On Mixing Times of Metropolized Algorithm With Optimization Step (MAO) : A New Framework

The ability to draw samples from a distribution is at the heart of many applications within the Bayesian paradigm and, more generally, in computational statistics. Markov Chain Monte Carlo pioneered by Metropolis et al. [1953], is often considered among practitioners as the default method for obtaining samples from distributions in a high-dimensional setting. In practice, variants of the Metropolis-Hastings enjoy tremendous success, notably in posterior exploration within a Bayesian setting Carpenter et al. [2017], Smith [2014]. In addition, Monte Carlo methods are commonly deployed in several applications: estimating the posterior mean, computing expectations of quantities of interest, and volumes of particular sets. Recently the research community has been interested in a noticeable manner in sampling methods and their interplay with the more established field of optimization Ma et al. [2019]. More specifically, due to the asymptotic nature of MCMC methods, a more tractable characterization of the dimension dependency of the convergence is an essential step in order to develop a better understanding of the convergence of this class of algorithms and to practical guidelines for practitioners.