  Second-order methods, which utilize gradients as well as Hessians to optimize
a given function, are of major importance in mathematical optimization. In this
work, we prove tight bounds on the oracle complexity of such methods for smooth
convex functions, or equivalently, the worst-case number of iterations required
to optimize such functions to a given accuracy. In particular, these bounds
indicate when such methods can or cannot improve on gradient-based methods,
whose oracle complexity is much better understood. We also provide
generalizations of our results to higher-order methods.
