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We propose a new method called the Metropolis-adjusted Mirror Langevinalgorithm for approximate sampling from distributions whose support is acompact and convex set. This algorithm adds an accept-reject filter to theMarkov chain induced by a single step of the mirror Langevin algorithm (Zhanget 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 thetarget, while known discretisations of the mirror Langevin dynamics includingthe mirror Langevin algorithm have an asymptotic bias. We give upper bounds forthe mixing time of the proposed algorithm when the potential is relativelysmooth, convex, and Lipschitz with respect to a self-concordant mirrorfunction. As a consequence of the reversibility of the Markov chain induced bythe algorithm, we obtain an exponentially better dependence on the errortolerance for approximate sampling. We also present numerical experiments thatcorroborate our theoretical findings.

Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin Algorithm