  In this paper, we study Combinatorial Semi-Bandits (CSB) that is an extension
of classic Multi-Armed Bandits (MAB) under Differential Privacy (DP) and
stronger Local Differential Privacy (LDP) setting. Since the server receives
more information from users in CSB, it usually causes additional dependence on
the dimension of data, which is a notorious side-effect for privacy preserving
learning. However for CSB under two common smoothness assumptions
\cite{kveton2015tight,chen2016combinatorial}, we show it is possible to remove
this side-effect. In detail, for $B_{\infty}$-bounded smooth CSB under either
$\varepsilon$-LDP or $\varepsilon$-DP, we prove the optimal regret bound is
$\Theta(\frac{mB^2_{\infty}\ln T } {\Delta\epsilon^2})$ or
$\tilde{\Theta}(\frac{mB^2_{\infty}\ln T} { \Delta\epsilon})$ respectively,
where $T$ is time period, $\Delta$ is the gap of rewards and $m$ is the number
of base arms, by proposing novel algorithms and matching lower bounds. For
$B_1$-bounded smooth CSB under $\varepsilon$-DP, we also prove the optimal
regret bound is $\tilde{\Theta}(\frac{mKB^2_1\ln T} {\Delta\epsilon})$ with
both upper bound and lower bound, where $K$ is the maximum number of feedback
in each round. All above results nearly match corresponding non-private optimal
rates, which imply there is no additional price for (locally) differentially
private CSB in above common settings.
