  This paper considers the distributed optimization problem where each node of
a peer-to-peer network minimizes a finite sum of objective functions by
communicating with its neighboring nodes. In sharp contrast to the existing
literature where the fastest distributed algorithms converge either with a
global linear or a local superlinear rate, we propose a distributed adaptive
Newton (DAN) algorithm with a global quadratic convergence rate. Our key idea
lies in the design of a finite-time set-consensus method with Polyak's adaptive
stepsize. Moreover, we introduce a low-rank matrix approximation (LA) technique
to compress the innovation of Hessian matrix so that each node only needs to
transmit message of dimension $\mathcal{O}(p)$ (where $p$ is the dimension of
decision vectors) per iteration, which is essentially the same as that of
first-order methods. Nevertheless, the resulting DAN-LA converges to an optimal
solution with a global superlinear rate. Numerical experiments on logistic
regression problems are conducted to validate their advantages over existing
methods.
