  The problem of minimizing a sum of local convex objective functions over a
networked system captures many important applications and has received much
attention in the distributed optimization field. Most of existing work focuses
on development of fast distributed algorithms under the presence of a central
clock. The only known algorithms with convergence guarantees for this problem
in asynchronous setup could achieve either sublinear rate under totally
asynchronous setting or linear rate under partially asynchronous setting (with
bounded delay). In this work, we built upon existing literature to develop and
analyze an asynchronous Newton based approach for solving a penalized version
of the problem. We show that this algorithm converges almost surely with global
linear rate and local superlinear rate in expectation. Numerical studies
confirm superior performance against other existing asynchronous methods.
