authors aim at obtaining optimal-utility randomization mechanisms while preserving differential privacy. The authors propose adding noise to the output of the query according to the geometric mechanism. Their framework is very interesting in the sense it provides a general definition of utility for a mechanism @math that captures any possible side information and preference (defined as a loss function) the users of @math may have. They prove that the geometric mechanism is optimal in the particular case of counting queries. Our results in Section do not restrict to counting queries, but on the other hand we only consider the case of binary loss function.
@cite_3
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A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism M* -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user u, no matter what its side information and preferences, derives as much utility from M* as from interacting with a differentially private mechanism Mu that is optimally tailored to u. More precisely, for every user u there is an optimal mechanism Mu for it that factors into a user-independent part (the geometric mechanism M*) followed by user-specific post-processing that can be delegated to the user itself. The first part of our proof of this result characterizes the optimal differentially private mechanism for a fixed but arbitrary user in terms of a certain basic feasible solution to a linear program with constraints that encode differential privacy. The second part shows that all of the relevant vertices of this polytope (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range. A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether a given database row is included. The goal of this paper is to formulate and provide strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a symmetric and monotone loss function). Our main result is the following: for each fixed count query and differential privacy level, there is a geometric mechanism @math ---a discrete variant of the simple and well-studi... A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Privacy can be rigorously quantified using the framework of differential privacy , which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism @math -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user @math , no matter what its side information and preferences, derives as much utility from @math as from interacting with a differentially private mechanism @math that is optimally tailored to @math . Differential privacy mechanism design has traditionally been tailored for a scalar-valued query function. Although many mechanisms such as the Laplace and Gaussian mechanisms can be extended to a matrix-valued query function by adding i.i.d. noise to each element of the matrix, this method is often suboptimal as it forfeits an opportunity to exploit the structural characteristics typically associated with matrix analysis. To address this challenge, we propose a novel differential privacy mechanism called the Matrix-Variate Gaussian (MVG) mechanism, which adds a matrix-valued noise drawn from a matrix-variate Gaussian distribution, and we rigorously prove that the MVG mechanism preserves (e,δ)-differential privacy. Furthermore, we introduce the concept of directional noise made possible by the design of the MVG mechanism. Directional noise allows the impact of the noise on the utility of the matrix-valued query function to be moderated. Finally, we experimentally demonstrate the performance of our mechanism using three matrix-valued queries on three privacy-sensitive datasets. We find that the MVG mechanism can notably outperforms four previous state-of-the-art approaches, and provides comparable utility to the non-private baseline. Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where both the data providers and data analysts want to maximize the utility of statistical analyses performed on the released data, we study the fundamental trade-off between local differential privacy and utility. This trade-off is formulated as a constrained optimization problem: maximize utility subject to local differential privacy constraints. We introduce a combinatorial family of extremal privatization mechanisms, which we call staircase mechanisms, and show that it contains the optimal privatization mechanisms for a broad class of information theoretic utilities such as mutual information and @math -divergences. We further prove that for any utility function and any privacy level, solving the privacy-utility maximization problem is equivalent to solving a finite-dimensional linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the size of the alphabet the data lives in. To account for this, we show that two simple privatization mechanisms, the binary and randomized response mechanisms, are universally optimal in the low and high privacy regimes, and well approximate the intermediate regime. Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where both the data providers and data analysts want to maximize the utility of statistical analyses performed on the released data, we study the fundamental trade-off between local differential privacy and utility. This trade-off is formulated as a constrained optimization problem: maximize utility subject to local differential privacy constraints. We introduce a combinatorial family of extremal privatization mechanisms, which we call staircase mechanisms, and show that it contains the optimal privatization mechanisms for a broad class of information theoretic utilities such as mutual information and f-divergences. We further prove that for any utility function and any privacy level, solving the privacy-utility maximization problem is equivalent to solving a finite-dimensional linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the size of the alphabet the data lives in. To account for this, we show that two simple privatization mechanisms, the binary and randomized response mechanisms, are universally optimal in the low and high privacy regimes, and well approximate the intermediate regime. Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where the data providers and data analysts want to maximize the utility of statistical inferences performed on the released data, we study the fundamental tradeoff between local differential privacy and information theoretic utility functions. We introduce a family of extremal privatization mechanisms, which we call staircase mechanisms, and prove that it contains the optimal privatization mechanism that maximizes utility. We further show that for all information theoretic utility functions studied in this paper, maximizing utility is equivalent to solving a linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the data size. To account for this, we show that two simple staircase mechanisms, the binary and randomized response mechanisms, are universally optimal in the high and low privacy regimes, respectively, and well approximate the intermediate regime. We study binary classification in the presence of class-conditional random noise, where the learner gets to see labels that are flipped independently with some probability, and where the flip probability depends on the class. Our goal is to devise learning algorithms that are efficient and statistically consistent with respect to commonly used utility measures. In particular, we look at a family of measures motivated by their application in domains where cost-sensitive learning is necessary (for example, when there is class imbalance). In contrast to most of the existing literature on consistent classification that are limited to the classical 0-1 loss, our analysis includes more general utility measures such as the AM measure (arithmetic mean of True Positive Rate and True Negative Rate). For this problem of cost-sensitive learning under class-conditional random noise, we develop two approaches that are based on suitably modifying surrogate losses. First, we provide a simple unbiased estimator of any loss, and obtain performance bounds for empirical utility maximization in the presence of i.i.d. data with noisy labels. If the loss function satis_es a simple symmetry condition, we show that using unbiased estimator leads to an efficient algorithm for empirical maximization. Second, by leveraging a reduction of risk minimization under noisy labels to classification with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain strong utility bounds. This approach implies that methods already used in practice, such as biased SVM and weighted logistic regression, are provably noise-tolerant. For two practically important measures in our family, we show that the proposed methods are competitive with respect to recently proposed methods for dealing with label noise in several benchmark data sets. In a companion paper we described what intuitively would seem to be the most general possible way to generate Coarsening at Random mechanisms, a sequential procedure called randomized monotone coarsening. Counter-examples showed that CAR mechanisms exist which cannot be represented in this way. Here, we further develop these results in two directions. Firstly, we consider what happens when data is coarsened at random in two or more phases. We show that the resulting coarsening mechanism is not CAR anymore, but under suitable assumptions is identified and can provide interesting alternative analysis of data under a non-CAR model. Secondly, we look at sequential mechanisms for generating MAR data, missing components of a multivariate random vector. Randomised monotone missingness schemes, in which one variable at a time is observed and depending on its value, another variable is chosen or the procedure is terminated, supply in our opinion the broadest class of physically interpretable MAR mechanisms. We show that every randomised monotone missingness scheme can be represented by a Markov monotone missingness scheme, in which the choice of which variable to observe next only depends on the set of previously observed variables and their values, not on the sequence in which they were measured. We also show that MAR mechanisms exist which cannot be represented sequentially. We consider statistical data analysis in the interactive setting. In this setting a trusted curator maintains a database of sensitive information about individual participants, and releases privacy-preserving answers to queries as they arrive. Our primary contribution is a new differentially private multiplicative weights mechanism for answering a large number of interactive counting (or linear) queries that arrive online and may be adaptively chosen. This is the first mechanism with worst-case accuracy guarantees that can answer large numbers of interactive queries and is efficient (in terms of the runtime's dependence on the data universe size). The error is asymptotically in its dependence on the number of participants, and depends only logarithmically on the number of queries being answered. The running time is nearly linear in the size of the data universe. As a further contribution, when we relax the utility requirement and require accuracy only for databases drawn from a rich class of databases, we obtain exponential improvements in running time. Even in this relaxed setting we continue to guarantee privacy for any input database. Only the utility requirement is relaxed. Specifically, we show that when the input database is drawn from a smooth distribution — a distribution that does not place too much weight on any single data item — accuracy remains as above, and the running time becomes poly-logarithmic in the data universe size. The main technical contributions are the application of multiplicative weights techniques to the differential privacy setting, a new privacy analysis for the interactive setting, and a technique for reducing data dimensionality for databases drawn from smooth distributions. We study a central problem in Algorithmic Mechanism Design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a non-trivial approximation ratio of @math [STOC'06], where @math is the number of items. This was subsequently improved to @math [Dobzinski, APPROX'07] and then to @math [Krysta and Vocking, ICALP'12]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of @math . Similarly to previous constructions, our mechanism uses polynomially many value and demand queries, and in fact provides the same approximation ratio for the larger class of XOS (a.k.a. fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although in general computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive. We design an expected polynomial time, truthful in expectation, (1-1 e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions and matroid weighted-rank functions. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P ≠ NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1 e)-approximation of the optimal welfare. This paper investigates the relation between three different notions of privacy: identifiability, differential privacy, and mutual-information privacy. Under a unified privacy-distortion framework, where the distortion is defined to be the expected Hamming distance between the input and output databases, we establish some fundamental connections between these three privacy notions. Given a maximum allowable distortion @math , we define the privacy-distortion functions @math , @math , and @math to be the smallest (most private best) identifiability level, differential privacy level, and mutual information between the input and the output, respectively. We characterize @math and @math , and prove that @math for @math within certain range, where @math is a constant determined by the prior distribution of the original database @math , and diminishes to zero when @math is uniformly distributed. Furthermore, we show that @math and @math can be achieved by the same mechanism for @math within certain range, i.e., there is a mechanism that simultaneously minimizes the identifiability level and achieves the best mutual-information privacy. Based on these two connections, we prove that this mutual-information optimal mechanism satisfies @math -differential privacy with @math . The results in this paper reveal some consistency between two worst case notions of privacy, namely, identifiability and differential privacy, and an average notion of privacy, mutual-information privacy. We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions. Our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least half that of the value of the fractional solution. This approximation ratio of 1 2 improves over an Ω(1 log m) ratio of Dobzinski, Nisan and Schapira [STOC 2005]. We also show an approximation ratio of 1 - 1 e when utility functions are fractionally subadditive. A result similar to this last result was previously obtained by Dobzinski and Schapira [Soda 2006], but via a different rounding technique that requires the use of a so called "XOS oracle".The randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation, but have no access to the actual utility functions of the players. This allows us to suggest new incentive compatible mechanisms for combinatorial auctions, extending previous work of Lavi and Swamy [FOCS 2005]. We study mechanism design problems in the ordinal setting wherein the preferences of agents are described by orderings over outcomes, as opposed to specific numerical values associated with them. This setting is relevant when agents can compare outcomes, but aren't able to evaluate precise utilities for them. Such a situation arises in diverse contexts including voting and matching markets. Our paper addresses two issues that arise in ordinal mechanism design. To design social welfare maximizing mechanisms, one needs to be able to quantitatively measure the welfare of an outcome which is not clear in the ordinal setting. Second, since the impossibility results of Gibbard and Satterthwaite Gibbard73,Satterthwaite75 force one to move to randomized mechanisms, one needs a more nuanced notion of truthfulness. We propose rank approximation as a metric for measuring the quality of an outcome, which allows us to evaluate mechanisms based on worst-case performance, and lex-truthfulness as a notion of truthfulness for randomized ordinal mechanisms. Lex-truthfulness is stronger than notions studied in the literature, and yet flexible enough to admit a rich class of mechanisms circumventing classical impossibility results . We demonstrate the usefulness of the above notions by devising lex-truthful mechanisms achieving good rank-approximation factors, both in the general ordinal setting, as well as structured settings such as (one-sided) matching markets , and its generalizations, matroid and scheduling markets. We study the design of truthful mechanisms that do not use payments for the generalized assignment problem (GAP) and its variants. An instance of the GAP consists of a bipartite graph with jobs on one side and machines on the other. Machines have capacities and edges have values and sizes; the goal is to construct a welfare maximizing feasible assignment. In our model of private valuations, motivated by impossibility results, the value and sizes on all job-machine pairs are public information; however, whether an edge exists or not in the bipartite graph is a job's private information. That is, the selfish agents in our model are the jobs, and their private information is their edge set. We want to design mechanisms that are truthful without money (henceforth strategyproof), and produce assignments whose welfare is a good approximation to the optimal omniscient welfare. We study several variants of the GAP starting with matching. For the unweighted version, we give an optimal strategyproof mechanism. For maximum weight bipartite matching, we show that no strategyproof mechanism, deterministic or randomized, can be optimal, and present a 2-approximate strategyproof mechanism along with a matching lowerbound. Next we study knapsack-like problems, which, unlike matching, are NP-hard. For these problems, we develop a general LP-based technique that extends the ideas of Lavi and Swamy [14] to reduce designing a truthful approximate mechanism without money to designing such a mechanism for the fractional version of the problem. We design strategyproof approximate mechanisms for the fractional relaxations of multiple knapsack, size-invariant GAP, and value-invariant GAP, and use this technique to obtain, respectively, 2, 4 and 4-approximate strategyproof mechanisms for these problems. We then design an O(log n)-approximate strategyproof mechanism for the GAP by reducing, with logarithmic loss in the approximation, to our solution for the value-invariant GAP. Our technique may be of independent interest for designing truthful mechanisms without money for other LP-based problems. This paper discusses two advancements in the theory of designing truthful randomized mechanisms. Our first contribution is a new framework for developing truthful randomized mechanisms. The framework enables the construction of mechanisms with polynomially small failure probability. This is in contrast to previous mechanisms that fail with constant probability. Another appealing feature of the new framework is that bidding truthfully is a stronglydominant strategy. The power of the framework is demonstrated by an @math -mechanism for combinatorial auctions that succeeds with probability @math . The other major result of this paper is an O(logmloglogm) randomized truthful mechanism for combinatorial auction with subadditivebidders. The best previously-known truthful mechanism for this setting guaranteed an approximation ratio of @math . En route, the new mechanism also provides the best approximation ratio for combinatorial auctions with submodularbidders currently achieved by truthful mechanisms. We study a mechanism design problem of privacy- preserving data collection with privacy protection uncertainty. A data collector wants to collect enough data to perform a certain computation that benefits the individuals who contribute the data, with the possibility of individual privacy leakage. The data collector adopts a privacy-preserving mechanism by adding some random noise to the computation result, which reduces the accuracy of the computation. Individuals decide whether to contribute data based on the potential benefit and the possible privacy cost induced by the mechanism. Due to the intrinsic uncertainty involved in privacy protection, we model individuals' privacy-aware participation using the prospect theory, which more accurately models individuals' behavior under uncertainty than the traditional expected utility theory. We show that the data collector's utility maximization problem involves a polynomial of high and fractional order, which is difficult to solve analytically. We get around this issue by proposing an approximation method, which allows us to obtain a closed form unique solution of the data collector's decision problem. We numerically show that the approximation error is small when the number of individuals is large. By comparing with the results under the expected utility theory, we conclude that a data collector who considers the more realistic prospect theory modeling should adopt a stricter privacy-preserving mechanism to boost her utility. We define a new interactive differentially private mechanism --- the median mechanism --- for answering arbitrary predicate queries that arrive online. Given fixed accuracy and privacy constraints, this mechanism can answer exponentially more queries than the previously best known interactive privacy mechanism (the Laplace mechanism, which independently perturbs each query result). With respect to the number of queries, our guarantee is close to the best possible, even for non-interactive privacy mechanisms. Conceptually, the median mechanism is the first privacy mechanism capable of identifying and exploiting correlations among queries in an interactive setting. We also give an efficient implementation of the median mechanism, with running time polynomial in the number of queries, the database size, and the domain size. This efficient implementation guarantees privacy for all input databases, and accurate query results for almost all input distributions. The dependence of the privacy on the number of queries in this mechanism improves over that of the best previously known efficient mechanism by a super-polynomial factor, even in the non-interactive setting. This paper studies how to enforce differential privacy by using the randomized response in the data collection scenario. Given a client’s value, the randomized algorithm executed by the client reports to the untrusted server a perturbed value. The use of randomized response in surveys enables easy estimations of accurate population statistics while preserving the privacy of the individual respondents. We compare the randomized response with the standard Laplace mechanism which is based on query-output independent adding of Laplace noise. Our research starts from the simple case with one single binary attribute and extends to the general case with multiple polychotomous attributes. We measure utility preservation in terms of the mean squared error of the estimate for various calculations including individual value estimate, proportion estimate, and various derived statistics. We theoretically derive the explicit formula of the mean squared error of various derived statistics based on the randomized response theory and prove the randomized response outperforms the Laplace mechanism. We evaluate our algorithms on YesiWell database including sensitive biomarker data and social network relationships of patients. Empirical evaluation results show effectiveness of our proposed techniques. Especially the use of the randomized response for collecting data incurs fewer utility loss than the output perturbation when the sensitivity of functions is high.