  We consider Linear Stochastic Approximation (LSA) with a constant stepsize
and Markovian data. Viewing the joint process of the data and LSA iterate as a
time-homogeneous Markov chain, we prove its convergence to a unique limiting
and stationary distribution in Wasserstein distance and establish
non-asymptotic, geometric convergence rates. Furthermore, we show that the bias
vector of this limit admits an infinite series expansion with respect to the
stepsize. Consequently, the bias is proportional to the stepsize up to higher
order terms. This result stands in contrast with LSA under i.i.d. data, for
which the bias vanishes. In the reversible chain setting, we provide a general
characterization of the relationship between the bias and the mixing time of
the Markovian data, establishing that they are roughly proportional to each
other.
  While Polyak-Ruppert tail-averaging reduces the variance of the LSA iterates,
it does not affect the bias. The above characterization allows us to show that
the bias can be reduced using Richardson-Romberg extrapolation with $m\ge 2$
stepsizes, which eliminates the $m-1$ leading terms in the bias expansion. This
extrapolation scheme leads to an exponentially smaller bias and an improved
mean squared error, both in theory and empirically. Our results immediately
apply to the Temporal Difference learning algorithm with linear function
approximation, Markovian data, and constant stepsizes.
