Accuracy vs. Simplicity: A Complex Trade-Off∗

Inductive learning aims at finding general rules that hold true in a database. Targeted learning seeks rules for the prediction of the value of a variable based on the values of others, as in the case of linear or non-parametric regression analysis. Non-targeted learning finds regularities without a specific prediction goal. We model the product of non-targeted learning as rules that state that a certain phenomenon never happens, or that certain conditions necessitate another. For all types of rules, there is a trade-off between the rule’s accuracy and its simplicity. Thus rule selection can be viewed as a choice problem, among pairs of degree of accuracy and degree of complexity. However, one cannot in general tell what is the feasible set in the accuracycomplexity space. Formally, we show that finding out whether a point belongs to this set is computationally hard. In particular, in the context of linear regression, finding a small set of variables that obtain a certain value of R2 is computationally hard. Computational complexity may explain why a person is not always aware of rules that, if ∗Earlier versions of this paper circulated under the title “From Cases to Rules: Induction and Regression.” We thank Hal Cole, Joe Halpern, Bart Lipman, Yishay Mansour, and Nimrod Megiddo for conversations and references. †Institut d’Anàlisi Econòmica, C.S.I.C. [email protected] ‡Tel-Aviv University and Cowles Foundation, Yale University. Gilboa gratefully acknowledges support from the Israel Science Foundation. [email protected] §University of Pennsylvania; Postlewaite gratefully acknowledges support from the National Science Foundation. [email protected] ¶Tel-Aviv University and the Ohio State University.Schmeidler gratefully acknowledges support from the Israel Science Foundation. [email protected]