canonical setting

Log-smooth + PI or LSI

$\nabla V$ is $\beta$ -Lipschitz; $C_{\rm PI}$ or $C_{\rm LSI}\le1/\alpha$ ; $\kappa=\beta/\alpha$ .

Updated Oct 1, 2026

Upper and lower bounds

Comparison table

6 scoped results. Tildes suppress logarithmic factors.

Best-known and comparison results for Log-smooth + PI or LSI
Result	Algorithm or model	Complexity	Guarantee	Oracle / start	Assumptions and notes	Review	Sources
best upper	Gaussian cloud sampler	$\widetilde O(\kappa d^{1/5}\,\operatorname{polylog}(1/\varepsilon))$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$ under LSI	$\nabla V$ (expected queries)Given $x_{\rm ref}$ with $\lVert\nabla V(x_{\rm ref})\rVert\le\sqrt{\alpha d}$ ; no warm start	$0<\varepsilon\le1/4$ ; log-Sobolev inequality.Exact bound: $O\bigl(\kappa\{d+\log(\kappa d/\varepsilon)\}^{1/5}\log^{13/2}(\kappa d/\varepsilon)\bigr)$ , via the proximal sampler around a FORS-corrected smoothed Picard HMC. Gradient-only; improves the dimension dependence of the proximal-sampler row from $\sqrt d$ to $d^{1/5}$ at the same $\kappa$ dependence. The cost of finding $x_{\rm ref}$ is not included.	Checkedpreprint	Chen–Chewi–Lu–Zhang (2026), Theorem 1.1 ↗
best upper	Implemented proximal sampler	$\widetilde O(\kappa\sqrt d\,\operatorname{polylog}(\sqrt{\operatorname{KL}_0}/\varepsilon^2))$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$ under LSI	$V,\nabla V$ plus proximal subroutineFinite initial KL; implementation warm start	Log-Sobolev inequality.Checked high-accuracy baseline from a finite initial KL; kept on the frontier because its initialization model differs from the Gaussian cloud sampler row.	Checkedmonograph	Chewi (2026), Corollary 8.6.3 ↗
lower unknown	Matching PI/LSI oracle lower bound	Unknown	Same functional-inequality class	$V,\nabla V$ Must be normalized	No matching result located in the reference corpus.Open literature-check item.	Unverifiedno primary source	—
upper	LMC	$\widetilde O(\kappa^2d/\varepsilon^2)$	$\sqrt{\operatorname{KL}}\le\varepsilon$ under LSI	$\nabla V$ Finite initial KL	Log-Sobolev inequality.Classical direct-discretization baseline.	Checkedmonograph	Chewi (2026), Theorem 4.2.7 ↗
upper	LMC Rényi interpolation	$\widetilde O(\kappa^2dq\,\varepsilon^{-2}\log\mathcal R_2(\mu_0\Vert\pi))$	$\sqrt{\mathcal R_q}\le\varepsilon$ under LSI	$\nabla V$ Finite initial Rényi divergence	$q\ge2$ and theorem step-size condition.Rényi order appears explicitly in the cost.	Checkedmonograph	Chewi (2026), Theorem 6.1.2 ↗
upper	ULMC warm-start construction	$\widetilde O(\kappa^{3/2}\sqrt d/\varepsilon)$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$ under LSI	$\nabla V$ $\mathcal R_2(\mu_0\Vert\pi)=\widetilde O(d)$	ULMC parameter choices from the corollary.Useful warm-start row; worse accuracy dependence than Metropolized methods.	Checkedmonograph	Chewi (2026), Corollary 6.3.2 ↗

Scope

Convexity is not required. The functional inequality supplies the mixing scale. PI and LSI are not interchangeable, so the guarantee column states which divergence is controlled.

Lower bounds

No matching general oracle lower bound for the PI/LSI-only class is recorded in the current monograph. Strongly-convex lower bounds are not automatically copied into this larger normalization.