A Hill - Climbing Approach for OptimizingClassi cation

We consider the problem of minimizing the expected cost of determining the correct value of a binary-valued function when it is costly to inspect the values of its arguments. This type of problem arises in distributed computing, in the design of diagnostic expert systems, in reliability analysis of multi-component systems, and in many other applications. Any feasible solution to the problem can be described by a sequential inspection procedure which is usually represented by a binary classiication tree. In this paper, we propose an eecient hill-climbing algorithm to search for the optimal or near-optimal classiication trees. Computational results show that the hill-climbing approach was able to nd optimal solutions for 95% of the cases tested. The following minimum expected-cost classiication problem arises in many logical inference , reliability testing, pattern recognition, and statistical classiication applications. Suppose that X = (x 1 ; ; x n) is a binary pattern vector with n components and that (X) is a function mapping pattern vectors into f0; 1g. If pattern vectors are generated randomly according to a probability n-vector (p 1 ; ; p n), where p i is the probability that x i = 1 and 1?p i is the probability that x i = 0; and if component x i can be inspected at a cost c i to determine its true value (0 or 1), then the minimum expected-cost classiication problem is to nd an adaptive sequential inspection strategy that minimizes the expected cost of determining (X). An adaptive sequential inspection strategy can be described as a binary classiication tree or decision tree. The leaf nodes of the tree represent the results of the inspection and are labeled by the corresponding values of the () function (instead of \0" or \1", we shall use \F" or \W" to label the leaf nodes to avoid confusion with the component indices). The non-leaf nodes represent inspection actions and are labeled by indices of the component to be inspected. Each non-leaf node v has exactly two sons called 0-son and 1-son of v, and v is called the father of its sons. Suppose node v is labeled i, its 0-son v 0 has label j and its 1-son v 1 has label k, then component j (or k) will be inspected next if component i is found to be equal to 0 (or 1). The rst component to be inspected is the root …