Learning from non-irreducible Markov chains

Most of the existing literature on supervised machine learning problems focuses case when training data set is drawn from an i.i.d. sample. However, many practical are characterized by temporal dependence and strong correlation between marginals data-generating process, suggesting that assumption not always justified. This problem has been already considered in context Markov chains satisfying Doeblin condition. condition, among other things, implies chain singular its behavior, i.e. it irreducible. In this article, we focus a necessarily irreducible chain. Under uniformly ergodic with respect to L1-Wasserstein distance, certain regularity assumptions hypothesis class state space chain, first obtain uniform convergence result for corresponding sample error, then conclude learnability approximate error minimization algorithm find generalization bounds. At end, relative also discussed.