Consider a gambler who repeatedly bets a fraction of her wealth on the outcome of an adversarially chosen coin, and suppose we ask only that her wealth grow at a certain rate relative to that of the best fixed betting fraction in hindsight. This talk develops the observation, due to Orabona and Pál, that this elementary game is not merely an analogy for online learning but is equivalent to it: a betting algorithm with a suitable wealth guarantee mechanically yields parameter-free algorithms for online linear optimization over Hilbert spaces and for learning with expert advice, with regret bounds that adapt to the unknown comparator without oracle tuning of a learning rate.
