Best-of-three-worlds Analysis for Linear Bandits with Follow-the-regularized-leader Algorithm
This provides a simpler solution for optimizing linear bandits in stochastic, adversarial, and mixed environments, which is incremental but improves algorithmic design.
The paper tackles the linear bandit problem by proving that the Follow-the-regularized-leader (FTRL) algorithm with a negative entropy regularizer achieves best-of-three-world results, matching or nearly matching the regret bounds of prior detect-switch algorithms with a simpler design.
The linear bandit problem has been studied for many years in both stochastic and adversarial settings. Designing an algorithm that can optimize the environment without knowing the loss type attracts lots of interest. \citet{LeeLWZ021} propose an algorithm that actively detects the loss type and then switches between different algorithms specially designed for specific settings. However, such an approach requires meticulous designs to perform well in all environments. Follow-the-regularized-leader (FTRL) is another type of popular algorithm that can adapt to different environments. This algorithm is of simple design and the regret bounds are shown to be optimal in traditional multi-armed bandit problems compared with the detect-switch type. Designing an FTRL-type algorithm for linear bandits is an important question that has been open for a long time. In this paper, we prove that the FTRL algorithm with a negative entropy regularizer can achieve the best-of-three-world results for the linear bandit problem. Our regret bounds achieve the same or nearly the same order as the previous detect-switch type algorithm but with a much simpler algorithmic design.