Analytic Theory to Differential Privacy

The purpose of this paper is to develop a mathematical analysis theory to solve differential privacy problems. The heart of our approaches is to use analytic tools to characterize the correlations among the outputs of different datasets, which makes it feasible to represent a differentially private mechanism with minimal number of parameters. These results are then used to construct differentially private mechanisms analytically. Furthermore, our approaches are universal to almost all query functions. We believe that the approaches and results of this paper are indispensable complements to the current studies of differential privacy that are ruled by the ad hoc and algorithmic approaches.

[1]  Anand D. Sarwate,et al.  Differentially Private Empirical Risk Minimization , 2009, J. Mach. Learn. Res..

[2]  Cynthia Dwork,et al.  Private False Discovery Rate Control , 2015, ArXiv.

[3]  David D. Jensen,et al.  Accurate Estimation of the Degree Distribution of Private Networks , 2009, 2009 Ninth IEEE International Conference on Data Mining.

[4]  Claude Castelluccia,et al.  Differentially private sequential data publication via variable-length n-grams , 2012, CCS.

[5]  Li Zhang,et al.  Analyze gauss: optimal bounds for privacy-preserving principal component analysis , 2014, STOC.

[6]  Anand D. Sarwate,et al.  Signal Processing and Machine Learning with Differential Privacy: Algorithms and Challenges for Continuous Data , 2013, IEEE Signal Processing Magazine.

[7]  Larry A. Wasserman,et al.  Differential privacy for functions and functional data , 2012, J. Mach. Learn. Res..

[8]  Aaron Roth,et al.  Beating randomized response on incoherent matrices , 2011, STOC '12.

[9]  Aleksandar Nikolov,et al.  The geometry of differential privacy: the sparse and approximate cases , 2012, STOC '13.

[10]  Xing Xie,et al.  PrivTree: A Differentially Private Algorithm for Hierarchical Decompositions , 2016, SIGMOD Conference.

[11]  Kamalika Chaudhuri,et al.  The Large Margin Mechanism for Differentially Private Maximization , 2014, NIPS.

[12]  Sofya Raskhodnikova,et al.  Analyzing Graphs with Node Differential Privacy , 2013, TCC.

[13]  Cynthia Dwork,et al.  Calibrating Noise to Sensitivity in Private Data Analysis , 2006, TCC.

[14]  Philip S. Yu,et al.  Differentially Private Data Publishing and Analysis: A Survey , 2017, IEEE Transactions on Knowledge and Data Engineering.

[15]  Benjamin C. M. Fung,et al.  Differentially private transit data publication: a case study on the montreal transportation system , 2012, KDD.

[16]  Shuigeng Zhou,et al.  Recursive mechanism: towards node differential privacy and unrestricted joins , 2013, SIGMOD '13.

[17]  Aaron Roth,et al.  The Algorithmic Foundations of Differential Privacy , 2014, Found. Trends Theor. Comput. Sci..

[18]  Andrew McGregor,et al.  The matrix mechanism: optimizing linear counting queries under differential privacy , 2015, The VLDB Journal.

[19]  P. Siarry,et al.  Multiobjective Optimization: Principles and Case Studies , 2004 .

[20]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[21]  Kamalika Chaudhuri,et al.  A Stability-based Validation Procedure for Differentially Private Machine Learning , 2013, NIPS.

[22]  Catuscia Palamidessi,et al.  Broadening the Scope of Differential Privacy Using Metrics , 2013, Privacy Enhancing Technologies.

[23]  Aaron Roth,et al.  Max-Information, Differential Privacy, and Post-selection Hypothesis Testing , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[24]  Sofya Raskhodnikova,et al.  Lipschitz Extensions for Node-Private Graph Statistics and the Generalized Exponential Mechanism , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[25]  Yin Yang,et al.  Functional Mechanism: Regression Analysis under Differential Privacy , 2012, Proc. VLDB Endow..

[26]  Piotr Indyk,et al.  Algorithmic applications of low-distortion geometric embeddings , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[27]  Liwei Wang,et al.  Efficient Algorithm for Privately Releasing Smooth Queries , 2013, NIPS.

[28]  Kamalika Chaudhuri,et al.  Privacy-preserving logistic regression , 2008, NIPS.

[29]  Divesh Srivastava,et al.  Accurate and efficient private release of datacubes and contingency tables , 2012, 2013 IEEE 29th International Conference on Data Engineering (ICDE).

[30]  Úlfar Erlingsson,et al.  RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response , 2014, CCS.

[31]  Benjamin C. M. Fung,et al.  Publishing set-valued data via differential privacy , 2011, Proc. VLDB Endow..

[32]  Ashwin Machanavajjhala,et al.  No free lunch in data privacy , 2011, SIGMOD '11.

[33]  Cynthia Dwork,et al.  Differential Privacy , 2006, ICALP.

[34]  Kunal Talwar,et al.  On differentially private low rank approximation , 2013, SODA.

[35]  Sofya Raskhodnikova,et al.  Private analysis of graph structure , 2011, Proc. VLDB Endow..

[36]  Philip S. Yu,et al.  Differentially private data release for data mining , 2011, KDD.

[37]  Ashwin Machanavajjhala,et al.  Principled Evaluation of Differentially Private Algorithms using DPBench , 2015, SIGMOD Conference.

[38]  Salil P. Vadhan,et al.  The Complexity of Differential Privacy , 2017, Tutorials on the Foundations of Cryptography.

[39]  Ilya Mironov,et al.  Differentially private recommender systems: building privacy into the net , 2009, KDD.

[40]  Pramod Viswanath,et al.  The Optimal Noise-Adding Mechanism in Differential Privacy , 2012, IEEE Transactions on Information Theory.

[41]  Jun Zhang,et al.  PrivBayes: private data release via bayesian networks , 2014, SIGMOD Conference.

[42]  Anand D. Sarwate,et al.  A near-optimal algorithm for differentially-private principal components , 2012, J. Mach. Learn. Res..

[43]  Sampath Kannan,et al.  Privacy-Preserving Data Analysis for the Federal Statistical Agencies , 2017, ArXiv.

[44]  Katrina Ligett,et al.  A Simple and Practical Algorithm for Differentially Private Data Release , 2010, NIPS.

[45]  Sofya Raskhodnikova,et al.  Smooth sensitivity and sampling in private data analysis , 2007, STOC '07.

[46]  Aaron Roth,et al.  Beyond worst-case analysis in private singular vector computation , 2012, STOC '13.

[47]  Elisa Bertino,et al.  Differentially Private K-Means Clustering , 2015, CODASPY.

[48]  Douglas J. Leith,et al.  Differential privacy in metric spaces: Numerical, categorical and functional data under the one roof , 2015, Inf. Sci..

[49]  Cynthia Dwork,et al.  Differential privacy and robust statistics , 2009, STOC '09.

[50]  Ling Huang,et al.  Learning in a Large Function Space: Privacy-Preserving Mechanisms for SVM Learning , 2009, J. Priv. Confidentiality.

[51]  Pramod Viswanath,et al.  Optimal Noise Adding Mechanisms for Approximate Differential Privacy , 2016, IEEE Transactions on Information Theory.

[52]  K. Athreya,et al.  Measure Theory and Probability Theory , 2006 .

[53]  Sofya Raskhodnikova,et al.  Testing the Lipschitz Property over Product Distributions with Applications to Data Privacy , 2013, TCC.

[54]  Divesh Srivastava,et al.  Private Release of Graph Statistics using Ladder Functions , 2015, SIGMOD Conference.

[55]  Moni Naor,et al.  Pure Differential Privacy for Rectangle Queries via Private Partitions , 2015, ASIACRYPT.

[56]  Kunal Talwar,et al.  Mechanism Design via Differential Privacy , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[57]  Aditya Bhaskara,et al.  Unconditional differentially private mechanisms for linear queries , 2012, STOC '12.

[58]  Tim Roughgarden,et al.  Interactive privacy via the median mechanism , 2009, STOC '10.

[59]  Guy N. Rothblum,et al.  A Multiplicative Weights Mechanism for Privacy-Preserving Data Analysis , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[60]  Kunal Talwar,et al.  On the geometry of differential privacy , 2009, STOC '10.

[61]  Aaron Roth,et al.  Iterative Constructions and Private Data Release , 2011, TCC.