Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performances, these methods are supported by a well-developed theory. Kernel methods use an implicit mapping of the input data into a high dimensional feature space defined by a kernel function, i.e., a function returning the inner product between the images of two data points in the feature space. Central to any kernel method is the kernel matrix, which is built by evaluating the kernel function on a given sample dataset. In this paper, we initiate the study of non-asymptotic spectral theory of random kernel matrices. These are n x n random matrices whose (i,j)th entry is obtained by evaluating the kernel function on @math and @math , where @math are a set of n independent random high-dimensional vectors. Our main contribution is to obtain tight upper bounds on the spectral norm (largest eigenvalue) of random kernel matrices constructed by commonly used kernel functions based on polynomials and Gaussian radial basis. As an application of these results, we provide lower bounds on the distortion needed for releasing the coefficients of kernel ridge regression under attribute privacy, a general privacy notion which captures a large class of privacy definitions. Kernel ridge regression is standard method for performing non-parametric regression that regularly outperforms traditional regression approaches in various domains. Our privacy distortion lower bounds are the first for any kernel technique, and our analysis assumes realistic scenarios for the input, unlike all previous lower bounds for other release problems which only hold under very restrictive input settings.
The goal of is to release global, statistical properties of a database while protecting the privacy of the individuals whose information the database contains. Differential privacy @cite_4 is a formal notion of privacy tailored to private data analysis. Differential privacy requires, roughly, that any single individual's data have little effect on the outcome of the analysis. A lot of recent research has gone in developing differentially private algorithms for various applications, including kernel methods @cite_15 . A typical objective here is to release as accurate an approximation as possible to some function @math evaluated on a database @math .
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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. 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 . One goal of statistical privacy research is to construct a data release mechanism that protects individual privacy while preserving information content. An example is a random mechanism that takes an input database X and outputs a random database Z according to a distribution Qn(⋅|X). Differential privacy is a particular privacy requirement developed by computer scientists in which Qn(⋅|X) is required to be insensitive to changes in one data point in X. This makes it difficult to infer from Z whether a given individual is in the original database X. We consider differential privacy from a statistical perspective. We consider several data-release mechanisms that satisfy the differential privacy requirement. We show that it is useful to compare these schemes by computing the rate of convergence of distributions and densities constructed from the released data. We study a general privacy method, called the exponential mechanism, introduced by McSherry and Talwar (2007). We show that the accuracy of this meth... We introduce a new, generic framework for private data analysis.The goal of private data analysis is to release aggregate information about a data set while protecting the privacy of the individuals whose information the data set contains.Our framework allows one to release functions f of the data withinstance-based additive noise. That is, the noise magnitude is determined not only by the function we want to release, but also bythe database itself. One of the challenges is to ensure that the noise magnitude does not leak information about the database. To address that, we calibrate the noise magnitude to the smoothsensitivity of f on the database x --- a measure of variabilityof f in the neighborhood of the instance x. The new frameworkgreatly expands the applicability of output perturbation, a technique for protecting individuals' privacy by adding a smallamount of random noise to the released statistics. To our knowledge, this is the first formal analysis of the effect of instance-basednoise in the context of data privacy. Our framework raises many interesting algorithmic questions. Namely,to apply the framework one must compute or approximate the smoothsensitivity of f on x. We show how to do this efficiently for several different functions, including the median and the cost ofthe minimum spanning tree. We also give a generic procedure based on sampling that allows one to release f(x) accurately on manydatabases x. This procedure is applicable even when no efficient algorithm for approximating smooth sensitivity of f is known orwhen f is given as a black box. We illustrate the procedure by applying it to k-SED (k-means) clustering and learning mixtures of Gaussians. Existing differential privacy (DP) studies mainly consider aggregation on data sets where each entry corresponds to a particular participant to be protected. In many situations, a user may pose a relational algebra query on a database with sensitive data, and desire differentially private aggregation on the result of the query. However, no existing work is able to release such aggregation when the query contains unrestricted join operations. This severely limits the applications of existing DP techniques because many data analysis tasks require unrestricted joins. One example is subgraph counting on a graph. Furthermore, existing methods for differentially private subgraph counting support only edge DP and are subject to very simple subgraphs. Until recent, whether any nontrivial graph statistics can be released with reasonable accuracy for arbitrary kind of input graphs under node DP was still an open problem. In this paper, we propose a novel differentially private mechanism that supports unrestricted joins, to release an approximation of a linear statistic of the result of some positive relational algebra calculation over a sensitive database. The error bound of the approximate answer is roughly proportional to the empirical sensitivity of the query --- a new notion that measures the maximum possible change to the query answer when a participant withdraws its data from the sensitive database. For subgraph counting, our mechanism provides a solution to achieve node DP, for any kind of subgraphs. Differential privacy is a strong notion for protecting individual privacy in privacy preserving data analysis or publishing. In this paper, we study the problem of differentially private histogram release based on an interactive differential privacy interface. We propose two multidimensional partitioning strategies including a baseline cell-based partitioning and an innovative kd-tree based partitioning. In addition to providing formal proofs for differential privacy and usefulness guarantees for linear distributive queries, we also present a set of experimental results and demonstrate the feasibility and performance of our method. Differential privacy has emerged as an important standard for privacy preserving computation over databases containing sensitive information about individuals. Research on differential privacy spanning a number of research areas, including theory, security, database, networks, machine learning, and statistics, over the last decade has resulted in a variety of privacy preserving algorithms for a number of analysis tasks. Despite maturing research efforts, the adoption of differential privacy by practitioners in industry, academia, or government agencies has so far been rare. Hence, in this tutorial, we will first describe the foundations of differentially private algorithm design that cover the state of the art in private computation on tabular data. In the second half of the tutorial we will highlight real world applications on complex data types, and identify research challenges in applying differential privacy to real world applications. The problem of privacy-preserving data analysis has a long history spanning multiple disciplines. As electronic data about individuals becomes increasingly detailed, and as technology enables ever more powerful collection and curation of these data, the need increases for a robust, meaningful, and mathematically rigorous definition of privacy, together with a computationally rich class of algorithms that satisfy this definition. Differential Privacy is such a definition.After motivating and discussing the meaning of differential privacy, the preponderance of this monograph is devoted to fundamental techniques for achieving differential privacy, and application of these techniques in creative combinations, using the query-release problem as an ongoing example. A key point is that, by rethinking the computational goal, one can often obtain far better results than would be achieved by methodically replacing each step of a non-private computation with a differentially private implementation. Despite some astonishingly powerful computational results, there are still fundamental limitations — not just on what can be achieved with differential privacy but on what can be achieved with any method that protects against a complete breakdown in privacy. Virtually all the algorithms discussed herein maintain differential privacy against adversaries of arbitrary computational power. Certain algorithms are computationally intensive, others are efficient. Computational complexity for the adversary and the algorithm are both discussed.We then turn from fundamentals to applications other than queryrelease, discussing differentially private methods for mechanism design and machine learning. The vast majority of the literature on differentially private algorithms considers a single, static, database that is subject to many analyses. Differential privacy in other models, including distributed databases and computations on data streams is discussed.Finally, we note that this work is meant as a thorough introduction to the problems and techniques of differential privacy, but is not intended to be an exhaustive survey — there is by now a vast amount of work in differential privacy, and we can cover only a small portion of it. In distributed optimization and iterative consensus literature, a standard problem is for @math agents to minimize a function @math over a subset of Euclidean space, where the cost function is expressed as a sum @math . In this paper, we study the private distributed optimization (PDOP) problem with the additional requirement that the cost function of the individual agents should remain differentially private. The adversary attempts to infer information about the private cost functions from the messages that the agents exchange. Achieving differential privacy requires that any change of an individual's cost function only results in unsubstantial changes in the statistics of the messages. We propose a class of iterative algorithms for solving PDOP, which achieves differential privacy and convergence to the optimal value. Our analysis reveals the dependence of the achieved accuracy and the privacy levels on the the parameters of the algorithm. We observe that to achieve @math -differential privacy the accuracy of the algorithm has the order of @math . Differential privacy is a powerful tool for providing privacy-preserving noisy query answers over statistical databases. It guarantees that the distribution of noisy query answers changes very little with the addition or deletion of any tuple. It is frequently accompanied by popularized claims that it provides privacy without any assumptions about the data and that it protects against attackers who know all but one record. In this paper we critically analyze the privacy protections offered by differential privacy. First, we use a no-free-lunch theorem, which defines non-privacy as a game, to argue that it is not possible to provide privacy and utility without making assumptions about how the data are generated. Then we explain where assumptions are needed. We argue that privacy of an individual is preserved when it is possible to limit the inference of an attacker about the participation of the individual in the data generating process. This is different from limiting the inference about the presence of a tuple (for example, Bob's participation in a social network may cause edges to form between pairs of his friends, so that it affects more than just the tuple labeled as "Bob"). The definition of evidence of participation, in turn, depends on how the data are generated -- this is how assumptions enter the picture. We explain these ideas using examples from social network research as well as tabular data for which deterministic statistics have been previously released. In both cases the notion of participation varies, the use of differential privacy can lead to privacy breaches, and differential privacy does not always adequately limit inference about participation. The definition of differential privacy has recently emerged as a leading standard of privacy guarantees for algorithms on statistical databases. We offer several relaxations of the definition which require privacy guarantees to hold only against efficient--i.e., computationally-bounded--adversaries. We establish various relationships among these notions, and in doing so, we observe their close connection with the theory of pseudodense sets by [1]. We extend the dense model theorem of to demonstrate equivalence between two definitions (indistinguishability- and simulatability-based) of computational differential privacy. Our computational analogues of differential privacy seem to allow for more accurate constructions than the standard information-theoretic analogues. In particular, in the context of private approximation of the distance between two vectors, we present a differentially-private protocol for computing the approximation, and contrast it with a substantially more accurate protocol that is only computationally differentially private. Over the past five years a new approach to privacy-preserving data analysis has born fruit [13, 18, 7, 19, 5, 37, 35, 8, 32]. This approach differs from much (but not all!) of the related literature in the statistics, databases, theory, and cryptography communities, in that a formal and ad omnia privacy guarantee is defined, and the data analysis techniques presented are rigorously proved to satisfy the guarantee. The key privacy guarantee that has emerged is differential privacy. Roughly speaking, this ensures that (almost, and quantifiably) no risk is incurred by joining a statistical database. In this survey, we recall the definition of differential privacy and two basic techniques for achieving it. We then show some interesting applications of these techniques, presenting algorithms for three specific tasks and three general results on differentially private learning. In this work we analyze the sample complexity of classification by differentially private algorithms. Differential privacy is a strong and well-studied notion of privacy introduced by (2006) that ensures that the output of an algorithm leaks little information about the data point provided by any of the participating individuals. Sample complexity of private PAC and agnostic learning was studied in a number of prior works starting with (, 2008) but a number of basic questions still remain open, most notably whether learning with privacy requires more samples than learning without privacy. We show that the sample complexity of learning with (pure) differential privacy can be arbitrarily higher than the sample complexity of learning without the privacy constraint or the sample complexity of learning with approximate differential privacy. Our second contribution and the main tool is an equivalence between the sample complexity of (pure) differentially private learning of a concept class @math (or @math ) and the randomized one-way communication complexity of the evaluation problem for concepts from @math . Using this equivalence we prove the following bounds: 1. @math , where @math is the Littlestone's (1987) dimension characterizing the number of mistakes in the online-mistake-bound learning model. Known bounds on @math then imply that @math can be much higher than the VC-dimension of @math . 2. For any @math , there exists a class @math such that @math but @math . 3. For any @math , there exists a class @math such that the sample complexity of (pure) @math -differentially private PAC learning is @math but the sample complexity of the relaxed @math -differentially private PAC learning is @math . This resolves an open problem of (2013b). 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. In large-scale statistical learning, data collection and model fitting are moving increasingly toward peripheral devices---phones, watches, fitness trackers---away from centralized data collection. Concomitant with this rise in decentralized data are increasing challenges of maintaining privacy while allowing enough information to fit accurate, useful statistical models. This motivates local notions of privacy---most significantly, local differential privacy, which provides strong protections against sensitive data disclosures---where data is obfuscated before a statistician or learner can even observe it, providing strong protections to individuals' data. Yet local privacy as traditionally employed may prove too stringent for practical use, especially in modern high-dimensional statistical and machine learning problems. Consequently, we revisit the types of disclosures and adversaries against which we provide protections, considering adversaries with limited prior information and ensuring that with high probability, ensuring they cannot reconstruct an individual's data within useful tolerances. By reconceptualizing these protections, we allow more useful data release---large privacy parameters in local differential privacy---and we design new (minimax) optimal locally differentially private mechanisms for statistical learning problems for privacy levels. We thus present practicable approaches to large-scale locally private model training that were previously impossible, showing theoretically and empirically that we can fit large-scale image classification and language models with little degradation in utility. 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... 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. Learning problems form an important category of computational tasks that generalizes many of the computations researchers apply to large real-life data sets. We ask: what concept classes can be learned privately, namely, by an algorithm whose output does not depend too heavily on any one input or specific training example? More precisely, we investigate learning algorithms that satisfy differential privacy, a notion that provides strong confidentiality guarantees in the contexts where aggregate information is released about a database containing sensitive information about individuals. We present several basic results that demonstrate general feasibility of private learning and relate several models previously studied separately in the contexts of privacy and standard learning. 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. Private regression has received attention from both database and security communities. Recent work by (USENIX Security 2014) analyzed the functional mechanism ( VLDB 2012) for training linear regression models over medical data. Unfortunately, they found that model accuracy is already unacceptable with differential privacy when @math . We address this issue, presenting an explicit connection between differential privacy and stable learning theory through which a substantially better privacy utility tradeoff can be obtained. Perhaps more importantly, our theory reveals that the most basic mechanism in differential privacy, output perturbation, can be used to obtain a better tradeoff for all convex-Lipschitz-bounded learning tasks. Since output perturbation is simple to implement, it means that our approach is potentially widely applicable in practice. We go on to apply it on the same medical data as used by Encouragingly, we achieve accurate models even for @math . In the last part of this paper, we study the impact of our improved differentially private mechanisms on model inversion attacks, a privacy attack introduced by We observe that the improved tradeoff makes the resulting differentially private model more susceptible to inversion attacks. We analyze this phenomenon formally. Differential privacy offers a way to answer queries about sensitive information while providing strong, provable privacy guarantees, ensuring that the presence or absence of a single individual in the database has a negligible statistical effect on the query's result. Proving that a given query has this property involves establishing a bound on the query's sensitivity---how much its result can change when a single record is added or removed. A variety of tools have been developed for certifying that a given query differentially private. In one approach, Reed and Pierce [34] proposed a functional programming language, Fuzz, for writing differentially private queries. Fuzz uses linear types to track sensitivity and a probability monad to express randomized computation; it guarantees that any program with a certain type is differentially private. Fuzz can successfully verify many useful queries. However, it fails when the sensitivity analysis depends on values that are not known statically. We present DFuzz, an extension of Fuzz with a combination of linear indexed types and lightweight dependent types. This combination allows a richer sensitivity analysis that is able to certify a larger class of queries as differentially private, including ones whose sensitivity depends on runtime information. As in Fuzz, the differential privacy guarantee follows directly from the soundness theorem of the type system. We demonstrate the enhanced expressivity of DFuzz by certifying differential privacy for a broad class of iterative algorithms that could not be typed previously.