  We design differentially private algorithms for the bandit convex
optimization problem in the projection-free setting. This setting is important
whenever the decision set has a complex geometry, and access to it is done
efficiently only through a linear optimization oracle, hence Euclidean
projections are unavailable (e.g. matroid polytope, submodular base polytope).
This is the first differentially-private algorithm for projection-free bandit
optimization, and in fact our bound of $\widetilde{O}(T^{3/4})$ matches the
best known non-private projection-free algorithm (Garber-Kretzu, AISTATS `20)
and the best known private algorithm, even for the weaker setting when
projections are available (Smith-Thakurta, NeurIPS `13).
