Multitask Online Mirror Descent

Published: 19 Sept 2022, Last Modified: 28 Feb 2023Accepted by TMLREveryoneRevisionsBibTeX
Abstract: 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 $\sqrt{1 + \sigma^2(N-1)}\sqrt{T}$, where $\sigma^2$ 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 $\sigma^2 \le 1$, our method improves upon the $\sqrt{NT}$ 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.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: Added: - remark on strongly convex and exp-concave losses - remark on connection to dynamic regret - new regret bound when several agents are active at each time step - pseudo-codes
Assigned Action Editor: ~Roi_Livni1
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 198