/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

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.

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The Algorithmic Foundations of Differential Privacy

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Convex bodies: the brunn–minkowski theory

Randomized Aggregatable Privacy-Preserving Ordinal Response, or RAPPOR, is a technology for crowdsourcing statistics from end-user client software, anonymously, with strong privacy guarantees. In short, RAPPORs allow the forest of client data to be studied, without permitting the possibility of looking at individual trees. By applying randomized response in a novel manner, RAPPOR provides the mechanisms for such collection as well as for efficient, high-utility analysis of the collected data. In particular, RAPPOR permits statistics to be collected on the population of client-side strings with strong privacy guarantees for each client, and without linkability of their reports. This paper describes and motivates RAPPOR, details its differential-privacy and utility guarantees, discusses its practical deployment and properties in the face of different attack models, and, finally, gives results of its application to both synthetic and real-world data.

RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response

We study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries and has been the focus of a long line of work. For a given set of d linear queries over a database x ∈ RN, we seek to find the differentially private mechanism that has the minimum mean squared error. For pure differential privacy, [5, 32] give an O(log2 d) approximation to the optimal mechanism. Our first contribution is to give an efficient O(log2 d) approximation guarantee for the case of (e,δ)-differential privacy. Our mechanism adds carefully chosen correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of [44], using tools from convex geometry. We next consider the sparse case when the number of queries exceeds the number of individuals in the database, i.e. when d > n Δ |x|1. The lower bounds used in the previous approximation algorithm no longer apply --- in fact better mechanisms are known in this setting [7, 27, 28, 31, 49]. Our second main contribution is to give an efficient (e,δ)-differentially private mechanism that, for any given query set A and an upper bound n on |x|1, has mean squared error within polylog(d,N) of the optimal for A and n. This approximation is achieved by coupling the Gaussian noise addition approach with linear regression over the l1 ball. Additionally, we show a similar polylogarithmic approximation guarantee for the optimal e-differentially private mechanism in this sparse setting. Our work also shows that for arbitrary counting queries, i.e. A with entries in {0,1}, there is an e-differentially private mechanism with expected error ~O(√n) per query, improving on the ~O(n2/3) bound of [7] and matching the lower bound implied by [15] up to logarithmic factors.The connection between the hereditary discrepancy and the privacy mechanism enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix A.

The geometry of differential privacy: the sparse and approximate cases

We give algorithms for geometric graph problems in the modern parallel models inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes a $(1+\epsilon)$-approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms). In contrast, for general graphs, achieving the same result for MST (or even connectivity) remains a challenging open problem, despite drawing significant attention in recent years. 
We develop a general algorithmic framework that, besides MST, also applies to Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic framework has implications beyond the MapReduce model. For example it yields a new algorithm for computing EMD cost in the plane in near-linear time, $n^{1+o_\epsilon(1)}$. We note that while recently Sharathkumar and Agarwal developed a near-linear time algorithm for $(1+\epsilon)$-approximating EMD, our algorithm is fundamentally different, and, for example, also solves the transportation (cost) problem, raised as an open question in their work. Furthermore, our algorithm immediately gives a $(1+\epsilon)$-approximation algorithm with $n^{\delta}$ space in the streaming-with-sorting model with $1/\delta^{O(1)}$ passes. As such, it is tempting to conjecture that the parallel models may also constitute a concrete playground in the quest for efficient algorithms for EMD (and other similar problems) in the vanilla streaming model, a well-known open problem.

Parallel Algorithms for Geometric Graph Problems

We give algorithms for geometric graph problems in the modern parallel models such as MapReduce. For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes a (1 + e)-approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms). In contrast, for general graphs, achieving the same result for MST (or even connectivity) remains a challenging open problem [9], despite drawing significant attention in recent years. We develop a general algorithmic framework that, besides MST, also applies to Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic framework has implications beyond the MapReduce model. For example it yields a new algorithm for computing EMD cost in the plane in near-linear time, n1+oe(1). We note that while recently [33] have developed a near-linear time algorithm for (1 + e)-approximating EMD, our algorithm is fundamentally different, and, for example, also solves the transportation (cost) problem, raised as an open question in [33]. Furthermore, our algorithm immediately gives a (1+ e)-approximation algorithm with nδ space in the streaming-with-sorting model with 1/δO(1) passes. As such, it is tempting to conjecture that the parallel models may also constitute a concrete playground in the quest for efficient algorithms for EMD (and other similar problems) in the vanilla streaming model, a well-known open problem.

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Parallel algorithms for geometric graph problems

Described herein is a method and system for providing privacy guarantees with an improved privacy-accuracy trade-off Dynamic data can be accessed from a database A sum model is selected from window sum, exponential decay sum, and polynomial decay sum An algorithm is initiated that produces polylogarithmic bounded error in the range of a sum function associated with the selected sum model and independent of time steps The data can be assembled in a dyadic tree structure A non-linearity component can be added to nodes of the dyadic tree structure For example, this can be a noise components or a weight applied to the update This can be done, for example, to different nodes differently Differential private estimators can be constructed for fixed steps of time The differential private estimators can be applied to a query means or filtering system to enhance privacy protection from potential adversaries

Private decayed sum estimation under continual observation

Consider fully dynamic data, where we track data as it gets inserted and deleted. There are well developed notions of private data analyses with dynamic data, for example, using differential privacy. We want to go beyond privacy, and consider privacy together with security, formulated recently as pan-privacy by Dwork et al. (ICS 2010). Informally, pan-privacy preserves differential privacy while computing desired statistics on the data, even if the internal memory of the algorithm is compromised (say, by a malicious break-in or insider curiosity or by fiat by the government or law).We study pan-private algorithms for basic analyses, like estimating distinct count, moments, and heavy hitter count, with fully dynamic data. We present the first known pan-private algorithms for these problems in the fully dynamic model. Our algorithms rely on sketching techniques popular in streaming: in some cases, we add suitable noise to a previously known sketch, using a novel approach of calibrating noise to the underlying problem structure and the projection matrix of the sketch; in other cases, we maintain certain statistics on sketches; in yet others, we define novel sketches. We also present the first known lower bounds explicitly for pan privacy, showing our results to be nearly optimal for these problems. Our lower bounds are stronger than those implied by differential privacy or dynamic data streaming alone and hold even if unbounded memory and/or unbounded processing time are allowed. The lower bounds use a noisy decoding argument and exploit a connection between pan-private algorithms and data sanitization.

http://www.cs.rutgers.edu/~rebecca.wright/Publications/pods11.pdf

Aleksandar Nikolov

Papers

The geometry of differential privacy: the sparse and approximate cases

Parallel Algorithms for Geometric Graph Problems

Parallel algorithms for geometric graph problems

Private decayed sum estimation under continual observation

Pan-private algorithms via statistics on sketches