Sharper bounds on R in the non-strongly convex case

Develop improved (tighter) bounds on the quantity R^2 = max(KL(μ^{(0)}|π), 2·sup_{n≥0} E[||X−x*||_L^2]) used in Theorem 5 to control the O(1/n) convergence rate of the Gibbs sampler when the potential is convex but not strongly convex, with particular attention to reducing or eliminating linear dependence on the warm-start constant C.

Background

When strong convexity is absent (λ* = 0), the authors show polynomial (O(1/n)) convergence rates for the Gibbs sampler in terms of KL divergence, controlled by a parameter R that depends on the initial distribution and second moments under π. Under warm-start assumptions, R can scale linearly with the warm-start constant C, leading to mixing-time bounds that grow with C.

The authors explicitly state that they do not claim the linear dependence on C is tight and leave the development of better bounds on R for future work, highlighting the need for sharper moment/initialization control in the non-strongly convex regime.