We consider the problem of minimizing a convex function under convex constraints, where some or all of the decision variables are constrained to be integer, with access to first-order oracles for the objective function and separation oracles for the constraints. We focus on the information complexity of the problem, i.e., the minimum number of oracle calls to solve the problem to a desired accuracy. We will present nearly tight upper and lower bounds for the problem, extending classical results from continuous convex optimization. We also initiate the study of, and obtain the first set of results on, information complexity under oracles that only reveal partial first-order information, e.g., where one can only make a binary query on the function value or (sub)gradient at a given point. Our bounds in these new settings show that these oracles are provably less informative than full first-order oracles for the purpose of optimization. We will close the talk with some open questions.