  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.
