  This paper examines online distributed Alternating Direction Method of
Multipliers (ADMM). The goal is to distributively optimize a global objective
function over a network of decision makers under linear constraints. The global
objective function is composed of convex cost functions associated with each
agent. The local cost functions, on the other hand, are assumed to have been
decomposed into two distinct convex functions, one of which is revealed to the
decision makers over time and one known a priori. In addition, the agents must
achieve consensus on the global variable that relates to the private local
variables via linear constraints. In this work, we extend online ADMM to a
distributed setting based on dual-averaging and distributed gradient descent.
We then propose a performance metric for such online distributed algorithms and
explore the performance of the sequence of decisions generated by the algorithm
as compared with the best fixed decision in hindsight. This performance metric
is called the social regret. A sub-linear upper bound on the social regret of
the proposed algorithm is then obtained that underscores the role of the
underlying network topology and certain condition measures associated with the
linear constraints. The online distributed ADMM algorithm is then applied to a
formation acquisition problem demonstrating the application of the proposed
setup in distributed robotics.
