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Bounds for a small number of standard sampling problems.

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Each page fixes a target class and oracle model, then lists upper bounds, lower bounds, initialization, metric, and caveats in one table.

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\pi\propto e^{-V}

\alpha I\preceq\nabla^2V\preceq\beta I

\kappa=\beta/\alpha

Best high-accuracy upper $\widetilde O(\kappa d^{1/5}\,\operatorname{polylog}(1/\varepsilon))$ Gaussian cloud samplerChecked

Best low-accuracy upper $\widetilde O(\kappa^{7/6}d^{1/6}/\varepsilon^{1/3})$ Smoothed Picard HMC + recursive warm startChecked

Best lower $\widetilde\Omega(\min\{\sqrt\kappa\log d,d\})$ General first-order oracle lower boundChecked

\pi=\operatorname{unif}(K)

B(0,1)\subseteq K

, membership queries;

\Lambda=\lVert\operatorname{Cov}(\pi)\rVert_{\rm op}

Best upper $\widetilde O(qd^2\Lambda\log^6(1/\varepsilon))$ Proximal / In-and-Out with restartCited

Lower bound unknownUnknownGeneral sampling lower boundUnverified

\pi\propto e^{-V}

0\preceq\nabla^2V\preceq\beta I

R_0^2=W_2^2(\mu_0,\pi)

Best upper $\widetilde O(\beta\sqrt d\,R_0^2/\varepsilon^2)$ Implemented proximal samplerChecked

Lower bound unknownUnknownMatching first-order lower boundUnverified

\nabla V

\beta

-Lipschitz;

C_{\rm PI}

C_{\rm LSI}\le1/\alpha

;

\kappa=\beta/\alpha

Best upper $\widetilde O(\kappa d^{1/5}\,\operatorname{polylog}(1/\varepsilon))$ Gaussian cloud samplerChecked

Lower bound unknownUnknownMatching PI/LSI oracle lower boundUnverified

\nabla V

\beta

-Lipschitz,

K_0=\operatorname{KL}(\mu_0\Vert\pi)<\infty

; target

\operatorname{FI}\le\varepsilon^2

Best upper $\widetilde O(\beta d^{1/3} K_0/\varepsilon^2)$ Exact ULD / FORSChecked

Best lower $\varepsilon^{-2+o(1)}$ General first-order Fisher lower boundChecked

V=m^{-1}\sum_{i=1}^m f_i

or unbiased noisy gradients; oracle cost counts component or stochastic queries.

Best upper $\widetilde O((\kappa\sqrt d+\sigma_\psi^2/\alpha)\operatorname{polylog}(1/\varepsilon))$ FORSChecked

Best lower $\Omega(\sigma^2/(\alpha\varepsilon))$ Bounded-variance stochastic-gradient lower boundChecked

Best finite-sum upper $\widetilde O((m+\kappa\sqrt{dm})\operatorname{polylog}(1/\varepsilon))$ Variance-reduced high-accuracy samplerChecked

V

convex,

\lVert\nabla V(x)-\nabla V(y)\rVert\le\beta_s\lVert x-y\rVert^s

s\in[0,1]

Best upper $\widetilde O(\beta_s^{2/(1+s)}d^{s/(1+s)}R_0^2/\varepsilon^2)$ FORS + proximal samplerCited

Lower bound unknownUnknownMatching Hölder-model lower boundUnverified