arXiv subjects

Brendan Bialy

Publications and source records attributed to Brendan Bialy.

4 recordsLinked to original sources

arXiv2026

Differentially Private Data-Driven Markov Chain Modeling

Markov chains model a wide range of user behaviors. However, generating accurate Markov chain models requires substantial user data, and sharing these models without privacy protections may reveal sensitive information about the underlying user data. We introduce a method for protecting user data used to formulate a Markov chain model. First, we develop a method for privatizing database queries whose outputs are elements of the unit simplex, and we prove that this method is differentially private. We quantify its accuracy by bounding the expected KL divergence between private and non-private queries. We extend this method to privatize stochastic matrices whose rows are each a simplex-valued query of a database, which includes data-driven Markov chain models. To assess their accuracy, we analytically bound the change in the stationary distribution and the change in the convergence rate between a non-private Markov chain model and its private form. Simulations show that under a typical privacy implementation, our method yields less than 2% error in the stationary distribution, indicating that our approach to private modeling faithfully captures the behavior of the systems we study.

cs.CR↗

arXiv2025

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

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.

math.OC↗

arXiv2024

Guaranteed Feasibility in Differentially Private Linearly Constrained Convex Optimization

Convex programming with linear constraints plays an important role in the operation of a number of everyday systems. However, absent any additional protections, revealing or acting on the solutions to such problems may reveal information about their constraints, which can be sensitive. Therefore, in this paper, we introduce a method for solving convex programs while keeping linear constraints private. First, we prove that this method is differentially private and always generates a feasible optimization problem (i.e., one whose solution exists). Then we show that the solution to the privatized problem also satisfies the original, non-private constraints. Next, we bound the expected loss in performance from privacy, which is measured by comparing the cost with privacy to that without privacy. Simulation results apply this framework to constrained policy synthesis in a Markov decision process, and they show that a typical privacy implementation induces only an approximately $9\%$ loss in solution quality.

math.OC↗

arXiv2024

Differentially Private Reward Functions in Policy Synthesis for Markov Decision Processes

Markov decision processes often seek to maximize a reward function, but onlookers may infer reward functions by observing the states and actions of such systems, revealing sensitive information. Therefore, in this paper we introduce and compare two methods for privatizing reward functions in policy synthesis for multi-agent Markov decision processes, which generalize Markov decision processes. Reward functions are privatized using differential privacy, a statistical framework for protecting sensitive data. The methods we develop perturb either (1) each agent's individual reward function or (2) the joint reward function shared by all agents. We show that approach (1) provides better performance. We then develop a polynomial-time algorithm for the numerical computation of the performance loss due to privacy on a case-by-case basis. Next, using approach (1), we develop guidelines for selecting reward function values to preserve ``goal" and ``avoid" states while still remaining private, and we quantify the increase in computational complexity needed to compute policies from privatized rewards. Numerical simulations are performed on three classes of systems and they reveal a surprising compatibility with privacy: using reasonably strong privacy ($ε=1.3$) on average induces as little as a~$5\%$ decrease in total accumulated reward and a $0.016\%$ increase in computation time.

eess.SY↗