Learning Proximal Operators with Gaussian Processes

Several distributed-optimization setups involve a group of agents coordinated by a central entity (coordinator), altogether operating in a collaborative framework. In such environments, it is often common that the agents solve proximal minimization problems that are hidden from the central coordinator. We develop a scheme for reducing communication between the agents and the coordinator based on learning the agents' proximal operators with Gaussian Processes. The scheme learns a Gaussian Process model of the proximal operator associated with each agent from historical data collected at past query points. These models enable probabilistic predictions of the solutions to the local proximal minimization problems. Based on the predictive variance returned by a model, representative of its prediction confidence, an adaptive mechanism allows the coordinator to decide whether to communicate with the associated agent. The accuracy of the Gaussian Process models results in significant communication reduction, as demonstrated in simulations of a distributed optimal power dispatch application.