Distributed Personalized Gradient Tracking With Convex Parametric Models
Ivano Notarnicola, Andrea Simonetto, Francesco Farina, Giuseppe Notarstefano
Abstract
We present a distributed optimization algorithm for solving online personalized optimization problems over a network of computing and communicating nodes, each of which linked to a specific user. The local objective functions are assumed to have a composite structure and to consist of a known time-varying (engineering) part and an unknown (user-specific) part. Regarding the unknown part, it is assumed to have a known parametric (e.g., quadratic) structure <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> , whose parameters are to be learned <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">along with</i> the evolution of the algorithm. The algorithm is composed of two intertwined components: 1) a dynamic gradient tracking scheme for finding local solution estimates and 2) a recursive least squares scheme for estimating the unknown parameters via user’s noisy feedback on the local solution estimates. The algorithm is shown to exhibit a bounded regret under suitable assumptions. Finally, a numerical example corroborates the theoretical analysis.