Non-Adaptive Stochastic Score Classification and Explainable Halfspace Evaluation

TL;DR

This paper introduces a non-adaptive, constant-factor approximation algorithm for evaluating complex score classification functions and halfspaces in sequential testing, improving theoretical bounds and demonstrating practical efficiency.

Contribution

It provides the first constant factor approximation for general score classification functions and extends the approach to halfspace evaluation and batched testing scenarios.

Findings

Algorithm achieves within 50% of the lower bound in experiments.

Provides an $O(d^2 \log d)$-approximation for halfspace evaluation.

Improves approximation bounds from logarithmic to constant factor for complex functions.

Abstract

Sequential testing problems involve a complex system with several components, each of which is "working" with some independent probability. The outcome of each component can be determined by performing a test, which incurs some cost. The overall system status is given by a function $f$ of the outcomes of its components. The goal is to evaluate this function $f$ by performing tests at the minimum expected cost. While there has been extensive prior work on this topic, provable approximation bounds are mainly limited to simple functions like ``k-out-of-n'' and halfspaces. We consider significantly more general "score classification" functions, and provide the first constant factor approximation algorithm (improving over a previous logarithmic approximation ratio). Moreover, our policy is non adaptive: it just involves performing tests in an a priori fixed order. We also consider the related halfspace evaluation problem, where we want to evaluate some function on $d$ halfspaces (e.g., intersection of halfspaces). We show that our approach provides an $O(d^2\log d)$-approximation algorithm for this problem. Our algorithms also extend to the setting of "batched'' tests, where multiple tests can be performed simultaneously while incurring an extra setup cost. 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.