A Unified View of Regularized Dual Averaging and Mirror Descent with Implicit Updates

TL;DR

This paper unifies and analyzes several online convex optimization algorithms, revealing their equivalence and providing improved regret bounds, while also extending their applicability to more general objectives and implicit updates.

Contribution

It proves the equivalence of FTRL-Proximal, RDA, and composite mirror descent, and offers a unified analysis with improved regret bounds and extensions to implicit updates.

Findings

FTRL-Proximal outperforms FOBOS and RDA on real datasets.

Unified analysis yields regret bounds matching or surpassing previous results.

Implicit updates extend the algorithms' applicability to more general settings.

Abstract

We study three families of online convex optimization algorithms: follow-the-proximally-regularized-leader (FTRL-Proximal), regularized dual averaging (RDA), and composite-objective mirror descent. We first prove equivalence theorems that show all of these algorithms are instantiations of a general FTRL update. This provides theoretical insight on previous experimental observations. In particular, even though the FOBOS composite mirror descent algorithm handles L1 regularization explicitly, it has been observed that RDA is even more effective at producing sparsity. Our results demonstrate that FOBOS uses subgradient approximations to the L1 penalty from previous rounds, leading to less sparsity than RDA, which handles the cumulative penalty in closed form. The FTRL-Proximal algorithm can be seen as a hybrid of these two, and outperforms both on a large, real-world dataset. Our second contribution is a unified analysis which produces regret bounds that match (up to logarithmic terms) or improve the best previously known bounds. This analysis also extends these algorithms in two important ways: we support a more general type of composite objective and we analyze implicit updates, which replace the subgradient approximation of the current loss function with an exact optimization.