Implementable tensor methods in unconstrained convex optimization

In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330-348, 2017; Lu et al. in SIOPT 28(1):333354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level O(1/k(4)), where k is the number of iterations. This is very close to the lower bound of the order O(1/k(5)), which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.

Automated summary

In this paper, new tensor methods are developed for unconstrained convex optimization. These methods solve an auxiliary problem of minimizing convex multivariate polynomial at each iteration. The simplest scheme and its accelerated version are analyzed, with comparison of convergence rates to lower complexity bounds for corresponding problem classes. Furthermore, an efficient technique based on the relative smoothness condition is suggested for solving the auxiliary problem in third-order methods, resulting in fast implementation and convergence rates close to the lower bound.