Personalized Online Optimization of Networked Systems via Gaussian Processes

Abstract

I. Abstract We consider a time-varying optimization problem associated with a physical system or a networked system with human-in-the-loop. The optimization problem features a cost comprising a known time-varying function capturing performance metrics of the system, and unknown functions associated with peo-ple/users interacting with such a system. In particular, the latter may capture discomfort, dissatisfaction, or sense of safety perceived by the users. Based on this optimization model, we investigate the development of learning-based online optimization methods with two main objectives: 1) continuously learn each of the cost functions based on feedback from the users; and, 2) utilize the learned functions to solve the target optimization problem in an online fashion. In particular, we focus on the synthesis of first-order online methods where feedback from the user is leveraged to learn the unknown functions via Gaussian Processes (GPs) or shape-constrained GPs concurrently with the execution of the online algorithm; and, measurements of the output of the system are used to estimate the gradient of the known engineering cost. The performance of the online algorithm is analyzed using metrics such as the tracking error and the dynamic regret. We also provide extensions to cases where a network resource is shared among different users, and develop an online consensus-based learning framework that strikes a balance between operational objectives and the perceived comfort and satisfaction of the various users. The upshot of the proposed method is that it learns the cost functions of each individual user, instead of relying on generic synthetic comfort or satisfaction models. As an application example, the proposed algorithms are applied to solve a real-time demand response problem in power grids. In this setting, the proposed online algorithms enable aggregations of distributed energy resources to provide services to the grid (in the form of, e.g., frequency and voltage support) while accounting for the comfort and satisfaction of the users. As an application example, the proposed algorithms are applied to solve a real-time demand response problem in power grids. In this setting, the proposed online algorithms enable aggregations of distributed energy resources to provide services to the grid (in the form of, e.g., frequency and voltage support) while accounting for the comfort and satisfaction of the users.