  Online optimization has emerged as powerful tool in large scale optimization.
In this pa- per, we introduce efficient online optimization algorithms based on
the alternating direction method (ADM), which can solve online convex
optimization under linear constraints where the objective could be non-smooth.
We introduce new proof techniques for ADM in the batch setting, which yields a
O(1/T) convergence rate for ADM and forms the basis for regret anal- ysis in
the online setting. We consider two scenarios in the online setting, based on
whether an additional Bregman divergence is needed or not. In both settings, we
establish regret bounds for both the objective function as well as constraints
violation for general and strongly convex functions. We also consider inexact
ADM updates where certain terms are linearized to yield efficient updates and
show the stochastic convergence rates. In addition, we briefly discuss that
online ADM can be used as projection- free online learning algorithm in some
scenarios. Preliminary results are presented to illustrate the performance of
the proposed algorithms.
