  We consider the online version of the isotonic regression problem. Given a
set of linearly ordered points (e.g., on the real line), the learner must
predict labels sequentially at adversarially chosen positions and is evaluated
by her total squared loss compared against the best isotonic (non-decreasing)
function in hindsight. We survey several standard online learning algorithms
and show that none of them achieve the optimal regret exponent; in fact, most
of them (including Online Gradient Descent, Follow the Leader and Exponential
Weights) incur linear regret. We then prove that the Exponential Weights
algorithm played over a covering net of isotonic functions has a regret bounded
by $O\big(T^{1/3} \log^{2/3}(T)\big)$ and present a matching $\Omega(T^{1/3})$
lower bound on regret. We provide a computationally efficient version of this
algorithm. We also analyze the noise-free case, in which the revealed labels
are isotonic, and show that the bound can be improved to $O(\log T)$ or even to
$O(1)$ (when the labels are revealed in isotonic order). Finally, we extend the
analysis beyond squared loss and give bounds for entropic loss and absolute
loss.
