Hoeffding's lemma for Markov Chains and its applications to statistical learning

We establish the counterpart of Hoeffding's lemma for Markov dependent random variables. Specifically, if a stationary Markov chain $\{X_i\}_{i \ge 1}$ with invariant measure $\pi$ admits an $\mathcal{L}_2(\pi)$-spectral gap $1-\lambda$, then for any bounded functions $f_i: x \mapsto [a_i,b_i]$, the sum of $f_i(X_i)$ is sub-Gaussian with variance proxy $\frac{1+\lambda}{1-\lambda} \cdot \sum_i \frac{(b_i-a_i)^2}{4}$. The counterpart of Hoeffding's inequality immediately follows. Our results assume none of reversibility, countable state space and time-homogeneity of Markov chains. They are optimal in terms of the multiplicative coefficient $(1+\lambda)/(1-\lambda)$, and cover Hoeffding's lemma and inequality for independent random variables as special cases with $\lambda = 0$. We illustrate the utility of these results by applying them to six problems in statistics and machine learning. They are linear regression, lasso regression, sparse covariance matrix estimation with Markov-dependent samples; Markov chain Monte Carlo estimation; respondence driven sampling; and multi-armed bandit problems with Markovian rewards.