Dual Adaptivity: Universal Algorithms for Minimizing the Adaptive Regret of Convex Functions

TL;DR

This paper introduces universal dual adaptive algorithms for online convex optimization that automatically adapt to various function types and changing environments, minimizing adaptive regret without prior parameter knowledge.

Contribution

It proposes a meta-expert framework with second-order bounds and sleeping experts, enabling universal adaptation to multiple convex function types and environment dynamics.

Findings

Algorithms achieve minimized adaptive regret across different convex functions.

Framework handles switching between function types during rounds.

Extends to online composite optimization with universal adaptive regret minimization.

Abstract

To deal with changing environments, a new performance measure -- adaptive regret, defined as the maximum static regret over any interval, was proposed in online learning. Under the setting of online convex optimization, several algorithms have been successfully developed to minimize the adaptive regret. However, existing algorithms lack universality in the sense that they can only handle one type of convex functions and need apriori knowledge of parameters, which hinders their application in real-world scenarios. To address this limitation, this paper investigates universal algorithms with dual adaptivity, which automatically adapt to the property of functions (convex, exponentially concave, or strongly convex), as well as the nature of environments (stationary or changing). Specifically, we propose a meta-expert framework for dual adaptive algorithms, where multiple experts are created dynamically and aggregated by a meta-algorithm. The meta-algorithm is required to yield a second-order bound, which can accommodate unknown function types. We further incorporate the technique of sleeping experts to capture the changing environments. For the construction of experts, we introduce two strategies (increasing the number of experts or enhancing the capabilities of experts) to achieve universality. Theoretical analysis shows that our algorithms are able to minimize the adaptive regret for multiple types of convex functions simultaneously, and also allow the type of functions to switch between rounds. Moreover, we extend our meta-expert framework to online composite optimization, and develop a universal algorithm for minimizing the adaptive regret of composite functions.