Multitask Online Mirror Descent

TL;DR

This paper introduces MT-OMD, a multitask online mirror descent algorithm that shares information across tasks, improving regret bounds when tasks are similar, with practical algorithms and experimental validation.

Contribution

The paper develops a multitask extension of Online Mirror Descent with theoretical regret bounds, closed-form updates, and empirical evidence of improved performance for similar tasks.

Findings

Regret bound of order √(1 + σ²(N-1))√T

Improved bounds when tasks are similar (σ² ≤ 1)

Algorithms with closed-form updates for practical use

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.