canonical setting

Log-concave + log-smooth

$\pi\propto e^{-V}$ , $0\preceq\nabla^2V\preceq\beta I$ , $R_0^2=W_2^2(\mu_0,\pi)$ .

Updated Aug 6, 2026

Upper and lower bounds

Comparison table

5 scoped results. Tildes suppress logarithmic factors.

Best-known and comparison results for Log-concave + log-smooth
Result	Algorithm or model	Complexity	Guarantee	Oracle / start	Assumptions and notes	Review	Sources
best upper	Implemented proximal sampler	$\widetilde O(\beta\sqrt d\,R_0^2/\varepsilon^2)$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$	$V,\nabla V$ plus proximal subroutine $W_2(\mu_0,\pi)=R_0$	Implementation and suppressed logarithms as in the cited corollary.Current checked headline in this normalization.	Checkedmonograph	Chewi (2026), Corollary 8.6.3 ↗
lower unknown	Matching first-order lower bound	Unknown	$\operatorname{KL}\le\varepsilon^2$ in the same model	$V,\nabla V$ $W_2(\mu_0,\pi)=R_0$	No matching result located in the reference corpus.Open literature-check item; do not infer a zero lower bound.	Unverifiedno primary source	—
upper	FORS-implemented proximal sampler	$\widetilde O(\beta\sqrt d\,R_0^2/\varepsilon^2)$	$\operatorname{KL}\le\varepsilon^2$	Gradient + proximal access $W_2(\mu_0,\pi)=R_0$	Expected-query statement; see implementation conditions.Same leading normalization with a KL guarantee; needs human theorem check.	Citedpreprint	Chen et al. (2026), Theorem G.1(3) ↗
upper	Averaged LMC	$O(\beta dR_0^2/\varepsilon^4)$	$\sqrt{\operatorname{KL}}\le\varepsilon$	$\nabla V$ $W_2(\mu_0,\pi)=R_0$	Constant-step LMC with averaging.Classical baseline with worse accuracy dependence.	Checkedmonograph	Chewi (2026), Theorem 4.3.6 ↗
upper	Ideal proximal chain	$\operatorname{KL}(\mu_n\Vert\pi)\le R_0^2/(nh)$	KL convergence per ideal step	Exact restricted Gaussian oracle $W_2(\mu_0,\pi)=R_0$	Abstract oracle; not a first-order query count.Useful iteration benchmark; implementation cost is separate.	Checkedmonograph	Chewi (2026), Theorem 8.4.1 ↗

Scope

This setting drops strong convexity. Bounds therefore retain a scale such as $R_0^2=W_2^2(\mu_0,\pi)$ . The table distinguishes an abstract restricted-Gaussian oracle from implementations using gradients and proximal evaluations.

Lower bounds

No matching lower bound in the same $(\beta,R_0,\operatorname{KL})$ first-order model is recorded in the reference monograph. The explicit unknown row is intentional; lower bounds from strongly convex targets or from volume estimation are not silently substituted.