  This paper studies Hoeffding's inequality for Markov chains under the
generalized concentrability condition defined via integral probability metric
(IPM). The generalized concentrability condition establishes a framework that
interpolates and extends the existing hypotheses of Markov chain Hoeffding-type
inequalities. The flexibility of our framework allows Hoeffding's inequality to
be applied beyond the ergodic Markov chains in the traditional sense. We
demonstrate the utility by applying our framework to several non-asymptotic
analyses arising from the field of machine learning, including (i) a
generalization bound for empirical risk minimization with Markovian samples,
(ii) a finite sample guarantee for Ployak-Ruppert averaging of SGD, and (iii) a
new regret bound for rested Markovian bandits with general state space.
