Non-Adaptive Stochastic Score Classification and Explainable Halfspace Evaluation

We consider the stochastic score classification problem. There are several binary tests, where each test $i$ is associated with a probability $p_i$ of being positive and a cost $c_i$. The score of an outcome is a weighted sum of all positive tests, and the range of possible scores is partitioned into intervals corresponding to different classes. The goal is to perform tests sequentially (and possibly adaptively) so as to identify the class at the minimum expected cost. We provide the first constant-factor approximation algorithm for this problem, which improves over the previously-known logarithmic approximation ratio. Moreover, our algorithm is $non$ $adaptive$: it just involves performing tests in a $fixed$ order until the class is identified. Our approach also extends to the $d$-dimensional score classification problem and the "explainable" stochastic halfspace evaluation problem (where we want to evaluate some function on $d$ halfspaces). We obtain an $O(d^2\log d)$-approximation algorithm for both these extensions. Finally, we perform computational experiments that demonstrate the practical performance of our algorithm for score classification. We observe that, for most instances, the cost of our algorithm is within $50\%$ of an information-theoretic lower bound on the optimal value.