Accelerating the cubic regularization of Newton's method on convex problems

In this paper we propose an accelerated version of the cubic regularization of Newton's method ( Nesterov and Polyak, in Math Program 108( 1): 177 - 205, 2006). The original version, used for minimizing a convex function with Lipschitz- continuous Hessian, guarantees a global rate of convergence of order O(1/k(2)), where k is the iteration counter. Our modified version converges for the same problem class with order O(1/k(3)) , keeping the complexity of each iteration unchanged. We study the complexity of both schemes on different classes of convex problems. In particular, we argue that for the second- order schemes, the class of non- degenerate problems is different from the standard class.