Many problems in machine learning can be abstracted to the sequential design task of finding the minimum of an unknown erratic and possibly discontinuous function on the basis of noisy measurements. In the present work, it is presumed that there is no penalty for bad choices during the experimental stage, and at some time, not known to the decision maker, or under his control, the experimentation will be terminated, and the decision maker mill need to specify the point considered best, on the basis of the experimentation. In this note, we seek the best trade-off between i) acquiring new test points, and ii) retesting at points previously selected so as to improve the estimates of relative performance. The algorithm is shown to achieve a performance standard described herein. This decision setting would seem natural for function minimization in a simulation context or for tuning up a production process prior to putting it into service.