Multitask Online Mirror Descent

06/04/2021

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by   Nicolò Cesa-Bianchi, et al.

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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 √(1 + σ^2(N-1))√(T), where σ^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, σ^2 ≤ 1, this improves upon the √(NT) bound obtained by running independent OMDs on each task. Our multitask extensions of Online Gradient Descent and Exponentiated Gradient, two important instances of OMD, are shown to enjoy closed-form updates, making them easy to use in practice. Finally, we provide numerical experiments on four real-world datasets which support our theoretical findings.