Mirror Langevin Monte Carlo: the Case Under Isoperimetry

Motivated by the connection between sampling and optimization, we study a mirror descent analogue of Langevin dynamics and analyze three different discretization schemes, giving nonasymptotic convergence rate under functional inequalities such as Log-Sobolev in the corresponding metric. Compared to the Euclidean setting, the result reveals intricate relationship between the underlying geometry and the target distribution and suggests that care might need to be taken in order for the discretized algorithm to achieve vanishing bias with diminishing stepsize for sampling from potentials under weaker smoothness/convexity regularity conditions.