Suboptimal Performance of the Bayes Optimal Algorithm in Frequentist Best Arm Identification

We consider the fixed-budget best arm identification problem with rewards following normal distributions. In this problem, the forecaster is given $K$ arms (or treatments) and $T$ time steps. The forecaster attempts to find the arm with the largest mean, via an adaptive experiment conducted using an algorithm. The algorithm's performance is evaluated by simple regret, reflecting the quality of the estimated best arm. While frequentist simple regret can decrease exponentially with respect to $T$, Bayesian simple regret decreases polynomially. This paper demonstrates that the Bayes optimal algorithm, which minimizes the Bayesian simple regret, does not yield an exponential decrease in simple regret under certain parameter settings. This contrasts with the numerous findings that suggest the asymptotic equivalence of Bayesian and frequentist approaches in fixed sampling regimes. Although the Bayes optimal algorithm is formulated as a recursive equation that is virtually impossible to compute exactly, we lay the groundwork for future research by introducing a novel concept termed the expected Bellman improvement.