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The change-point detection problem seeks to identify distributional changes at an unknown change-point k^∗ in a stream of data. This problem appears in many important practical settings involving personal data, including biosurveillance, fault detection, finance, signal detection, and security systems. The field of differential privacy offers data analysis tools that provide powerful worst-case privacy guarantees. We study the statistical problem of change-point detection through the lens of differential privacy. We give private algorithms for both online and offline change-point detection, analyze these algorithms theoretically, and provide empirical validation of our results. 

In this work we consider the problem of online submodular maximization under a cardinality constraint with differential privacy (DP). A stream of T submodular functions over a common finite ground set U arrives online, and at each time-step the decision maker must choose at most k elements of U before observing the function. The decision maker obtains a profit equal to the function evaluated on the chosen set and aims to learn a sequence of sets that achieves low expected regret. In the full-information setting, we develop an (𝜀,𝛿)-DP algorithm with expected (1-1/e)-regret bound of 𝑂(𝑘2log|𝑈|𝑇log𝑘/𝛿√𝜀). This algorithm contains k ordered experts that learn the best marginal increments for each item over the whole time horizon while maintaining privacy of the functions. In the bandit setting, we provide an (𝜀,𝛿+𝑂(𝑒−𝑇1/3))-DP algorithm with expected (1-1/e)-regret bound of 𝑂(log𝑘/𝛿√𝜀(𝑘(|𝑈|log|𝑈|)1/3)2𝑇2/3). One challenge for privacy in this setting is that the payoff and feedback of expert i depends on the actions taken by her i-1 predecessors. This particular type of information leakage is not covered by post-processing, and new analysis is required. Our techniques for maintaining privacy with feedforward may be of independent interest. 

We consider the design of private prediction markets , financial markets designed to elicit predictions about uncertain events without revealing too much information about market participants’ actions or beliefs. Our goal is to design market mechanisms in which participants’ trades or wagers influence the market’s behavior in a way that leads to accurate predictions, yet no single participant has too much influence over what others are able to observe. We study the possibilities and limitations of such mechanisms using tools from differential privacy. We begin by designing a private one-shot wagering mechanism in which bettors specify a belief about the likelihood of a future event and a corresponding monetary wager. Wagers are redistributed among bettors in a way that more highly rewards those with accurate predictions. We provide a class of wagering mechanisms that are guaranteed to satisfy truthfulness, budget balance on expectation, and other desirable properties while additionally guaranteeing ε-joint differential privacy in the bettors’ reported beliefs, and analyze the trade-off between the achievable level of privacy and the sensitivity of a bettor’s payment to her own report. We then ask whether it is possible to obtain privacy in dynamic prediction markets, focusing our attention on the popular cost-function framework in which securities with payments linked to future events are bought and sold by an automated market maker. We show that under general conditions, it is impossible for such a market maker to simultaneously achieve bounded worst-case loss and ε-differential privacy without allowing the privacy guarantee to degrade extremely quickly as the number of trades grows (at least logarithmically in number of trades), making such markets impractical in settings in which privacy is valued. We conclude by suggesting several avenues for potentially circumventing this lower bound.