In this paper, we use Proximal Cubic regularized Newton Methods (PCNM) to optimize the sum of a smooth convex function and a non-smooth convex function, where we use inexact gradient and Hessian, and an inexact subsolver for the cubic regularized second-order subproblem. We propose inexact variants of PCNM and accelerated PCNM respectively, and show that both variants can achieve the same convergence rate as in the exact case, provided that the errors in the inexact gradient, Hessian and subsolver decrease at appropriate rates. Meanwhile, in the online stochastic setting where data comes endlessly, we give the overall complexity of the proposed algorithms and show that they are as competitive as the stochastic gradient descent. Moreover, we give the overall complexity of the proposed algorithms in the finite-sum setting and show that it is as competitive as the state of the art variance reduced algorithms. Finally, we propose an efficient algorithm for the cubic regularized second-order subproblem, which can converge to an enough small neighborhood of the optimal solution in a superlinear rate.
We provide convergence rates for Krylov subspace solutions to the trust-region and cubic-regularized (nonconvex) quadratic problems. Such solutions may be efficiently computed by the Lanczos method and have long been used in practice. We prove error bounds of the form @math and @math , where @math is a condition number for the problem, and @math is the Krylov subspace order (number of Lanczos iterations). We also provide lower bounds showing that our analysis is sharp.