  We consider two formulations for distributed optimization wherein $N$ agents
in a generic connected network solve a problem of common interest: distributed
personalized optimization and consensus optimization. A new method termed DINAS
(Distributed Inexact Newton method with Adaptive Stepsize) is proposed. DINAS
employs large adaptively computed step-sizes, requires a reduced global
parameters knowledge with respect to existing alternatives, and can operate
without any local Hessian inverse calculations nor Hessian communications. When
solving personalized distributed learning formulations, DINAS achieves
quadratic convergence with respect to computational cost and linear convergence
with respect to communication cost, the latter rate being independent of the
local functions condition numbers or of the network topology. When solving
consensus optimization problems, DINAS is shown to converge to the global
solution. Extensive numerical experiments demonstrate significant improvements
of DINAS over existing alternatives. As a result of independent interest, we
provide for the first time convergence analysis of the Newton method with the
adaptive Polyak's step-size when the Newton direction is computed inexactly in
centralized environment.
