Haussler, Littlestone, and Warmuth described a general-purpose algorithm fo
r learning according to the prediction model, and proved an upper bound on
the probability that their algorithm makes a mistake in terms of the number
of examples seen and the Vapnik-Chervonenkis (VC) dimension of the concept
class being learned. We show that their bound is within a factor of 1 + o(
1) of the best possible such bound for any algorithm.