  This work addresses decentralized online optimization in non-stationary
environments. A network of agents aim to track the minimizer of a global
time-varying convex function. The minimizer evolves according to a known
dynamics corrupted by an unknown, unstructured noise. At each time, the global
function can be cast as a sum of a finite number of local functions, each of
which is assigned to one agent in the network. Moreover, the local functions
become available to agents sequentially, and agents do not have a prior
knowledge of the future cost functions. Therefore, agents must communicate with
each other to build an online approximation of the global function. We propose
a decentralized variation of the celebrated Mirror Descent, developed by
Nemirovksi and Yudin. Using the notion of Bregman divergence in lieu of
Euclidean distance for projection, Mirror Descent has been shown to be a
powerful tool in large-scale optimization. Our algorithm builds on Mirror
Descent, while ensuring that agents perform a consensus step to follow the
global function and take into account the dynamics of the global minimizer. To
measure the performance of the proposed online algorithm, we compare it to its
offline counterpart, where the global functions are available a priori. The gap
between the two is called dynamic regret. We establish a regret bound that
scales inversely in the spectral gap of the network, and more notably it
represents the deviation of minimizer sequence with respect to the given
dynamics. We then show that our results subsume a number of results in
distributed optimization. We demonstrate the application of our method to
decentralized tracking of dynamic parameters and verify the results via
numerical experiments.
