Dimension dependence of MALA for higher-order smooth distributions

Determine how the total-variation mixing time of the Metropolis-adjusted Langevin algorithm (MALA) scales with the ambient dimension d for general probability distributions on R^d whose log-densities satisfy higher-order smoothness assumptions, beyond the special case of product distributions.

Background

The paper reviews that for strongly log-concave targets with second-order smoothness, the minimax dimension dependence of MALA is settled at O(d^{1/2}) up to logarithms. It also notes scaling limits d^{1/3} for sufficiently regular product distributions under stronger smoothness assumptions.

However, outside the product structure, rigorous dimension dependence under higher-order smoothness is not established. The authors explicitly state that extending such results to general distributions remains an open question, motivating their broader investigation of HMC under isoperimetry and higher-order smoothness.