On the Optimal Combination of Tensor Optimization Methods
Abstract
We consider the minimization problem of a sum of a number of functions having Lipshitz $p$th order derivatives with different Lipschitz constants. In this case, to accelerate optimization, we propose a general framework allowing to obtain nearoptimal oracle complexity for each function in the sum separately, meaning, in particular, that the oracle for a function with lower Lipschitz constant is called a smaller number of times. As a building block, we extend the current theory of tensor methods and show how to generalize nearoptimal tensor methods to work with inexact tensor step. Further, we investigate the situation when the functions in the sum have Lipschitz derivatives of a different order. For this situation, we propose a generic way to separate the oracle complexity between the parts of the sum. Our method is not optimal, which leads to an open problem of the optimal combination of oracles of a different order.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2002.01004
 arXiv:
 arXiv:2002.01004
 Bibcode:
 2020arXiv200201004K
 Keywords:

 Mathematics  Optimization and Control