Decision Trees for Function Evaluation - Simultaneous Optimization of Worst and Expected Cost
This work addresses a central challenge in automatic diagnosis and active learning, providing an optimal approximation for cost-efficient function evaluation, though it is incremental in improving upon existing methods for this known bottleneck.
The paper tackles the problem of adaptively evaluating discrete functions with variable query costs, aiming to minimize both worst-case and expected costs. The proposed algorithm constructs a decision tree that achieves a logarithmic approximation for both cost measures, which is optimal under the assumption that P ≠ NP.
In several applications of automatic diagnosis and active learning a central problem is the evaluation of a discrete function by adaptively querying the values of its variables until the values read uniquely determine the value of the function. In general, the process of reading the value of a variable might involve some cost, computational or even a fee to be paid for the experiment required for obtaining the value. This cost should be taken into account when deciding the next variable to read. The goal is to design a strategy for evaluating the function incurring little cost (in the worst case or in expectation according to a prior distribution on the possible variables' assignments). Our algorithm builds a strategy (decision tree) which attains a logarithmic approxima- tion simultaneously for the expected and worst cost spent. This is best possible under the assumption that $P \neq NP.$