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The Gaussian mechanism is one differential privacy mechanism commonly used to protect numerical data. However, it may be ill-suited to some applications because it has unbounded support and thus can produce invalid numerical answers to queries, such as negative ages or human heights in the tens of meters. One can project such private values onto valid ranges of data, though such projections lead to the accumulation of private query responses at the boundaries of such ranges, thereby harming accuracy. Motivated by the need for both privacy and accuracy over bounded domains, we present a bounded Gaussian mechanism for differential privacy, which has support only on a given region. We present both univariate and multivariate versions of this mechanism and illustrate a significant reduction in variance relative to comparable existing work. 

This paper develops a framework for privatizing the spectrum of the Laplacian of an undirected graph using differential privacy. We consider two privacy formulations. The first obfuscates the presence of edges in the graph and the second obfuscates the presence of nodes. We compare these two privacy formulations and show that the privacy formulation that considers edges is better suited to most engineering applications. We use the bounded Laplace mechanism to provide (epsilon, delta)-differential privacy to the eigenvalues of a graph Laplacian, and we pay special attention to the algebraic connectivity, which is the Laplacian's the second smallest eigenvalue. Analytical bounds are presented on the accuracy of the mechanisms and on certain graph properties computed with private spectra. A suite of numerical examples confirms the accuracy of private spectra in practice. 

In cooperative multi-agent reinforcement learning (Co-MARL), a team of agents must jointly optimize the team's longterm rewards to learn a designated task. Optimizing rewards as a team often requires inter-agent communication and data sharing, leading to potential privacy implications. We assume privacy considerations prohibit the agents from sharing their environment interaction data. Accordingly, we propose Privacy-Engineered Value Decomposition Networks (PE-VDN), a Co-MARL algorithm that models multi-agent coordination while provably safeguarding the confidentiality of the agents' environment interaction data. We integrate three privacy-engineering techniques to redesign the data flows of the VDN algorithm-an existing Co-MARL algorithm that consolidates the agents' environment interaction data to train a central controller that models multi-agent coordination-and develop PE-VDN. In the first technique, we design a distributed computation scheme that eliminates Vanilla VDN's dependency on sharing environment interaction data. Then, we utilize a privacy-preserving multi-party computation protocol to guar-antee that the data flows of the distributed computation scheme do not pose new privacy risks. Finally, we enforce differential privacy to preempt inference threats against the agents' training data-past environment interactions-when they take actions based on their neural network predictions. We implement PE-VDN in StarCraft Multi-Agent Competition (SMAC) and show that it achieves 80% of Vanilla VDN's win rate while maintaining differential privacy levels that provide meaningful privacy guarantees. The results demonstrate that PE-VDN can safeguard the confidentiality of agents' environment interaction data without sacrificing multi-agent coordination. 

Stochastic matrices are commonly used to analyze Markov chains, but revealing them can leak sensitive information. Therefore, in this paper we introduce a technique to privatize stochastic matrices in a way that (i) conceals the probabilities they contain, and (ii) still allows for accurate analyses of Markov chains. Specifically, we use differential privacy, which is a statistical framework for protecting sensitive data. To implement it, we introduce the Matrix Dirichlet Mechanism, which is a probabilistic mapping that perturbs a stochastic matrix to provide privacy. We prove that this mechanism provides differential privacy, and we quantify the error induced in private stochastic matrices as a function of the strength of privacy being provided. We then bound the distance between the stationary distribution of the underlying, sensitive stochastic matrix and the stationary distribution of its privatized form. Numerical results show that, under typical conditions, privacy introduces error as low as 5.05% in the stationary distribution of a stochastic matrix.