(I could have a go at explaining newton's method, but I don't understand the arxiv article. In simple terms newton raphson iteration is an iterative method of finding the "zeros" of a function. You do it by evaluating the derivative of a function, and iteratively following that gradient until you find the zero point.)
Even in convex functions, the whole domain is not necessarily sufficiently close.
e.g. y = X^2 - 1 choose 'starting position' X = 0.
Newton raphson is used to find where f(x) = 0, not where the derivative f'(x) = 0.
For the function given above f(x) = 0 when x = 1 and x = -1. Not when the derivative is at 0 (when x = 0).
You'd be done in 0 steps to find the minimum of the function, or the point of 0 derivative, but that's not what you're looking for with newton raphson.
The article says "Unfortunately, Newton’s method is unstable: it works only for strongly convex problems and may diverge exponentially when initialized not very close to the optimum." And I was asking for an example or description of such a case.
You'll also note there are several comments in this discussion that mention the hessian/2nd order derivatives, and these all show up because this is being used in the context of optimization by looking for f'(x) = 0.
Example from Boyd.
https://math.stackexchange.com/questions/3408436/newton-rhap...