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.

中文翻译：

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张量方法的局部收敛


在本文中，我们研究了解决复合目标凸优化问题的高阶张量方法的局部收敛性。我们在光滑分量均匀凸性的假设下证明局部超线性收敛，具有 Lipschitz 连续高阶导数。函数值和最小次梯度范数均收敛。还讨论了凸和均匀凸情况下复合张量方法的全局复杂性界限。最后，我们展示了如何使用不精确的近端迭代来全局化方法的局部收敛。