This paper considers how to obtain MCMC quantitative convergence bounds which
can be translated into tight complexity bounds in high-dimensional settings. We
propose a modified drift-and-minorization approach, which establishes a
generalized drift condition defined in a subset of the state space. The subset
is called the ``large set'' and is chosen to rule out some ``bad'' states which
have poor drift property when the dimension gets large. Using the ``large set''
together with a ``centered'' drift function, a quantitative bound can be
obtained which can be translated into a tight complexity bound. As a
demonstration, we analyze a certain realistic Gibbs sampler algorithm and
obtain a complexity upper bound for the mixing time, which shows that the
number of iterations required for the Gibbs sampler to converge is constant
under certain conditions on the observed data and the initial state. It is our
hope that this modified drift-and-minorization approach can be employed in many
other specific examples to obtain complexity bounds for high-dimensional Markov
chains.Comment: 42 page