Relative entropy convergence for the kinetic Langevin algorithm under a log-Sobolev hypothesis

Establish whether the relative-entropy-based convergence framework developed for the overdamped Langevin algorithm by Vempala and Wibisono can be adapted to the kinetic (underdamped) Langevin algorithm. Specifically, prove convergence guarantees for the kinetic Langevin algorithm (the Euler discretization of the underdamped Langevin diffusion with friction parameter β and time step η) measured in relative entropy under the assumption that the target distribution μ satisfies a log-Sobolev inequality, with both the warm-start condition and the error quantified in relative entropy.

Background

The paper establishes non-asymptotic total variation bounds for the kinetic (underdamped) Langevin algorithm under Poincaré and gradient-Lipschitz assumptions on the target measure μ, leveraging hypocoercive estimates and a discretization analysis. The authors note a loss of regularity from hypothesis to conclusion (chi-square warm start leading to total variation bounds) and express interest in stronger conclusions measured by chi-square or relative entropy.

They highlight that in the overdamped Langevin setting, Vempala and Wibisono proved analogous results in relative entropy under a log-Sobolev assumption. The open question is whether such relative-entropy-based analysis extends to the kinetic Langevin algorithm, which is inherently degenerate and requires hypocoercive techniques, making a direct adaptation nontrivial.