A Communication Efficient Quasi-Newton Method for Large-Scale Distributed Multi-Agent Optimization

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 $\mathcal{O}\left( {{d^3}} \right)$ to $\mathcal{O}\left( {cd} \right)$ 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.