Amplification of weak learning under the uniform distribution

Let 3 be a class of boolean functions, such as A(7° or NC1. We show that if X satisfies certain closure properties, then a weak learning algorithm for 7 over the uniform distribution can be amplified to a strong learning algorithm. This result can be used as an effective tool in the construction of learning algorithms over the uniform distribution. We present two methods. The first using the XO R lemma of [Yao82], and the second using a theorem from the extremal theory of finite sets. The second method suggests a simple proof of the XOR lemma.