Differentially Private Online Submodular Maximization

Abstract

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 $(\varepsilon,\delta)$-DP algorithm with expected (1-1/e)-regret bound of $O( \frac{k^2\log |U|\sqrt{T \log k/\delta}}{\varepsilon} )$. 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 $(\varepsilon,\delta+ O(e^{-T^{1/3}}))$-DP algorithm with expected (1-1/e)-regret bound of $O( \frac{\sqrt{\log k/\delta}}{\varepsilon} (k (|U| \log |U|)^{1/3})^2 T^{2/3} )$. 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.