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Chawla et al. introduced a way to use the Markov chain Monte Carlo method to estimate weighted sums in multiplicative weight update algorithms when the number of inputs is exponential. But their algorithm still required extensive simulation of the Markov chain in order to get accurate estimates of the weighted sums. We propose an optimized version of Chawla et al.’s algorithm, which produces exactly the same classifications while often using fewer Markov chain simulations. We also apply two other sampling techniques and empirically compare them with Chawla et al.’s Metropolis sampler to determine how effective each is in drawing good samples in terms of accuracy of weighted sum estimates and total time. We perform our analyses in the context of learning DNF formulas using littlestone’s Winnow algorithm.