  In this paper we develop the first algorithms for online submodular
minimization that preserve differential privacy under full information feedback
and bandit feedback. A sequence of $T$ submodular functions over a collection
of $n$ elements arrive online, and at each timestep the algorithm must choose a
subset of $[n]$ before seeing the function. The algorithm incurs a cost equal
to the function evaluated on the chosen set, and seeks to choose a sequence of
sets that achieves low expected regret.
  Our first result is in the full information setting, where the algorithm can
observe the entire function after making its decision at each timestep. We give
an algorithm in this setting that is $\epsilon$-differentially private and
achieves expected regret
$\tilde{O}\left(\frac{n^{3/2}\sqrt{T}}{\epsilon}\right)$. This algorithm works
by relaxing submodular function to a convex function using the Lovasz
extension, and then simulating an algorithm for differentially private online
convex optimization.
  Our second result is in the bandit setting, where the algorithm can only see
the cost incurred by its chosen set, and does not have access to the entire
function. This setting is significantly more challenging because the algorithm
does not receive enough information to compute the Lovasz extension or its
subgradients. Instead, we construct an unbiased estimate using a single-point
estimation, and then simulate private online convex optimization using this
estimate. Our algorithm using bandit feedback is $\epsilon$-differentially
private and achieves expected regret
$\tilde{O}\left(\frac{n^{3/2}T^{3/4}}{\epsilon}\right)$.
