  We study learning algorithms for the classical Markovian bandit problem with
discount. We explain how to adapt PSRL [24] and UCRL2 [2] to exploit the
problem structure. These variants are called MB-PSRL and MB-UCRL2. While the
regret bound and runtime of vanilla implementations of PSRL and UCRL2 are
exponential in the number of bandits, we show that the episodic regret of
MB-PSRL and MB-UCRL2 is $\tilde{O}(S\sqrt{nK})$ where $K$ is the number of
episodes, $n$ is the number of bandits and $S$ is the number of states of each
bandit (the exact bound in S, n and K is given in the paper). Up to a factor
$\sqrt S$, this matches the lower bound of $\Omega(\sqrt{SnK})$ that we also
derive in the paper. MB-PSRL is also computationally efficient: its runtime is
linear in the number of bandits. We further show that this linear runtime
cannot be achieved by adapting classical non-Bayesian algorithms such as UCRL2
or UCBVI to Markovian bandit problems. Finally, we perform numerical
experiments that confirm that MB-PSRL outperforms other existing algorithms in
practice, both in terms of regret and of computation time.
