  In this paper we propose an efficient stochastic optimization algorithm to
search for Bayesian experimental designs such that the expected information
gain is maximized. The gradient of the expected information gain with respect
to experimental design parameters is given by a nested expectation, for which
the standard Monte Carlo method using a fixed number of inner samples yields a
biased estimator. In this paper, applying the idea of randomized multilevel
Monte Carlo (MLMC) methods, we introduce an unbiased Monte Carlo estimator for
the gradient of the expected information gain with finite expected squared
$\ell_2$-norm and finite expected computational cost per sample. Our unbiased
estimator can be combined well with stochastic gradient descent algorithms,
which results in our proposal of an optimization algorithm to search for an
optimal Bayesian experimental design. Numerical experiments confirm that our
proposed algorithm works well not only for a simple test problem but also for a
more realistic pharmacokinetic problem.
