  This paper develops a unified framework, based on iterated random operator
theory, to analyze the convergence of constant stepsize recursive stochastic
algorithms (RSAs). RSAs use randomization to efficiently compute expectations,
and so their iterates form a stochastic process. The key idea of our analysis
is to lift the RSA into an appropriate higher-dimensional space and then
express it as an equivalent Markov chain. Instead of determining the
convergence of this Markov chain (which may not converge under constant
stepsize), we study the convergence of the distribution of this Markov chain.
To study this, we define a new notion of Wasserstein divergence. We show that
if the distribution of the iterates in the Markov chain satisfy a contraction
property with respect to the Wasserstein divergence, then the Markov chain
admits an invariant distribution. We show that convergence of a large family of
constant stepsize RSAs can be understood using this framework, and we provide
several detailed examples.
