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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 three other sampling techniques and empirically compare them with Chawla et al.’sMetropolis sampler to determine how effective each is in drawing good samples in the least amount of time, in terms of accuracy of weighted sum estimates and in terms of Winnow’s prediction accuracy.