Sampling regime
Log-concave + log-smooth
Compare theorems by mathematical guarantee, rate, and source — not by raw crawler rows.
Theorem and frontier cases
best upper · Normalized SampleWiki statement
Implemented proximal sampler
Primary theorem audit pending — this is not presented as a verbatim paper theorem
Guarantee / accuracy
\[\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon\]
Complexity / rate
\[\widetilde O(\beta\sqrt d\,R_0^2/\varepsilon^2)\]
lower unknown · Open problem
Matching first-order lower bound
Matching lower bound not currently known in the pinned literature trail
Guarantee / accuracy
\[\operatorname{KL}\le\varepsilon^2\]
upper · Exact source theorem
FORS-implemented proximal sampler
Theorem G.1(3), specialized to s=1
Theorem statement
\[D_{\mathsf{KL}}(\widehat\mu\Vert\mu)\le\varepsilon^2,\qquad N=\widetilde O\!\left(\beta\sqrt d\frac{W_2^2(\mu_0,\mu)}{\varepsilon^2}\right)\]
upper · Exact source theorem
Averaged LMC
Theorem 4.3.6(1) (Durmus et al. 2019; weakly convex case)
Theorem statement
\[\begin{gathered}0\preceq\nabla^2V\preceq\beta I_d,\qquad h\asymp\frac{\varepsilon^2}{\beta d},\qquad \bar\mu_{Nh}:=\frac1N\sum_{n=1}^N\widehat\mu_{nh},\\ \sqrt{\operatorname{KL}(\bar\mu_{Nh}\Vert\pi)}\le\varepsilon\quad\text{after}\quad N=O\!\left(\frac{\beta d\,W_2^2(\widehat\mu_0,\pi)}{\varepsilon^4}\right).\end{gathered}\]
upper · Exact source theorem
Ideal proximal chain
Theorem 8.4.1 (Proximal sampler under log-concavity)
Theorem statement
\[\operatorname{KL}(\mu_n^X\Vert\pi^X)\le \frac{W_2^2(\mu_0^X,\pi^X)}{nh}\]