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Summary: The paper proposes a novel approach to computationally efficient maximum likelihood learning in exponential families. In general, finding the maximum likelihood solution is intractable. From a convex optimization perspective, the sticking point is the need to calculate an integral wrt the currently proposed EF parameter. By assuming that MCMC is fast-mixing for all allowed parameters, the author(s) are able to show that the integrals needed for proximal gradient descent can be calculated with sufficient precision that, when combined with the results of Schmidt et al. (2011), a fully-polynomial randomized approximation scheme for calculating the MLE can be obtained. Both the convex and strongly convex cases are considered, which lead to different types of guarantees: the former on the likelihood error, the latter on the parameter error.