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 .
@cite_15 @cite_4
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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.