  We propose a communication efficient quasi-Newton method for large-scale
multi-agent convex composite optimization. We assume the setting of a network
of agents that cooperatively solve a global minimization problem with strongly
convex local cost functions augmented with a non-smooth convex regularizer. By
introducing consensus variables, we obtain a block-diagonal Hessian and thus
eliminate the need for additional communication when approximating the
objective curvature information. Moreover, we reduce computational costs of
existing primal-dual quasi-Newton methods from $\mathcal{O}(d^3)$ to
$\mathcal{O}(cd)$ by storing $c$ pairs of vectors of dimension $d$. An
asynchronous implementation is presented that removes the need for
coordination. Global linear convergence rate in expectation is established, and
we demonstrate the merit of our algorithm numerically with real datasets.
