Multitask Online Mirror Descent

We introduce and analyze MT-OMD, a multitask generalization of Online Mirror Descent (OMD) which operates by sharing updates between tasks. We prove that the regret of MT-OMD is of order p 1 + 2(N − 1)p T, where 2 is the task variance according to the geometry induced by the regularizer, N is the number of tasks, and T is the time horizon. Whenever tasks are similar, that is 2 1, our method improves upon the p NT bound obtained by running independent OMDs on each task. We further provide a matching lower bound, and show that our multitask extensions of Online Gradient Descent and Exponentiated Gradient, two major instances of OMD, enjoy closed-form updates, making them easy to use in practice. Finally, we present experiments which support our theoretical findings.