Computing expectations in high-dimensional spaces is a key challenge in probabilistic inference and machine learning. Monte Carlo sampling, and importance sampling in particular, is one of the leading approaches. We propose a generalized importance sampling scheme based on randomly selecting (exponentially large) subsets of states rather than individual ones. By collecting a small number of extreme states in the sampled sets, we obtain estimates of statistics of interest, such as the partition function of an undirected graphical model. We incorporate this idea into a novel maximum likelihood learning algorithm based on cutting planes. We demonstrate empirically that our scheme provides accurate answers and scales to problems with up to a million variables.
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