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We propose a new method called the Metropolis-adjusted Mirror Langevin
algorithm for approximate sampling from distributions whose support is a
compact and convex set. This algorithm adds an accept-reject filter to the
Markov chain induced by a single step of the mirror Langevin algorithm (Zhang
et al., 2020), which is a basic discretisation of the mirror Langevin dynamics.
Due to the inclusion of this filter, our method is unbiased relative to the
target, while known discretisations of the mirror Langevin dynamics including
the mirror Langevin algorithm have an asymptotic bias. We give upper bounds for
the mixing time of the proposed algorithm when the potential is relatively
smooth, convex, and Lipschitz with respect to a self-concordant mirror
function. As a consequence of the reversibility of the Markov chain induced by
the algorithm, we obtain an exponentially better dependence on the error
tolerance for approximate sampling. We also present numerical experiments that
corroborate our theoretical findings.
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