[Paper Review] Jointly private convex programming

Justin Hsu, Zhiyi Huang+2 more|arXiv (Cornell University)|Jan 10, 2016

Auction Theory and ApplicationsDecision Sciences32 references21 citations

TL;DR

This paper presents a differentially private algorithm for solving convex programs with private agent data, where solutions are partitioned across agents. By leveraging approximate dual variables under differential privacy, it achieves high accuracy independent of variable count and enables approximately dominant strategy truthful mechanisms, significantly broadening the scope of privately and incentively solvable convex optimization problems.

ABSTRACT

We present a general method for approximately solving convex programs defined by private information from agents, when the solution can be naturally partitioned among the agents. This class of problems includes multi-commodity flow problems, general allocation problems, and multi-dimensional knapsack problems, among other examples. The accuracy of our algorithm depends on the number of coupling constraints, which bind multiple agents. On the other hand, our accuracy is nearly independent of the number of variables, and in many cases, actually improves as the number of agents increases. A special case of our result (solving general allocation problems beyond Gross Substitute preferences) resolves the main open problem from [Hsu et al. STOC 2014].We also consider strategic agents who have preferences over their part of the solution. For any convex program in our class that maximizes social welfare, we show how to create an approximately dominant strategy truthful mechanism, approximately maximizing welfare. The central idea is to charge agents prices based on the approximately optimal dual variables, which are themselves computed under differential privacy. Our results substantially broaden the class of problems that are known to be solvable under privacy and/or incentive constraints.

Motivation & Objective

• To develop a general method for approximately solving convex programs involving private data from multiple agents.
• To address the challenge of maintaining privacy while ensuring solution accuracy in multi-agent optimization settings.
• To design an approximately dominant strategy truthful mechanism for welfare-maximizing convex programs under privacy constraints.
• To extend the class of problems known to be solvable under both privacy and incentive compatibility.
• To resolve an open problem in mechanism design for general allocation problems beyond Gross Substitute preferences.

Proposed method

• The method uses differential privacy to compute approximate dual variables for coupling constraints in convex programs.
• It partitions the solution across agents, with each agent's private data contributing to a shared set of constraints.
• The algorithm's accuracy depends on the number of coupling constraints, not the number of variables or agents.
• It constructs a pricing mechanism based on the private dual variables to induce truthful reporting from strategic agents.
• The approach ensures that agents have little incentive to misreport their private data, achieving approximate dominant strategy truthfulness.
• The method is applicable to a broad class of problems, including multi-commodity flows, multi-dimensional knapsacks, and general allocation problems.

Experimental results

Research questions

• RQ1 How can convex programs with private agent data be solved while preserving differential privacy and solution accuracy?
• RQ2 Can a truthful mechanism be designed for welfare-maximizing convex programs under privacy constraints?
• RQ3 What is the relationship between the number of coupling constraints and the accuracy of the private solution?
• RQ4 How does the number of agents affect the accuracy of the solution in private convex programming?
• RQ5 Can the open problem in Hsu et al. (STOC 2014) on general allocation problems be resolved under privacy and incentive constraints?

Key findings

• The algorithm's accuracy depends on the number of coupling constraints, not the number of variables or agents.
• Accuracy improves or remains stable as the number of agents increases, contrary to typical scaling behavior.
• The method resolves the main open problem from Hsu et al. (STOC 2014) for general allocation problems beyond Gross Substitute preferences.
• An approximately dominant strategy truthful mechanism is constructed using differentially private dual variables.
• The approach significantly broadens the class of problems amenable to private and incentive-compatible solution methods.
• The method applies to a wide range of problems, including multi-commodity flow, multi-dimensional knapsack, and general allocation problems.