Distribution-Independent Evolvability of Linear Threshold Functions

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Vitaly Feldman ;
Proceedings of the 24th Annual Conference on Learning Theory, PMLR 19:253-272, 2011.

Abstract

Valiant’s model of evolvability models the evolutionary process of acquiring useful functionality as a restricted form of learning from random examples \citepValiant:09.Linear threshold functions and their various subclasses, such as conjunctions and decision lists, play a fundamental role in learning theory and hence their evolvabilityhas been the primary focus of research on Valiant’s framework. One of the main open problems regarding the model is whether conjunctions are evolvable distribution-independently\citepFeldmanValiant:08colt. We show that the answer is negative. Our proof is based on a new combinatorial parameter of a concept class that lower-bounds the complexity of learning fromcorrelations.We contrast the lower bound with a proof that linear threshold functions having a non-negligible margin on the data points are evolvable distribution-independently via a simple mutationalgorithm. Our algorithm relies on a non-linear loss function being used to select the hypotheses instead of 0-1 loss in Valiant’s original definition. The proof of evolvabilityrequires that the loss function satisfies several mild conditions that are, for example, satisfied by the quadratic loss function studied in several other works \citepMichael:07,Feldman:09sqd,Valiantp:11manu. An important property of our evolution algorithm is monotonicity, that is the algorithm guaranteesevolvability without any decreases in performance. Previously, monotone evolvability was only shown for conjunctions with quadratic loss \citepFeldman:09sqd or when the distribution on the domain is severely restricted \citepMichael:07,Feldman:09sqd,KanadeVV:10.

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