  We incorporate future information in the form of the estimated value of
future gradients in online convex optimization. This is motivated by demand
response in power systems, where forecasts about the current round, e.g., the
weather or the loads' behavior, can be used to improve on predictions made with
only past observations. Specifically, we introduce an additional predictive
step that follows the standard online convex optimization step when certain
conditions on the estimated gradient and descent direction are met. We show
that under these conditions and without any assumptions on the predictability
of the environment, the predictive update strictly improves on the performance
of the standard update. We give two types of predictive update for various
family of loss functions. We provide a regret bound for each of our predictive
online convex optimization algorithms. Finally, we apply our framework to an
example based on demand response which demonstrates its superior performance to
a standard online convex optimization algorithm.
