  We introduce a novel online multitask setting. In this setting each task is
partitioned into a sequence of segments that is unknown to the learner.
Associated with each segment is a hypothesis from some hypothesis class. We
give algorithms that are designed to exploit the scenario where there are many
such segments but significantly fewer associated hypotheses. We prove regret
bounds that hold for any segmentation of the tasks and any association of
hypotheses to the segments. In the single-task setting this is equivalent to
switching with long-term memory in the sense of [Bousquet and Warmuth; 2003].
We provide an algorithm that predicts on each trial in time linear in the
number of hypotheses when the hypothesis class is finite. We also consider
infinite hypothesis classes from reproducing kernel Hilbert spaces for which we
give an algorithm whose per trial time complexity is cubic in the number of
cumulative trials. In the single-task special case this is the first example of
an efficient regret-bounded switching algorithm with long-term memory for a
non-parametric hypothesis class.
