Adaptive Online Learning in Dynamic Environments

TL;DR

This paper introduces Ader, an adaptive online learning algorithm that achieves optimal dynamic regret bounds in changing environments, improving efficiency and extending to models with available dynamical sequences.

Contribution

The paper proposes Ader, a novel adaptive online learning method that attains the optimal dynamic regret bound and reduces gradient evaluations, also extending to dynamical models.

Findings

Ader achieves the optimal $O( ext{sqrt}(T(1+P_T)))$ dynamic regret.

The improved Ader reduces gradient evaluations from $O( ext{log} T)$ to 1.

Ader can incorporate sequences of dynamical models for better adaptation.

Abstract

In this paper, we study online convex optimization in dynamic environments, and aim to bound the dynamic regret with respect to any sequence of comparators. Existing work have shown that online gradient descent enjoys an $O(\sqrt{T}(1+P_T))$ dynamic regret, where $T$ is the number of iterations and $P_T$ is the path-length of the comparator sequence. However, this result is unsatisfactory, as there exists a large gap from the $\Omega(\sqrt{T(1+P_T)})$ lower bound established in our paper. To address this limitation, we develop a novel online method, namely adaptive learning for dynamic environment (Ader), which achieves an optimal $O(\sqrt{T(1+P_T)})$ dynamic regret. The basic idea is to maintain a set of experts, each attaining an optimal dynamic regret for a specific path-length, and combines them with an expert-tracking algorithm. Furthermore, we propose an improved Ader based on the surrogate loss, and in this way the number of gradient evaluations per round is reduced from $O(\log T)$ to $1$. Finally, we extend Ader to the setting that a sequence of dynamical models is available to characterize the comparators.