Jul 12, 2020
In this paper, we study (locally) differentially private Combinatorial Semi-Bandits (CSB). Compared with private Multi-Armed Bandits (MAB), since the server receives more information from the user, it usually leads to additional dependence over the dimension of feedback, which is a notorious problem in private learning. Somewhat surprisingly, we show that it is possible to remove this side-effect caused by privacy protection and nearly match corresponding non-private best results. In detail, for general CSB with B-bounded smooth reward function in the sense of Chen et al. 2016, we propose a novel algorithm that achieves regret bound Õ(mB^2log T / ϵ) over T rounds under ϵ-local differential privacy, where m is the number of base arms. However, for Linear CSB, B equals K, where K is the maximum number of feedback in each round, and above bound has an additional K compared with non-private optimal result. We then propose a different algorithm with nearly optimal regret bound Õ(mKlog T / ϵ) if one cares about ϵ-differential privacy rather than ϵ-local differential privacy. Besides, we also prove some lower bounds in each setting.
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