  Spurred by the enthusiasm surrounding the "Big Data" paradigm, the
mathematical and algorithmic tools of online optimization have found widespread
use in problems where the trade-off between data exploration and exploitation
plays a predominant role. This trade-off is of particular importance to several
branches and applications of signal processing, such as data mining,
statistical inference, multimedia indexing and wireless communications (to name
but a few). With this in mind, the aim of this tutorial paper is to provide a
gentle introduction to online optimization and learning algorithms that are
asymptotically optimal in hindsight - i.e., they approach the performance of a
virtual algorithm with unlimited computational power and full knowledge of the
future, a property known as no-regret. Particular attention is devoted to
identifying the algorithms' theoretical performance guarantees and to establish
links with classic optimization paradigms (both static and stochastic). To
allow a better understanding of this toolbox, we provide several examples
throughout the tutorial ranging from metric learning to wireless resource
allocation problems.
