Learning decision trees using the Fourier spectrum

Eyal Kushilevitz*, Yishay Mansour

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. This algorithm uses membership queries). The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over G F(2)). This paper shows how to learn in polynomial time any function that can be approximated (in norm L2) by a polynomially sparse function (i.e., a function with only polynomially many nonzero Fourier coefficients). The authors demonstrate that any function f whose L1-norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial L1-norm can be learned deterministically. The algorithm can also exactly identify a decision tree of depth d in time polynomial in 2d and n. This result implies that trees of logarithmic depth can be identified in polynomial time.

Original languageEnglish
Pages (from-to)1331-1348
Number of pages18
JournalSIAM Journal on Computing
Issue number6
StatePublished - 1993
Externally publishedYes


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