[Paper Review] Differentially Private Convex Optimization with Feasibility Guarantees

Vladimir Dvorkin, Ferdinando Fioretto+3 more|arXiv (Cornell University)|Jun 22, 2020

Privacy-Preserving Technologies in Data34 references4 citations

TL;DR

This paper proposes a differentially private framework for convex optimization with hard constraints, ensuring feasibility through a chance-constrained stochastic optimization reformulation. By making optimization variables a linear function of noise, it guarantees high-probability feasibility while enabling privacy-preserving release of identity and sum queries, achieving a tunable trade-off between optimality loss and solution variance.

ABSTRACT

This paper develops a novel differentially private framework to solve convex optimization problems with sensitive optimization data and complex physical or operational constraints. Unlike standard noise-additive algorithms, that act primarily on the problem data, objective or solution, and disregard the problem constraints, this framework requires the optimization variables to be a function of the noise and exploits a chance-constrained problem reformulation with formal feasibility guarantees. The noise is calibrated to provide differential privacy for identity and linear queries on the optimization solution. For many applications, including resource allocation problems, the proposed framework provides a trade-off between the expected optimality loss and the variance of optimization results.

Motivation & Objective

• To address the lack of feasibility guarantees in differentially private constrained optimization, especially when releasing solutions to sensitive data.
• To develop a method that ensures privacy for identity and sum queries on optimization solutions without violating physical or operational constraints.
• To provide formal feasibility guarantees through a chance-constrained reformulation that maintains constraint satisfaction with high probability.
• To enable a tunable trade-off between expected optimality loss and variance of the solution, particularly in resource allocation and energy systems.
• To overcome limitations of standard output or input perturbation methods that fail to preserve feasibility or require bilevel optimization for correction.

Proposed method

• Reformulates the original convex optimization problem as a stochastic chance-constrained program where the solution variables are linear functions of random noise.
• Uses a linear functional recourse z(ξ) = z̃ + f(ξ), where z̃ is the expected solution and f(ξ) is a noise-dependent term, to ensure feasibility with high probability.
• Calibrates noise to achieve (ε, δ)-differential privacy for identity and linear queries on the solution, independent of sensitive data d.
• Imposes constraints Gz = d (e.g., flow conservation) to be exactly satisfied, while allowing perturbations to be redistributed among non-released variables to reduce optimality loss.
• Solves a bilevel-like optimization problem to jointly optimize the expected solution z̃ and the functional recourse f(ξ), ensuring both privacy and feasibility.
• Employs a two-stage approach: first, optimize z̃ and f(ξ) under chance constraints; second, sample solutions from the resulting distribution to release.

Experimental results

Research questions

• RQ1 Can a differentially private optimization framework be designed that guarantees feasibility of solutions under complex physical or operational constraints?
• RQ2 How can differential privacy be achieved for identity and sum queries on optimization solutions without compromising constraint satisfaction?
• RQ3 What trade-offs exist between expected optimality loss and variance of the solution, and how can they be controlled?
• RQ4 Can the proposed framework outperform standard output perturbation methods in terms of solution quality and feasibility?
• RQ5 How does the separation of released and non-released variables affect the trade-off between optimality loss and solution variance?

Key findings

• The proposed framework guarantees feasibility of the solution with high probability by enforcing a chance constraint on the solution distribution, ensuring Gz = d is satisfied.
• For the 118_ieee test case, the PSQ-a algorithm achieved zero variance in optimality loss at ε = 10, with expected optimality loss E[∆c] increasing only moderately as ϕ increased.
• The overall solution variance Var[z(ξ)] was reduced to nearly Tr[Σ] (the total variance of the 9 perturbations), demonstrating effective variance control.
• The framework outperformed standard output perturbation algorithms in terms of both optimality loss and solution variance on benchmark energy networks.
• The method enables a tunable trade-off between expected optimality loss and solution variance by adjusting the trade-off parameter ϕ in the chance-constrained formulation.
• The approach achieves (ε, δ)-differential privacy for identity and sum queries by ensuring the functional recourse f(ξ) is independent of sensitive data d.