Differentially Private Linear Programming: Reduced Sub-Optimality and Guaranteed Constraint Satisfaction

TL;DR

This work addresses how to solve linear programs whose data-dependent components $A(D)$, $b(D)$, and $c(D)$ are privacy-sensitive, using differential privacy to perturb all three terms. It develops dedicated privacy mechanisms—Matrix-Variate Truncated Laplace for $A(D)$, Multivariate Truncated Laplace for $b(D)$, and the Laplace mechanism for $c(D)$—and introduces a post-processing step that tightens constraints to guarantee feasibility with respect to the original non-private problem. The authors prove the privatized LP is differentially private and that its solution remains feasible for the original problem, providing a quantitative bound on expected sub-optimality via the Hoffman constant-based analysis. Empirical results on a CMDP-like advertising problem show zero constraint violations and up to a $65\%$ reduction in sub-optimality compared to prior methods, with insights into budget allocation and scalability. The framework thus enables privacy-preserving optimization in sensitive, data-driven LPs while preserving feasibility and offering controllable accuracy loss.

Abstract

Linear programming is a fundamental tool in a wide range of decision systems. However, without privacy protections, sharing the solution to a linear program may reveal information about the underlying data used to formulate it, which may be sensitive. Therefore, in this paper we introduce an approach for protecting sensitive data while formulating and solving a linear program. First, we prove that this method perturbs objectives and constraints in a way that makes them differentially private. Then, we show that (i) privatized problems always have solutions, and (ii) their solutions satisfy the constraints in their corresponding original, non-private problems. The latter result solves an open problem in the literature. Next, we analytically bound the expected sub-optimality of solutions that is induced by privacy. Numerical simulations show that, under a typical privacy setup, the solution produced by our method yields a $65\%$ reduction in sub-optimality compared to the state of the art.

Figures (5)

• Figure 1: Performance loss with varying privacy strength. Combining Algorithm 5 with a privatized objective from hsu2014privately leads to constraint violation for all $\epsilon\in[0.25, 2]$. High constraint violation allows the solution to give the appearance of superior performance; however, such a solution leads to significant violation of some advertisers' budgets, which is unacceptable. Even when allowing this constraint violation, the solution produced by hsu2014privately still yields worse performance than that of Remark \ref{['rem:modify']} and Algorithm \ref{['algo:solve']}. We also compare to munoz2021private, and we emphasize that Munoz incurs lower sub-optimality because it privatizes only $b(D)$ in the constraints, while Algorithm 1 is used to privatize both $A(D)$ and $b(D)$. The approach in munoz2021private only incurs $0.5\%$ sub-optimality at $\epsilon = 2$, while the approach in Algorithm 1 incurs roughly $20$% sub-optimality, which indicates that privacy for $A(D)$ induces $19.5\%$ additional sub-optimality.
• Figure 2: Performance loss with varying $M$. As the number of variables increases, Algorithm 5 in hsu2014privately allows their solver to run for more iterations, leading to improvement in accuracy, though this leads to a dramatic increase in computation time. In Algorithm \ref{['algo:solve']}, we see only an $11\%$ decrease in optimal revenue with a $10\times$ increase in problem size, going from $13.3\%$ sub-optimality with $M = 10$ to $24\%$ sub-optimality at $M=100$. Performance remains roughly constant with increasing number of constraints for Algorithm \ref{['algo:solve']} and munoz2021private, while Algorithm 5 in hsu2014privately steadily improves in performance but still has much higher sub-optimality.
• Figure 3: Sub-optimality gap with varying privacy budget allocated to the cost function, with the remaining cost budget divided evenly among the constraints.
• Figure 4: Grid in which the agent starts at the blue state, its goal is the green state, and hazardous states are red.
• Figure 5: Cost of Privacy for privately generated policies using Algorithm \ref{['algo:solve']} and munoz2021private. The sub-optimality in munoz2021private remains constant, however their approach leaks private information about $D$ since their approach cannot privatize $A(D)$. The sub-optimality of Algorithm \ref{['algo:solve']} approaches that of munoz2021private as $\epsilon$ increases, indicating that at low privacy we recover the performance of munoz2021private without privacy leakage.

Theorems & Definitions (25)

• Remark 2.1
• Remark 2.3
• Definition 2.5: Adjacency
• Definition 2.6: Differential Privacy; dwork2014algorithmic
• Lemma 2.7: Sequential Composition of Private Mechanisms dwork2014algorithmic
• Lemma 2.8: Immunity to Post-Processing; dwork2014algorithmic
• Definition 3.1
• Lemma 3.2: Matrix-Variate Truncated Laplace Mechanism
• proof
• Definition 3.3