Learning from non-irreducible Markov chains
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Publication:6379807
DOI10.1016/J.JMAA.2023.127049arXiv2110.04338MaRDI QIDQ6379807
Publication date: 8 October 2021
Abstract: Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong correlation between the marginals of the data-generating process, suggesting that the i.i.d. assumption is not always justified. This problem has been already considered in the context of Markov chains satisfying the Doeblin condition. This condition, among other things, implies that the chain is not singular in its behavior, i.e. it is irreducible. In this article, we focus on the case when the training data set is drawn from a not necessarily irreducible Markov chain. Under the assumption that the chain is uniformly ergodic with respect to the -Wasserstein distance, and certain regularity assumptions on the hypothesis class and the state space of the chain, we first obtain a uniform convergence result for the corresponding sample error, and then we conclude learnability of the approximate sample error minimization algorithm and find its generalization bounds. At the end, a relative uniform convergence result for the sample error is also discussed.
Markov chains (discrete-time Markov processes on discrete state spaces) (60J10) Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) (68Q17) Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) (68Q87)
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