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Learning Monotone Decision Trees in Polynomial Time

By Ryan O&apos and Rocco A. Servedio


We give an algorithm that learns any monotone Boolean function f: {-1, 1}n! {-1, 1}to any constant accuracy, under the uniform distribution, in time polynomial in n and in thedecision tree size of f. This is the first algorithm that can learn arbitrary monotone Booleanfunctions to high accuracy, using random examples only, in time polynomial in a reasonable measure of the complexity of f. A key ingredient of the result is a new bound showing that theaverage sensitivity of any monotone function computed by a decision tree of size s must be atmost plog s. This bound has already proved to be of independent utility in the study of decisiontree complexity [27]. We generalize the basic inequality and learning result described above in various ways; specif-ically, to partition size (a stronger complexity measure than decision tree size), p-biased measuresover the Boolean cube (rather than just the uniform distribution), and real-valued (rather tha

Year: 2005
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