A program for sequential allocation of three Bernoulli populations

Abstract

We describe a program for optimizing and analyzing sequential allocation problems involving three Bernoulli populations. Previous researchers had considered this problem computationally intractable, and we know of no prior exact optimizations for such problems, even for very small sample sizes. Despite this, our program is currently able to solve problems of size 200 or more by using a parallel computer, and problems of size 100 on a workstation. We describe the program and the techniques used to enable it to scale to large sample sizes. As an illustrative example, the program is used to create an adaptive sampling procedure that is the optimal solution to a 3-arm bandit problem. The bandit procedure is then compared to two other allocation procedures along various metrics. We also indicate extensions of the program that enable it to solve a v ariety of related problems.

Scalable Algorithms for Adaptive Statistical Designs

“…Our goal is to reduce computational concerns to the point where they are not a key issue in the selection of appropriate designs. This paper has concentrated on the parallel computational aspects of this work, while other papers analyze the statistical and application impact [4][5][6].…”

“…Determining the range of s3 values assigned to each processor is nontrivial, because the number of states corresponding to a given value of s3 grows as (m − s3) 4 . Thus, simply assigning all processors an equal number of s3 values would result in massive load imbalance and poor scaling.…”

Parallel Computing Techniques

“…Hardwick et al (1999) used this technique to solve sequential allocation problems involving three Bernoulli populations. Christofides et al (1999) applied it to the problem of discretizing multidimensional probability functions.…”

Phase I Clinical Trials

“…Leung and Wang (2002) point out that the CRM is a myopic strategy and might not be globally optimal. A globally optimal strategy requires comparing all possible sets of actions that could be taken and this remains computationally formidable for designs having more than three dose levels (Hardwick, Oehmke and Stout, 1999).…”