85 research outputs found

    Learning Boolean Halfspaces with Small Weights from Membership Queries

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    We consider the problem of proper learning a Boolean Halfspace with integer weights {0,1,…,t}\{0,1,\ldots,t\} from membership queries only. The best known algorithm for this problem is an adaptive algorithm that asks nO(t5)n^{O(t^5)} membership queries where the best lower bound for the number of membership queries is ntn^t [Learning Threshold Functions with Small Weights Using Membership Queries. COLT 1999] In this paper we close this gap and give an adaptive proper learning algorithm with two rounds that asks nO(t)n^{O(t)} membership queries. We also give a non-adaptive proper learning algorithm that asks nO(t3)n^{O(t^3)} membership queries

    Distribution-Independent Evolvability of Linear Threshold Functions

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    Valiant's (2007) model of evolvability models the evolutionary process of acquiring useful functionality as a restricted form of learning from random examples. Linear threshold functions and their various subclasses, such as conjunctions and decision lists, play a fundamental role in learning theory and hence their evolvability has been the primary focus of research on Valiant's framework (2007). One of the main open problems regarding the model is whether conjunctions are evolvable distribution-independently (Feldman and Valiant, 2008). 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 from correlations. 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 mutation algorithm. Our algorithm relies on a non-linear loss function being used to select the hypotheses instead of 0-1 loss in Valiant's (2007) original definition. The proof of evolvability requires that the loss function satisfies several mild conditions that are, for example, satisfied by the quadratic loss function studied in several other works (Michael, 2007; Feldman, 2009; Valiant, 2010). An important property of our evolution algorithm is monotonicity, that is the algorithm guarantees evolvability without any decreases in performance. Previously, monotone evolvability was only shown for conjunctions with quadratic loss (Feldman, 2009) or when the distribution on the domain is severely restricted (Michael, 2007; Feldman, 2009; Kanade et al., 2010
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