Sampling regime
Weakly smooth log-concave
Compare theorems by mathematical guarantee, rate, and source — not by raw crawler rows.
Theorem and frontier cases
best upper · Exact source theorem
FORS + proximal sampler
Theorem G.1(3)
Theorem statement
\[D_{\mathsf{KL}}(\widehat\mu\Vert\mu)\le\varepsilon^2,\qquad N=\widetilde O\!\left(\beta_s^{2/(1+s)}d^{s/(1+s)}\frac{W_2^2(\mu_0,\mu)}{\varepsilon^2}\right)\]
lower unknown · Open problem
Matching Hölder-model lower bound
Matching lower bound not currently known in the pinned literature trail
Matching result remains open.
upper · Exact source theorem
Averaged LMC
Theorem 4.3.11 (Durmus et al. 2019)
Theorem statement
\[\begin{gathered}V\text{ is convex and }L\text{-Lipschitz},\qquad h\asymp\frac{\varepsilon^2}{L^2},\qquad \bar\mu_{Nh}:=\frac1N\sum_{n=1}^N\widehat\mu_{nh},\\ \sqrt{\operatorname{KL}(\bar\mu_{Nh}\Vert\pi)}\le\varepsilon\quad\text{after}\quad N=O\!\left(\frac{L^2W_2^2(\widehat\mu_0,\pi)}{\varepsilon^4}\right).\end{gathered}\]
upper · Normalized SampleWiki statement
Nonsmooth mirror-Langevin
Primary theorem audit pending — this is not presented as a verbatim paper theorem
Guarantee / accuracy
\[\sqrt{\operatorname{KL}}\le\varepsilon\]
Complexity / rate
\[O(L^2D_\phi(\pi,\mu_{0+})/\varepsilon^4)\]