Mixing-time bounds for weakly log-concave targets without isoperimetric assumptions

Establish a mixing-time upper bound for weakly log-concave probability distributions using only the mixing accuracy eplace eplace eplace?

Background

The paper analyzes the MARCO sampler under regularity assumptions that include an isoperimetric inequality for the joint target distribution. Under these assumptions, marginalization yields a lower-dimensional marginal target with no larger isoperimetric condition number, enabling sharper mixing-time upper bounds than direct MCMC sampling in the joint space.

The authors note that extending this analysis to weakly log-concave targets without assuming an isoperimetric inequality would require a specialized approach. Specifically, they identify the absence of a proven mixing-time upper bound depending only on the accuracy threshold, dimension, warm-start parameter, and smoothness constant as the unresolved obstacle preventing their comparison assumption from being instantiated in a practical example.

References

To the best of the authors' knowledge, a mixing time upper bound determined by the parameters $(\varepsilon; d,\warmstart{\pi0},L(\pi))$ has not been proven; hence \Cref{assump:mixing_time_comparison} could not be satisfied by any practical example.

— Exploiting Exact Conditionals Improves Conditioning: Provably Fast Mixing Time Bounds By Sampling from the Marginal  (2608.27884 - Chowdhary et al., 28 Aug 2026) in Section 4.2, immediately following the proof of Theorem \ref{thm:mixing_M_le_J}