  Bandit-style algorithms have been studied extensively in stochastic and
adversarial settings. Such algorithms have been shown to be useful in
multiplayer settings, e.g. to solve the wireless network selection problem,
which can be formulated as an adversarial bandit problem. A leading bandit
algorithm for the adversarial setting is EXP3. However, network behavior is
often repetitive, where user density and network behavior follow regular
patterns. Bandit algorithms, like EXP3, fail to provide good guarantees for
periodic behaviors. A major reason is that these algorithms compete against
fixed-action policies, which is ineffective in a periodic setting.
  In this paper, we define a periodic bandit setting, and periodic regret as a
better performance measure for this type of setting. Instead of comparing an
algorithm's performance to fixed-action policies, we aim to be competitive with
policies that play arms under some set of possible periodic patterns $F$ (for
example, all possible periodic functions with periods $1,2,\cdots,P$). We
propose Periodic EXP4, a computationally efficient variant of the EXP4
algorithm for periodic settings. With $K$ arms, $T$ time steps, and where each
periodic pattern in $F$ is of length at most $P$, we show that the periodic
regret obtained by Periodic EXP4 is at most $O\big(\sqrt{PKT \log K + KT \log
|F|}\big)$. We also prove a lower bound of $\Omega\big(\sqrt{PKT + KT
\frac{\log |F|}{\log K}} \big)$ for the periodic setting, showing that this is
optimal within log-factors. As an example, we focus on the wireless network
selection problem. Through simulation, we show that Periodic EXP4 learns the
periodic pattern over time, adapts to changes in a dynamic environment, and far
outperforms EXP3.
