Jointly Private Convex Programming

TL;DR

This paper introduces a highly general method for approximately solving convex programs with solutions distributed among agents under joint differential privacy, applicable to various complex problems, with accuracy depending mainly on the number of binding constraints.

Contribution

It presents a universal approach for differentially private convex programming that scales well with the number of agents and extends to strategic settings with incentive compatibility.

Findings

Algorithm's error depends on the number of binding constraints, not total variables.

Method applies to multi-commodity flow, allocation, and knapsack problems.

Achieves approximate social welfare maximization under privacy and strategic considerations.

Abstract

In this paper we present an extremely general method for approximately solving a large family of convex programs where the solution can be divided between different agents, subject to joint differential privacy. This class includes multi-commodity flow problems, general allocation problems, and multi-dimensional knapsack problems, among other examples. The accuracy of our algorithm depends on the \emph{number} of constraints that bind between individuals, but crucially, is \emph{nearly independent} of the number of primal variables and hence the number of agents who make up the problem. As the number of agents in a problem grows, the error we introduce often becomes negligible. We also consider the setting where agents are strategic and have preferences over their part of the solution. For any convex program in this class that maximizes \emph{social welfare}, there is a generic reduction that makes the corresponding optimization \emph{approximately dominant strategy truthful} by charging agents prices for resources as a function of the approximately optimal dual variables, which are themselves computed under differential privacy. Our results substantially expand the class of problems that are known to be solvable under both privacy and incentive constraints.