Newton's method is not for function approximation, but for optimization ("find the minimum"). Newton's method already has "good asymptotic time" (e.g. if you u

Newton's method is not for function approximation, but for optimization ("find the minimum"). Newton's method already has "good asymptotic time" (e.g. if you use it to compute a square root, you double the number of correct digits with each step!) - but only under certain conditions (e.g. "not too far from the solution"). This paper shows how to modify it so this sort of benefit is obtained under a wider range of conditions ("global"). There are other related methods with a similar goal, but some require more expensive computations. The method from the paper appears to fill a sweet spot in high-dimensional optimization by virtue of requiring "few" matrix inversions (or linear solves) while having very good convergence properties. When you are dealing with the kind of problem that this sort of method is applicable to, it appears to me that you might benefit from looking into using this method. But it's not a revolutionary improvement from what I can tell - there are many related methods, and chances are you could have used something similar for similar benefit already. And it's likely that someone already used more or less this method in practice. As the paper notes: > Our goal is twofold. On the one hand, we are interested in designing methods that are useful for applications and can be used without any change as black-box tools. On the other hand, we hope that our theory will serve as the basis for further study of globally-convergent second-order and quasi-Newton methods with superior rates. Although many of the ideas that we discuss in this paper are not new and have been observed by practitioners to help solve challenging optimization problems, our analysis, however, is the first of its kind. We hope that our theory will lead to appearance of new methods that are motivated by the theoretical insights of our work.