Adaptive and Efficient Algorithms for Tracking the Best Expert

TL;DR

This paper introduces two adaptive algorithms for prediction with expert advice in dynamic environments, achieving improved data-dependent tracking regret bounds and extending to online matrix prediction.

Contribution

The paper presents novel adaptive algorithms with data-dependent bounds for tracking regret, including the first for online matrix prediction.

Findings

Second-order tracking regret bound achieved

Path-length bound for slowly moving environments

Extension to online matrix prediction with data-dependent bounds

Abstract

In this paper, we consider the problem of prediction with expert advice in dynamic environments. We choose tracking regret as the performance metric and develop two adaptive and efficient algorithms with data-dependent tracking regret bounds. The first algorithm achieves a second-order tracking regret bound, which improves existing first-order bounds. The second algorithm enjoys a path-length bound, which is generally not comparable to the second-order bound but offers advantages in slowly moving environments. Both algorithms are developed under the online mirror descent framework and draw inspiration from existing algorithms that attain data-dependent bounds of static regret. The key idea is to use a clipped simplex in the updating step of online mirror descent. Finally, we extend our algorithms and analysis to online matrix prediction and provide the first data-dependent tracking regret bound for this problem.