Online convex optimization over Erdos-Rényi random networks

The work studies how node-to-node communications over an Erdos-Rényi random network influence distributed online convex optimization, which is vital in solving large-scale machine learning in antagonistic or changing environments. At per step, each node (computing unit) makes a local decision, experiences a loss evaluated with a convex function, and communicates the decision with other nodes over a network. The node-to-node communications are described by the Erdos-Rényi rule, where independently each link takes place with a probability p over a prescribed connected graph. The objective is to minimize the system-wide loss accumulated over a finite time horizon. We consider standard distributed gradient descents with full gradients, one-point bandits and two-points bandits for convex and strongly convex losses, respectively. We establish how the regret bounds scale with respect to time horizon T, network size N, decision dimension d, and an algebraic network connectivity. The regret bounds scaling with respect to T match those obtained by state-of-the-art algorithms and fundamental limits in the corresponding centralized online optimization problems, e.g., O(^vT) and O(ln(T)) regrets are established for convex and strongly convex losses with full gradient feedback and two-points information, respectively. For classical Erdos-Rényi networks over all-to-all possible node communications, the regret scalings with respect to the probability p are analytically established, based on which the tradeoff between the communication overhead and computation accuracy is clearly demonstrated. Numerical studies have validated the theoretical findings.