canonical setting

Smooth non-log-concave + Fisher accuracy

$\nabla V$ is $\beta$ -Lipschitz, $K_0=\operatorname{KL}(\mu_0\Vert\pi)<\infty$ ; target $\operatorname{FI}\le\varepsilon^2$ .

Updated Aug 11, 2026

Upper and lower bounds

Comparison table

5 scoped results. Tildes suppress logarithmic factors.

Edit table

Best-known and comparison results for Smooth non-log-concave + Fisher accuracy
Result	Algorithm or model	Complexity	Guarantee	Oracle / start	Assumptions and notes	Review	Sources
best upper	Exact ULD / FORS	$\widetilde O(\beta d^{1/3} K_0/\varepsilon^2)$	$\operatorname{FI}\le\varepsilon^2$	$V,\nabla V$ $\operatorname{KL}(\mu_0\Vert\pi)=K_0$	Exact-diffusion implementation assumptions from the paper.Checked against the cited preprint. The upper bound depends on the initial log Rényi divergence.	Checkedpreprint	Chen–Chewi–Rakhlin–Zhang (2026), §1.2 and §6 ↗Chewi–Wibisono (2026) ↗
best lower	General first-order Fisher lower bound	$\varepsilon^{-2+o(1)}$	$\operatorname{FI}\le\varepsilon^2$	First-order $K_0=1$	Dimension grows mildly with accuracy.High-accuracy lower bound in the stated growing-dimension regime.	Checkedpublished	Chewi–Gerber–Lee–Lu (ALT 2023) ↗Chewi (2026), Theorem 11.4.4 ↗
upper/lower	Averaged LMC	$\Theta(\beta^2 d K_0/\varepsilon^4)$	$\operatorname{FI}\le\varepsilon^2$ on average	$\nabla V$ $\operatorname{KL}(\mu_0\Vert\pi)=K_0$	The upper bound is checked. The matching lower bound is an AI-authored note and has not been independently checked.	Checkedmonograph	Chewi (2026), Theorem 11.2.1 ↗Balasubramanian–Chewi–Erdogdu–Salim–Zhang (COLT 2022) ↗Averaged LMC FI lower bound (AI note, Aug. 10, 2026) ↗
lower	One-dimensional first-order Fisher lower bound	$\Omega\bigl(\varepsilon^{-1}\sqrt{\log(1/\varepsilon)}\bigr)$	$\operatorname{FI}\le\varepsilon^2$	First-order $K_0=1$	$d=1$ ; sufficiently small $\varepsilon$ .Separated from the growing-dimension lower bound.	Checkedpublished	Chewi–Gerber–Lee–Lu (ALT 2023) ↗Chewi (2026), Theorem 11.4.4 ↗
lower	Large-initial-gap query complexity	$\Theta(\beta K_0/\varepsilon^2)$	$\operatorname{FI}\le\varepsilon^2$ at $\varepsilon^2=\beta d$	First-orderKL gap $K_0$	$\widetilde\Omega(K_0^{2/3})\le d\le\widetilde O(K_0)$ .Equivalent to $\Theta(K_0/d)$ at the stated Fisher scale.	Checkedpublished	Chewi–Gerber–Lee–Lu (ALT 2023) ↗Chewi (2026), Theorem 11.4.3 ↗

Scope

Without log-concavity, Fisher information is used as a local stationarity criterion. A small Fisher guarantee does not by itself imply global total variation accuracy without an additional functional inequality.