Characterizing scenarios where the weaker local-norm condition holds

Identify classes of target potentials and Riemannian metrics for which the weaker sufficient condition—uniform boundedness of the local norm x ↦ ||∇f(x)||_{M(x)^{-1}} on a large convex subset of the domain—holds, and analyze its implications for mixing-time bounds of MAPLA without requiring a global gradient upper bound.

Background

The analysis shows that MAPLA’s mixing-time guarantees can be derived under a weaker sufficient condition that does not require a global (β,M)-gradient upper bound, provided the local norm is uniformly bounded on a large convex subset. The authors point out that finding other concrete scenarios where this condition naturally holds is an open direction.

Such characterization would broaden the applicability of MAPLA’s theory and potentially reduce assumptions needed for practical problems, especially in constrained domains where global Lipschitz-type bounds may be hard to verify.