A unified scheme to accelerate adaptive cubic regularization and gradient methods for convex optimization
In this paper we propose a unified two phase scheme for convex optimization to accelerate: (1) the adaptive cubic regularization methods with exact/inexact Hessian matrices, and (2) the adaptive gradient method, without any knowledge of the Lipschitz constants for the gradient or the Hessian. This is achieved by tuning the parameters used in the algorithm \textit{adaptively} in its process of progression, which can be viewed as a relaxation over the existing algorithms in the literature. Under the assumption that the sub problems can be solved approximately, we establish overall iteration complexity bounds for three newly proposed algorithms to obtain an \epsilon$ approximate solution. Specifically, we show that the adaptive cubic regularization methods with the exact/inexact Hessian matrix both achieve an iteration complexity in the order of $O\left( 1 \epsilon^{1/3} \right) which matches that of the original accelerated cubic regularization method presented in [Nesterov 2008 Accelerating] assuming the availability of the exact Hessian information and the Lipschitz constants, and that the sub problems are solved to optimality. Under the same two phase adaptive acceleration framework, the gradient method achieves an iteration complexity in the order $O\left( 1 \epsilon^{1/2} \right) which is known to be best possible (cf. Nesterov 2013 Introductory) Our numerical experiment results show a clear effect of acceleration displayed in the adaptive Newton method with cubic regularization on a set of regularized logistic regression instances.