  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.
