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This paper introduces a multi-round interaction problem with privacyconstraints between two agents that observe correlated data. The agentsalternately share data with one another for a total of K rounds such that eachagent initiates sharing over K/2 rounds. The interactions are modeled as acollection of K random mechanisms (mappings), one for each round. The goal isto jointly design the K private mechanisms to determine the set of allachievable distortion-leakage pairs at each agent. Arguing that a mutualinformation-based leakage metric can be appropriate for streaming datasettings, this paper: (i) determines the set of all achievable distortion-leakage tuples ; (ii) shows that the K mechanisms allow for precisely composingthe total privacy budget over K rounds without loss; and (ii) developsconditions under which interaction reduces the net leakage at both agents andillustrates it for a specific class of sources. The paper then focuses onlog-loss distortion to better understand the effect on leakage of using acommonly used utility metric in learning theory. The resulting interactionproblem leads to a non-convex sum-leakage-distortion optimization problem thatcan be viewed as an interactive version of the information bottleneck problem.A new merge-and-search algorithm that extends the classical agglomerativeinformation bottleneck algorithm to the interactive setting is introduced todetermine a provable locally optimal solution. Finally, the benefit ofinteraction under log-loss is illustrated for specific source classes and theoptimality of one-shot is proved for Gaussian sources under both mean-squareand log-loss distortions constraints.

Privacy-guaranteed Two-Agent Interactions Using Information-Theoretic Mechanisms