Goodness-of-fit testing for H\"older continuous densities under local differential privacy

We address the problem of goodness-of-fit testing for Hölder continuous densities under local differential privacy constraints. We study minimax separation rates when only noninteractive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal up to at most a logarithmic factor for several choices of f0 including densities from uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In particular, we observe that the results are deteriorated in the private setting compared to the non-private one. Moreover, we show that sequentially interactive mechanisms improve upon the results obtained when considering only non-interactive privacy mechanisms.

[1]  A. Carpentier,et al.  Sharp Local Minimax Rates for Goodness-of-Fit Testing in Large Random Graphs, multivariate Poisson families and multinomials , 2020, 2012.13766.

[2]  Ryan M. Rogers,et al.  Differentially Private Chi-Squared Hypothesis Testing: Goodness of Fit and Independence Testing , 2016, ICML 2016.

[3]  Or Sheffet,et al.  Locally Private Hypothesis Testing , 2018, ICML.

[4]  Cristina Butucea,et al.  Locally private non-asymptotic testing of discrete distributions is faster using interactive mechanisms , 2020, NeurIPS.

[5]  Huanyu Zhang,et al.  Differentially Private Testing of Identity and Closeness of Discrete Distributions , 2017, NeurIPS.

[6]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[7]  Cristina Butucea,et al.  Interactive versus non-interactive locally differentially private estimation: Two elbows for the quadratic functional , 2020 .

[8]  Himanshu Tyagi,et al.  Test without Trust: Optimal Locally Private Distribution Testing , 2018, AISTATS.

[9]  Thomas B. Berrett,et al.  Classification under local differential privacy , 2019, 1912.04629.

[10]  Seth Neel,et al.  The Role of Interactivity in Local Differential Privacy , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[11]  V. Spokoiny.  Adaptive hypothesis testing using wavelets , 1996 .

[12]  Angelika Rohde,et al.  Geometrizing rates of convergence under local differential privacy constraints , 2020 .

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

[14]  Gregory Valiant,et al.  An Automatic Inequality Prover and Instance Optimal Identity Testing , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[15]  Daniel M. Kane,et al.  A New Approach for Testing Properties of Discrete Distributions , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[16]  L. Wasserman,et al.  A Statistical Framework for Differential Privacy , 2008, 0811.2501.

[17]  Marco Gaboardi,et al.  Local Private Hypothesis Testing: Chi-Square Tests , 2017, ICML.

[18]  Jean-Michel Loubes,et al.  Minimax optimal goodness-of-fit testing for densities under a local differential privacy constraint , 2020, ArXiv.

[19]  Pramod Viswanath,et al.  Extremal Mechanisms for Local Differential Privacy , 2014, J. Mach. Learn. Res..

[20]  Amandine Dubois,et al.  Local differential privacy: Elbow effect in optimal density estimation and adaptation over Besov ellipsoids , 2019, Bernoulli.

[21]  Martin J. Wainwright,et al.  Minimax Optimal Procedures for Locally Private Estimation , 2016, ArXiv.

[22]  Sivaraman Balakrishnan,et al.  Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates , 2017, The Annals of Statistics.

[23]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[24]  Constantinos Daskalakis,et al.  Priv'IT: Private and Sample Efficient Identity Testing , 2017, ICML.

[25]  Ronitt Rubinfeld,et al.  Differentially Private Identity and Equivalence Testing of Discrete Distributions , 2018, ICML.

[26]  Daniel Kifer,et al.  Revisiting Differentially Private Hypothesis Tests for Categorical Data , 2015 .