Local convergence of tensor methods

TL;DR

This paper analyzes the local convergence properties of high-order tensor methods for convex optimization, establishing superlinear convergence under certain conditions and discussing how to extend local results globally.

Contribution

It provides theoretical justification for local superlinear convergence of tensor methods and introduces techniques to globalize these convergence results.

Findings

Superlinear convergence under uniform convexity

Global complexity bounds for tensor methods

Method to extend local convergence to global convergence

Abstract

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations.