Online Convex Optimization Over Erdos-Renyi Random Networks

The work studies how node-to-node communications over an Erd\H{o}s-R\'enyi 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 Erd\H{o}s-R\'enyi 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., $\mathcal{O}(\sqrt{T}) $ and $\mathcal{O}(\ln(T)) $ regrets are established for convex and strongly convex losses with full gradient feedback and two-points information, respectively. For classical Erd\H{o}s-R\'enyi 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.