Optimal learning for sequential sampling with non-parametric beliefs

We propose a sequential learning policy for ranking and selection problems, where we use a non-parametric procedure for estimating the value of a policy. Our estimation approach aggregates over a set of kernel functions in order to achieve a more consistent estimator. Each element in the kernel estimation set uses a di erent bandwidth to achieve better aggregation. The nal estimate uses a weighting scheme with the inverse mean square errors of the kernel estimators as weights. This weighting scheme is shown to be optimal under independent kernel estimators. For choosing the measurement, we employ the knowledge gradient method, a myopic policy that relies on predictive distributions to calculate the optimal sampling point. Our method allows a setting where the beliefs are expected to be correlated but the correlation structure is unknown beforehand. This is an extension of the known knowledge gradient with correlated beliefs. Moreover, the proposed policy is asymptotically optimal.