In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Hölder continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.

在本文中，我们研究了求解具有一致凸目标的复合最小化问题的牛顿法三次正则化的迭代复杂度。我们引入了一定程度的二阶条件数的概念，并证明了具有正则化参数自适应估计的方法在非退化情况下的线性收敛速度。该算法在目标平滑部分的 Hölder 连续 Hessian 的一致凸目标函数的不同问题类之间自动实现最佳可能的全局复杂度界限。作为我们发展的副产品，