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1. Privacy-Preserving Policy Synthesis in Markov Decision Processes

   https://doi.org/10.1109/CDC42340.2020.9304015  

   Gohari, Parham; Hale, Matthew; Topcu, Ufuk (December 2020, Proceedings of the 2020 59th IEEE Conference on Decision and Control (CDC))

   null (Ed.)

   In decision-making problems, the actions of an agent may reveal sensitive information that drives its decisions. For instance, a corporation’s investment decisions may reveal its sensitive knowledge about market dynamics. To prevent this type of information leakage, we introduce a policy synthesis algorithm that protects the privacy of the transition probabilities in a Markov decision process. We use differential privacy as the mathematical definition of privacy. The algorithm first perturbs the transition probabilities using a mechanism that provides differential privacy. Then, based on the privatized transition probabilities, we synthesize a policy using dynamic programming. Our main contribution is to bound the "cost of privacy," i.e., the difference between the expected total rewards with privacy and the expected total rewards without privacy. We also show that computing the cost of privacy has time complexity that is polynomial in the parameters of the problem. Moreover, we establish that the cost of privacy increases with the strength of differential privacy protections, and we quantify this increase. Finally, numerical experiments on two example environments validate the established relationship between the cost of privacy and the strength of data privacy protections. 

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2. Differentially private binary- and matrix-valued data query: an XOR mechanism

   https://doi.org/10.14778/3446095.3446106  

   Ji, Tianxi; Li, Pan; Yilmaz, Emre; Ayday, Erman; Ye, Yanfang; Sun, Jinyuan (January 2021, Proceedings of the VLDB Endowment)

   null (Ed.)

   Differential privacy has been widely adopted to release continuous- and scalar-valued information on a database without compromising the privacy of individual data records in it. The problem of querying binary- and matrix-valued information on a database in a differentially private manner has rarely been studied. However, binary- and matrix-valued data are ubiquitous in real-world applications, whose privacy concerns may arise under a variety of circumstances. In this paper, we devise an exclusive or (XOR) mechanism that perturbs binary- and matrix-valued query result by conducting an XOR operation on the query result with calibrated noises attributed to a matrix-valued Bernoulli distribution. We first rigorously analyze the privacy and utility guarantee of the proposed XOR mechanism. Then, to generate the parameters in the matrix-valued Bernoulli distribution, we develop a heuristic approach to minimize the expected square query error rate under ϵ -differential privacy constraint. Additionally, to address the intractability of calculating the probability density function (PDF) of this distribution and efficiently generate samples from it, we adapt an Exact Hamiltonian Monte Carlo based sampling scheme. Finally, we experimentally demonstrate the efficacy of the XOR mechanism by considering binary data classification and social network analysis, all in a differentially private manner. Experiment results show that the XOR mechanism notably outperforms other state-of-the-art differentially private methods in terms of utility (such as classification accuracy and F 1 score), and even achieves comparable utility to the non-private mechanisms. 

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3. Sharp Composition Bounds for Gaussian Differential Privacy via Edgeworth Expansion

   Zheng, Qingqing; Dong, Jinshuo; Long, Qi; Su, Weijie (January 2020, International Conference on Machine Learning)

   Datasets containing sensitive information are often sequentially analyzed by many algorithms. This raises a fundamental question in differential privacy regarding how the overall privacy bound degrades under composition. To address this question, we introduce a family of analytical and sharp privacy bounds under composition using the Edgeworth expansion in the framework of the recently proposed f-differential privacy. In contrast to the existing composition theorems using the central limit theorem, our new privacy bounds under composition gain improved tightness by leveraging the refined approximation accuracy of the Edgeworth expansion. Our approach is easy to implement and computationally efficient for any number of compositions. The superiority of these new bounds is confirmed by an asymptotic error analysis and an application to quantifying the overall privacy guarantees of noisy stochastic gradient descent used in training private deep neural networks. 

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4. Virtual Markov Chains

   https://doi.org/10.53733/147  

   Evans, Steven; Jaffe, Adam (September 2021, New Zealand Journal of Mathematics)

   We introduce the space of virtual Markov chains (VMCs) as a projective limit of the spaces of all finite state space Markov chains (MCs), in the same way that the space of virtual permutations is the projective limit of the spaces of all permutations of finite sets.We introduce the notions of virtual initial distribution (VID) and a virtual transition matrix (VTM), and we show that the law of any VMC is uniquely characterized by a pair of a VID and VTM which have to satisfy a certain compatibility condition.Lastly, we study various properties of compact convex sets associated to the theory of VMCs, including that the Birkhoff-von Neumann theorem fails in the virtual setting. 

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5. Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

   https://doi.org/10.1017/apr.2021.20  

   Budhiraja, Amarjit; Fraiman, Nicolas; Waterbury, Adam (March 2022, Advances in Applied Probability)

   Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD. 

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