  The knowledge gradient (KG) algorithm is a popular and effective algorithm
for the best arm identification (BAI) problem. Due to the complex calculation
of KG, theoretical analysis of this algorithm is difficult, and existing
results are mostly about the asymptotic performance of it, e.g., consistency,
asymptotic sample allocation, etc. In this research, we present new theoretical
results about the finite-time performance of the KG algorithm. Under
independent and normally distributed rewards, we derive bounds for the sample
allocation of the algorithm. With these bounds, existing asymptotic results
become simple corollaries. Furthermore, we derive upper and lower bounds for
the probability of error and simple regret of the algorithm, and show the
performance of the algorithm for the multi-armed bandit (MAB) problem. These
developments not only extend the existing analysis of the KG algorithm, but can
also be used to analyze other improvement-based algorithms. Last, we use
numerical experiments to compare the bounds we derive and the performance of
the KG algorithm.
