  This paper deals with a network of computing agents aiming to solve an online
optimization problem in a distributed fashion, i.e., by means of local
computation and communication, without any central coordinator. We propose the
gradient tracking with adaptive momentum estimation (GTAdam) distributed
algorithm, which combines a gradient tracking mechanism with first and second
order momentum estimates of the gradient. The algorithm is analyzed in the
online setting for strongly convex cost functions with Lipschitz continuous
gradients. We provide an upper bound for the dynamic regret given by a term
related to the initial conditions and another term related to the temporal
variations of the objective functions. Moreover, a linear convergence rate is
guaranteed in the static setup. The algorithm is tested on a time-varying
classification problem, on a (moving) target localization problem, and in a
stochastic optimization setup from image classification. In these numerical
experiments from multi-agent learning, GTAdam outperforms state-of-the-art
distributed optimization methods.
