Minimax Learning of Ergodic Markov Chains

09/13/2018

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by   Geoffrey Wolfer, et al.

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We compute the finite-sample minimax (modulo logarithmic factors) sample complexity of learning the parameters of a finite Markov chain from a single long sequence of states. Our error metric is a natural variant of total variation. The sample complexity necessarily depends on the spectral gap and minimal stationary probability of the unknown chain - for which, at least in the reversible case, there are known finite-sample estimators with fully empirical confidence intervals. To our knowledge, this is the first PAC-type result with nearly matching (up to logs) upper and lower bounds for learning, in any metric in the context of Markov chains.