Lower Bounds on Learning Decision Lists and Trees

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k-Decision lists and decision trees play important roles in learning theory as well as in practical learning systems.k-Decision lists generalize classes such as monomials,k-DNF, andk-CNF, and like these subclasses they are polynomially PAC-learnable [R. Rivest,Mach. Learning2(1987), 229–246]. This leaves open the question of whetherk-decision lists can be learned as efficiently ask-DNF. We answer this question negatively in a certain sense, thus disproving a claim in a popular textbook [M. Anthony and N. Biggs, “Computational Learning Theory,” Cambridge Univ. Press, Cambridge, UK, 1992]. Decision trees, on the other hand, are not even known to be polynomially PAC-learnable, despite their widespread practical application. We will show that decision trees are not likely to be efficiently PAC-learnable. We summarize our specific results. The following problems cannot be approximated in polynomial time within a factor of 2logδ nfor anyδ<1, unlessNP⊂DTIME[2polylog n]: a generalized set cover,k-decision lists,k-decision lists by monotone decision lists, and decision trees. Decision lists cannot be approximated in polynomial time within a factor ofnδ, for some constantδ>0, unlessNP=P. Also,k-decision lists withl0–1 alternations cannot be approximated within a factor logl nunlessNP⊂DTIME[nO(log log n)] (providing an interesting comparison to the upper bound obtained by A. Dhagat and L. Hellerstein [in“FOCS '94,” pp. 64–74]).