Optimal Newton-type methods for nonconvex smooth optimization problems

We consider a general class of second-order iterations for unconstrained optimization that includes regularization and trust-region variants of Newton's method. For each method in this class, we exhibit a smooth, bounded-below objective function, whose gradient is globally Lipschitz continuous within an open convex set containing any iterates encountered and whose Hessian is $\alpha-$Holder continuous (for given $\alpha\in [0,1]$) on the path of the iterates, for which the method in question takes at least $\epsilon^{-(2+\alpha)/(1+\alpha)}$ function-evaluations to generate a first iterate whose gradient is smaller than $\epsilon$ in norm. This provides a lower bound on the evaluation complexity of second-order methods in our class when applied to smooth problems satisfying our assumptions. Furthermore, for $\alpha=1$, this lower bound is of the same order in $\epsilon$ as the upper bound on the evaluation complexity of cubic regularization, thus implying cubic regularization has optimal worst-case evaluation complexity within our class of second-order methods.