  This paper considers the optimal adaptive allocation of measurement effort
for identifying the best among a finite set of options or designs. An
experimenter sequentially chooses designs to measure and observes noisy signals
of their quality with the goal of confidently identifying the best design after
a small number of measurements. This paper proposes three simple and intuitive
Bayesian algorithms for adaptively allocating measurement effort, and
formalizes a sense in which these seemingly naive rules are the best possible.
One proposal is top-two probability sampling, which computes the two designs
with the highest posterior probability of being optimal, and then randomizes to
select among these two. One is a variant of top-two sampling which considers
not only the probability a design is optimal, but the expected amount by which
its quality exceeds that of other designs. The final algorithm is a modified
version of Thompson sampling that is tailored for identifying the best design.
We prove that these simple algorithms satisfy a sharp optimality property. In a
frequentist setting where the true quality of the designs is fixed, one hopes
the posterior definitively identifies the optimal design, in the sense that
that the posterior probability assigned to the event that some other design is
optimal converges to zero as measurements are collected. We show that under the
proposed algorithms this convergence occurs at an exponential rate, and the
corresponding exponent is the best possible among all allocation

Machine Learning