I am keen to know about the literature landscape for distributed convex optimization methods which use second order information like the Newton step. This is as such a less evolved area compared to first order methods, which have had most of the focus from the research community for large scale optimization (e.g., ADMM etc).
But I see papers like "A distributed newton method for network utility maximization" by Ermin Wei et al. I want to have an understanding of whether these methods have been extended to other areas like Machine Learning and what is the problem size that a state of the art distributed Newton method can handle?
The paper that you mentioned by E. Wei is proposed for network utility maximization (NUM) problems, but recently a new algorithm called Network Newton is proposed that is an approximation of distributed Newton method for solving general distributed convex optimization problems. The authors have also considered the application of algorithm for logistic regression.