The exponential mechanism is a general method to construct a randomized estimator that satisfies $(\varepsilon, 0)$-differential privacy. Recently, Wang et al. showed that the Gibbs posterior, which is a data-dependent probability distribution that contains the Bayesian posterior, is essentially equivalent to the exponential mechanism under certain boundedness conditions on the loss function. While the exponential mechanism provides a way to build an $(\varepsilon, 0)$-differential private algorithm, it requires boundedness of the loss function, which is quite stringent for some learning problems. In this paper, we focus on $(\varepsilon, \delta)$-differential privacy of Gibbs posteriors with convex and Lipschitz loss functions. Our result extends the classical exponential mechanism, allowing the loss functions to have an unbounded sensitivity.

Lipschitzness, which we make precise in Section 2 and in Assumption 3.1.

Our preliminary results suggest that the regularized exponential mechanism can effectively emulate previous empirical and population risk bounds, negating the need for smoothness assumptions for algorithms with polynomial running time.

Their algorithm has a time and space complexity of O ( dk) .

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However, it is important to consider users' privacy in the dataset, since results from data