  The expected improvement (EI) algorithm is a popular strategy for information
collection in optimization under uncertainty. The algorithm is widely known to
be too greedy, but nevertheless enjoys wide use due to its simplicity and
ability to handle uncertainty and noise in a coherent decision theoretic
framework. To provide rigorous insight into EI, we study its properties in a
simple setting of Bayesian optimization where the domain consists of a finite
grid of points. This is the so-called best-arm identification problem, where
the goal is to allocate measurement effort wisely to confidently identify the
best arm using a small number of measurements. In this framework, one can show
formally that EI is far from optimal. To overcome this shortcoming, we
introduce a simple modification of the expected improvement algorithm.
Surprisingly, this simple change results in an algorithm that is asymptotically
optimal for Gaussian best-arm identification problems, and provably outperforms
standard EI by an order of magnitude.
