  We consider unconstrained stochastic optimization problems with no available
gradient information. Such problems arise in settings from derivative-free
simulation optimization to reinforcement learning. We propose an adaptive
sampling quasi-Newton method where we estimate the gradients of a stochastic
function using finite differences within a common random number framework. We
develop modified versions of a norm test and an inner product quasi-Newton test
to control the sample sizes used in the stochastic approximations and provide
global convergence results to the neighborhood of the optimal solution. We
present numerical experiments on simulation optimization problems to illustrate
the performance of the proposed algorithm. When compared with classical
zeroth-order stochastic gradient methods, we observe that our strategies of
adapting the sample sizes significantly improve performance in terms of the
number of stochastic function evaluations required.
