Private Optimization Without Constraint Violations

We study the problem of differentially private optimization with linear constraints when the right-hand-side of the constraints depends on private data. This type of problem appears in many applications, especially resource allocation. Previous research provided solutions that retained privacy, but sometimes violated the constraints. In many settings, however, the constraints cannot be violated under any circumstances. To address this hard requirement, we present an algorithm that releases a nearly-optimal solution satisfying the problem's constraints with probability 1. We also prove a lower bound demonstrating that the difference between the objective value of our algorithm's solution and the optimal solution is tight up to logarithmic factors among all differentially private algorithms. We conclude with experiments on real and synthetic datasets demonstrating that our algorithm can achieve nearly optimal performance while preserving privacy.

[1]  Aaron Roth,et al.  Privacy and Truthful Equilibrium Selection for Aggregative Games , 2014, WINE.

[2]  Y. Rinott,et al.  Confidentiality and Differential Privacy in the Dissemination of Frequency Tables , 2018, Statistical Science.

[3]  Raef Bassily,et al.  Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds , 2014, 1405.7085.

[4]  Ashwin Machanavajjhala,et al.  ShrinkWrap: Efficient SQL Query Processing in Differentially Private Data Federations , 2018, Proc. VLDB Endow..

[5]  Tim Roughgarden,et al.  Private matchings and allocations , 2013, SIAM J. Comput..

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

[7]  Aaron Roth,et al.  Differentially private combinatorial optimization , 2009, SODA '10.

[8]  Pramod Viswanath,et al.  The Staircase Mechanism in Differential Privacy , 2015, IEEE Journal of Selected Topics in Signal Processing.

[9]  Raef Bassily,et al.  Private Stochastic Convex Optimization with Optimal Rates , 2019, NeurIPS.

[10]  Wu Li.  The Sharp Lipschitz-Constants for Feasible and Optimal-Solutions of a Perturbed Linear Program , 2018 .

[11]  Spyros Antonatos,et al.  The Bounded Laplace Mechanism in Differential Privacy , 2018, J. Priv. Confidentiality.

[12]  Wu Li.  The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program , 1993 .

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

[14]  Wei Shi,et al.  Differential Privacy via a Truncated and Normalized Laplace Mechanism , 2019, Journal of Computer Science and Technology.

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

[16]  Wei Ding,et al.  Tight Analysis of Privacy and Utility Tradeoff in Approximate Differential Privacy , 2018, AISTATS.

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

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

[19]  Tim Roughgarden,et al.  Privately Solving Linear Programs , 2014, ICALP.

[20]  O. Mangasarian,et al.  Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems , 1987 .

[21]  Daniel Kifer,et al.  Private Convex Optimization for Empirical Risk Minimization with Applications to High-dimensional Regression , 2012, COLT.

[22]  Francesco Cesarone,et al.  Real-world datasets for portfolio selection and solutions of some stochastic dominance portfolio models , 2016, Data in brief.

[23]  Ian Goodfellow,et al.  Deep Learning with Differential Privacy , 2016, CCS.