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# Attribute-Efficient Learning andWeight-Degree Tradeoffs for Polynomial Threshold Functions

*Proceedings of the 25th Annual Conference on Learning Theory*, PMLR 23:14.1-14.19, 2012.

#### Abstract

We study the challenging problem of learning decision lists attribute-efficiently, giving both positive and negative results. Our main positive result is a new tradeoff between the running time and mistake bound for learning length-\emphk decision lists over \emphn Boolean variables. When the allowed running time is relatively high, our new mistake bound improves significantly on the mistake bound of the best previous algorithm of Klivans and Servedio (Klivans and Servedio, 2006). Our main negative result is a new lower bound on the \emphweight of any degree-\emphd polynomial threshold function (PTF) that computes a particular decision list over \emphk variables (the “ODD-MAX-BIT” function). The main result of Beigel (Beigel, 1994) is a weight lower bound of 2^Ω(\emphk/\emphd^2), which was shown to be essentially optimal for \emphd ≤ \emphk^1/3 by Klivans and Servedio. Here we prove a 2^Ω(√\emphk/d) lower bound, which improves on Beigel’s lower bound for \emphd > \emphk^1/3. This lower bound establishes strong limitations on the effectiveness of the Klivans and Servedio approach and suggests that it may be difficult to improve on our positive result. The main tool used in our lower bound is a new variant of Markov’s classical inequality which may be of independent interest; it provides a bound on the derivative of a univariate polynomial in terms of both its degree \emphand the size of its coefficients.