Minimax Learning of Ergodic Markov Chains

Geoffrey Wolfer, Aryeh Kontorovich

Research output: Contribution to journalConference articlepeer-review


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 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 logarithmic factors) upper and lower bounds for learning, in any metric, in the context of Markov chains.

Original languageAmerican English
Pages (from-to)904-930
Number of pages27
JournalProceedings of Machine Learning Research
StatePublished - 1 Jan 2019
Event30th International Conference on Algorithmic Learning Theory, ALT 2019 - Chicago, United States
Duration: 22 Mar 201924 Mar 2019


  • ergodic Markov chain
  • learning
  • minimax

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability


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