  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 a priori, whose parameters are to be learned along with
the evolution of the algorithm. The algorithm is composed of two intertwined
components: (i) a dynamic gradient tracking scheme for finding local solution
estimates and (ii) 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.
