Accelerated Regularized Newton Methods for Minimizing Composite Convex Functions

In this paper, we study accelerated Regularized Newton Methods for minimizing objectives formed as a sum of two functions: one is convex and twice differentiable with Holder-continuous Hessian, and the other is a simple closed convex function. For the case in which the Holder parameter $\nu \in [0,1]$ is known, we propose methods that make at most $\cal {O}\left(\frac {1} {\epsilon^{1/(2+\nu)}}\right)$ iterations to reduce the funcitonal residual below a given precision $\epsilon > 0$. For the general case, in which the $\nu$ is not known, we proposeo a universal method that ensures the same precision in at most $\cal{O} \left(\frac {1} {\epsilon^{2/[3(1+\nu)]}}\right)$ iterations.