Toward Drawing an Atlas of Hypothesis Classes: Approximating a Hypothesis via Another Hypothesis Model

Abstract

Computational knowledge discovery can be considered to be a complicated human activity concerned with searching for something new from data with computer systems. The optimization of theentire process of computational knowledge discovery is a big challenge in computer science. If we had an atlas of hypothesis classes which describes prior and basic knowledge on relative relationship between the hypothesis classes, it would be helpful in selecting hypothesis classes to be searched in discovery processes. In this paper, to give a foundation for an atlas of various classes of hypotheses, we have defined a measure of approximation of a hypothesis class C ₁ to another class C ₂. The hypotheses we consider here are restricted to m-ary Boolean functions. For 0 ≤ ∈ ≤ 1, we say that C ₁ is (1-∈)-approximated to C ₂ if, for every distribution Dover (0, 1)^m, and for each hypothesis h ₁ ∈ C ₁, there exists a hypothesis h ₂ ∈ C ₂ such that, with the probability at most ∈, we have h ₁ x) ≠ h ₂ x) where x ∈ { 0, 1 } ^m is drawn randomly and independently according to D. Thus, we can use the approximation ratio of C ₁ to C ₂ as an index of how similar C ₁ is to C ₂. We discuss lower bounds of the approximation ratios among representative classes of hypotheses like decision lists, decision trees, linear discriminant functions and so on. This prior knowledge would come in useful when selecting hypothesis classes in the initial stage and the sequential stages involved in the entire discovery process.