canonical setting

Strongly log-concave + log-smooth

$\pi\propto e^{-V}$ , $\alpha I\preceq\nabla^2V\preceq\beta I$ , $\kappa=\beta/\alpha$ .

Updated Oct 6, 2026

Upper and lower bounds

Comparison table

13 scoped results. Tildes suppress logarithmic factors.

Best-known and comparison results for Strongly log-concave + log-smooth
Result	Algorithm or model	Complexity	Guarantee	Oracle / start	Assumptions and notes	Review	Sources
best upper	Gaussian cloud sampler	$\widetilde O(\kappa d^{1/5}\,\operatorname{polylog}(1/\varepsilon))$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$	$\nabla V$ (expected queries)Given $x_{\rm ref}$ with $\lVert\nabla V(x_{\rm ref})\rVert\le\sqrt{\alpha d}$ ; no warm start	$0<\varepsilon\le1/4$ ; only needs $\beta$ -smoothness and an LSI with constant $1/\alpha$ .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. Ignoring logarithms, the best checked rate for $\kappa\ll d^{1/10}$ and for $d^{1/5}\ll\kappa\ll d^{2/5}$ ; Picard warm start + Proximal BPS is better in between, and exact ULD / FORS for $\kappa\gg d^{2/5}$ .	Checkedpreprint	Chen–Chewi–Lu–Zhang (2026), Theorem 1.1 ↗
best upper	Picard warm start + Proximal BPS	$\widetilde O\bigl((\kappa^{7/6}d^{1/6}+\kappa^{1/2}d^{1/4})\,\operatorname{polylog}(1/\varepsilon)\bigr)$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$	$\nabla V$ (expected queries)Given $x_{\rm ref}$ with $\lVert\nabla V(x_{\rm ref})\rVert\le\sqrt{\alpha d}$	$0<\varepsilon<1/4$ ; standard curvature bounds.Includes warm start generation; trades condition-number dependence for dimension dependence versus FORS.	Checkedpreprint	Chen–Chewi–Lu–Zhang (2026), Theorem 1.3 ↗Proximal BPS companion paper ↗
best upper	Smoothed Picard HMC + recursive warm start	$\widetilde O(\kappa^{7/6}d^{1/6}/\varepsilon^{1/3})$	$\operatorname{KL}(\mu_N\Vert\pi)\le\varepsilon^2$	$\nabla V$ (expected queries)Given $x_{\rm ref}$ with $V(x_{\rm ref})-\inf V\le d$	$0<\varepsilon\le1/2$ ; standard curvature bounds.Generates its own initialization. The additive $\kappa$ term is dominated; there is no $\kappa^2$ term. Improves on FORS when $d\gtrsim\kappa^3/\varepsilon^2$ , ignoring logarithms.	Checkedpreprint	Chen–Chewi–Lu–Zhang (2026), Theorem 7.1(i) ↗
provisional upper	Proximal sampler + Picard warm start + Proximal BPS	$\widetilde O(\kappa d^{3/16}\,\operatorname{polylog}(1/\varepsilon))$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$	$\nabla V$ (expected queries)Known mode / Gaussian cold start; inner reference points computed by convex optimization	$\kappa\gtrsim d^{1/8}$ ; approximate RGO error budgeted across outer steps.Derived bound, not a stated theorem: inner condition number $t\asymp d^{1/8}$ gives $(\kappa/t)\,\bigl(t^{7/6}d^{1/6}+t^{1/2}d^{1/4}\bigr)=O(\kappa d^{3/16})$ . Improves the listed high-accuracy rates for $d^{1/8}\ll\kappa\ll d^{7/16}$ , ignoring logarithms; no matching general lower bound.	AI-only noteno primary source	Derived from Chen–Chewi–Lu–Zhang, Theorem 1.3 ↗Lee–Shen–Tian (2021), proximal reduction ↗
best upper	Exact ULD / FORS	$\widetilde O(\kappa^{2/3}d^{1/3}\,\operatorname{polylog}(1/\varepsilon))$	$\mathcal R_q\le\varepsilon^2$	$V,\nabla V$ Cold start under the paper's setup	Exact path-space rejection implementation; fixed Rényi order.Complementary high-accuracy frontier; theorem-level transcription still needs human checking.	Citedpreprint	Chen–Chewi–Rakhlin–Zhang (2026), Theorem 3.2 ↗
best lower	General first-order oracle lower bound	$\widetilde\Omega(\min\{\sqrt\kappa\log d,d\})$	Constant total-variation accuracy	$V,\nabla V$ queriesAny randomized algorithm in the stated model	Gaussian subclass; parameter ranges in the theorem.Fixed $d\ge2$ also has the tight $\Theta_d(\log\kappa)$ dependence.	Checkedpublished	Chewi et al. (2023), query lower bounds ↗
upper/lower	LMC / ULA	Upper: $\widetilde O(\kappa\sqrt d/\varepsilon)$ ; lower: $\widetilde\Omega\bigl(\kappa+\sqrt{\kappa d}/\varepsilon\bigr)$	$\sqrt\alpha\,W_2(\mu_N,\pi)\le\varepsilon$	$\nabla V$ ; fixed-step LMCAny finite-second-moment law; initial Wasserstein distance enters logarithmically	$0<\varepsilon\le1$ ; standard curvature bounds only.Lower bound: anisotropic Gaussian targets with one curvature- $\alpha$ direction and $d-1$ curvature- $\beta$ directions, for $d\ge2$ and $\kappa\ge2$ , under worst-case initialization. Stability and slow-direction relaxation require $\Omega(\kappa)$ steps; stationary bias and relaxation give $\widetilde\Omega(\sqrt{\kappa d}/\varepsilon)$ .	Checkedpreprint	Pedrotti–Whalley (2026), Theorem 1; Corollary 1.1 ↗
upper/lower	MALA (cold start)	$\widetilde\Theta(\kappa d)$	Constant TV accuracy; upper bound extends to high accuracy with polylogarithmic overhead	$V,\nabla V$ ; fixed-step MALAUpper: $\mathcal N(x_\star,\beta^{-1}I)$ ; lower: a cold law with $\chi^2(\mu_0\Vert\pi)\le e^d$	Known mode for the upper bound; lower bound is worst-case over cold initial laws.Matching dependence on dimension and condition number in the cold-start analysis; the lower bound does not assert hardness for the particular Gaussian upper-bound initialization.	Checkedmonograph	Chewi (2026), Theorem 7.3.4 ↗Chewi (2026), Theorem 7.5.7 ↗
upper	ULMC warm start + MALA	$\widetilde O(\kappa\sqrt d\,\operatorname{polylog}(1/\varepsilon))$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$	$\nabla V$ for ULMC; $V,\nabla V$ for MALAKnown mode $x_\star$ ; Gaussian cold start with $\mathcal R_2(\mu_0\Vert\pi)=\widetilde O(d)$	Standard strong convexity and smoothness only.ULMC generates $\mathcal R_2=O(1)$ , hence $\chi^2=O(1)$ , in $\widetilde O(\kappa\sqrt d)$ queries. The displayed cost includes initialization and the subsequent MALA run.	Checkedmonograph	Chewi (2026), Corollary 7.3.6 ↗ULMC Rényi guarantee: Corollary 6.3.2 ↗Warm-start MALA: Theorem 7.3.5 ↗
upper/lower	Randomized / Poisson midpoint ULMC	Upper: RM $\widetilde O(\kappa^{5/6}d^{1/3}/\varepsilon^{2/3})$ ; Poisson ( $p=3$ ) $\widetilde O(\kappa^{13/10}d^{11/30}/\varepsilon^{1/3})$ . Lower: $\Omega(d^{1/3}/\varepsilon^{1/3})$ iterations	RM upper: $\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$ ; Poisson upper and lower bound: $\sqrt\alpha\,W_2(\mu_N,\pi)\le\varepsilon$	$\nabla V$ (Poisson: expected queries); one random quadrature time shared across coordinatesUpper: known mode $x_\star$ ; position $\mathcal N(x_\star,\beta^{-1}I)$ and standard Gaussian momentum. Lower: position $\mathcal N(e_d,I)$ , standard Gaussian momentum	Poisson upper: $0<\varepsilon\le c\min\{\sqrt{d/\kappa},d/\sqrt\kappa\}$ ; Corollary 2 with $p=3$ . Lower: standard Gaussian target ( $\kappa=1$ ), $d\ge2$ , $0<\varepsilon\le\varepsilon_0$ , Poisson grid size $k\ge2$ .Poisson accuracy is rescaled from $W_2^2\le\varepsilon^2d/\alpha$ in the paper. Its Wasserstein guarantee and the RM TV guarantee are distinct; larger $p$ gives further tradeoffs. Lower bound: the shared quadrature time makes every coordinate's variance fluctuate together, giving stationary bias $\gtrsim\min\{dh^3,\sqrt d\,h^{3/2}\}$ . It matches Poisson ULMC for fixed $p>3$ at sufficiently small $\varepsilon$ , and also holds for overdamped RLMC and PLMC; it does not bound the RM TV guarantee.	Checkedmonograph	Chewi (2026), Theorem 5.3.17 (RM) ↗Srinivasan–Nagaraj (2025), Corollary 2 and Remark 4 (Poisson) ↗Chen–Chewi–Lu–Zhang (2026), Corollary 6.3 and Proposition 6.2 (lower bound) ↗
upper/lower	Heun (deterministic midpoint LMC)	Upper: $\widetilde O(\kappa^{4/3}d^{1/3}/\varepsilon^{2/3})$ ; lower: $\widetilde\Omega\bigl(\kappa\min\{d^{1/3}/\varepsilon^{2/3},\,d^{1/2}/\varepsilon^{1/2}\}\bigr)$	$\sqrt\alpha\,W_2(\mu_N,\pi)\le\varepsilon$	$\nabla V$ ; fixed-step Heun, two gradients per stepUpper: known mode, $X_0=x_\star$ ; lower: a law with $\mathbb E[V(X_0)-V(x_\star)]\le Cd$	Upper: $0<\varepsilon\le\sqrt d/2$ ; lower: $d\ge d_0$ , $\kappa\ge\kappa_0$ , $0<\varepsilon\le\varepsilon_0$ . No third-derivative bounds.Lower bound: step-size-dependent oscillatory targets built from cosine-perturbed blocks coupled to a curvature- $\alpha$ coordinate. For $d^{-1}\le\varepsilon\le\varepsilon_0$ it reads $\widetilde\Omega(\kappa d^{1/3}/\varepsilon^{2/3})$ , matching the $d$ and $\varepsilon$ powers with a $\kappa^{1/3}$ gap. Dominated by smoothed Picard HMC, whose KL guarantee gives $\sqrt\alpha\,W_2\le\sqrt2\,\varepsilon$ .	Checkedpreprint	Chen–Chewi–Lu–Zhang (2026), Theorems 3.1 and 5.2 ↗
upper/lower	Underdamped exponential midpoint	Upper: $\widetilde O(\kappa^{5/4}d^{1/4}/\varepsilon^{1/2})$ ; lower: $\widetilde\Omega\bigl(\kappa+\kappa^{3/4}d^{1/4}/\varepsilon^{1/2}\bigr)$	$\sqrt\alpha\,W_2(\mu_N,\pi)\le\varepsilon$ (position marginal)	$\nabla V$ ; fixed step, friction $\gamma=2\sqrt\beta$ , two gradients per stepUpper: known mode, $X_0=x_\star$ , $P_0\sim\mathcal N(0,I)$ ; lower: $X_0=\alpha^{-1/2}e_d$ , $P_0\sim\mathcal N(0,I)$	Upper: $0<\varepsilon\le\sqrt d/2$ ; lower: $d\ge2$ , $\kappa\ge2$ , $0<\varepsilon\le1/2$ . No third-derivative bounds.Lower bound: anisotropic Gaussian with $d-1$ curvature- $\beta$ directions and one curvature- $\alpha$ direction; matches the $d$ and $\varepsilon$ powers with a $\kappa^{1/2}$ gap. At bounded $\kappa$ , beats the $d^{1/3}/\varepsilon^{1/3}$ lower bound for every randomized midpoint method when $\varepsilon\gg d^{-1/2}$ . Dominated by smoothed Picard HMC, whose KL guarantee gives $\sqrt\alpha\,W_2\le\sqrt2\,\varepsilon$ .	Checkedpreprint	Chen–Chewi–Lu–Zhang (2026), Theorem 4.1; Proposition 5.3 ↗
upper	Block-Krylov Gaussian sampler	$O((d\wedge\sqrt\kappa)\log(d/\varepsilon^2))$	$\sqrt{\operatorname{KL}}\le\varepsilon$	Gaussian matrix-vector / gradientCentered Gaussian model	Gaussian subclass.Nearly tight for Gaussians, not a general-target upper bound.	Checkedmonograph	Chewi (2026), Theorem 9.4.1 ↗

Scope

The default oracle returns $V(x)$ and $\nabla V(x)$ . Rows are not directly comparable unless the error metric, initialization, and additional smoothness assumptions match.

Conventions

$\widetilde O$ and $\widetilde\Omega$ suppress logarithmic factors. Rows use an explicit initialization or include the cost of generating one. Algorithm-specific matching bounds do not imply a general oracle lower bound.