Controlling costs: Feature selection on a budget.

The traditional framework for feature selection treats all features as costing the same amount. However, in reality, a scientist often has considerable discretion regarding which variables to measure, and the decision involves a tradeoff between model accuracy and cost (where cost can refer to money, time, difficulty or intrusiveness). In particular, unnecessarily including an expensive feature in a model is worse than unnecessarily including a cheap feature. We propose a procedure, which we call cheap knockoffs, for performing feature selection in a cost-conscious manner. The key idea behind our method is to force higher cost features to compete with more knockoffs than cheaper features. We derive an upper bound on the weighted false discovery proportion associated with this procedure, which corresponds to the fraction of the feature cost that is wasted on unimportant features. We prove that this bound holds simultaneously with high probability over a path of selected variable sets of increasing size. A user may thus select a set of features based, for example, on the overall budget, while knowing that no more than a particular fraction of feature cost is wasted. We investigate, through simulation and a biomedical application, the practical importance of incorporating cost considerations into the feature selection process.