Concentration of the local norm to remove the gradient upper bound assumption

Prove concentration for the local norm x ↦ ||∇f(x)||_{M(x)^{-1}} under curvature lower and upper bounds ((μ,M) and (λ,M)) to eliminate the need for a global gradient upper bound in MAPLA’s mixing-time analysis, analogous to results established for MALA when M is the identity.

Background

The authors note that removing the (β,M)-gradient upper bound by establishing concentration of the local norm would strengthen the theoretical guarantees for MAPLA, paralleling recent advances for MALA in Euclidean settings.

If such concentration results can be shown under (μ,M)-curvature lower bounds and (λ,M)-curvature upper bounds, MAPLA’s mixing-time theory could be substantially generalized and simplified, providing sharper bounds and wider applicability.