A PDE-Based Analysis of the Symmetric Two-Armed Bernoulli Bandit

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 and the number of prediction periods approaches infinity, we obtain the leading order terms of the expected regret and pseudoregret for this problem by associating each of them with a solution of a linear parabolic partial differential equation. Our results improve upon the previously known results; specifically, we explicitly compute the leading order term of the optimal regret and pseudoregret in three different scaling regimes for the gap. Additionally, we obtain new non-asymptotic bounds for any given time horizon.