  This work addresses a version of the two-armed Bernoulli bandit problem where
the sum of the means of the arms is one (the symmetric two-armed Bernoulli
bandit). In a regime where the gap between these means goes to zero as the
number of prediction periods approaches infinity, i.e., the difficulty of
detecting the gap increases as the sample size increases, we obtain the leading
order terms of the minmax optimal regret and pseudoregret for this problem by
associating each of them with a solution of a linear heat equation. Our results
improve upon the previously known results; specifically, we explicitly compute
these leading order terms in three different scaling regimes for the gap.
Additionally, we obtain new non-asymptotic bounds for any given time horizon.
Although optimal player strategies are not known for more general bandit
problems, there is significant interest in considering how regret accumulates
under specific player strategies, even when they are not known to be optimal.
We expect that the methods of this paper should be useful in settings of that
type.

Machine Learning;Analysis of PDEs;Machine Learning