Improved sampling algorithms and functional inequalities for non-log-concave distributions

We study the problem of sampling from a distribution $μ$ with density $\propto e^{-V}$ for some potential function $V:\mathbb R^d\to \mathbb R$ with query access to $V$ and $\nabla V$. We start with the following standard assumptions: (1) $V$ is $L$-smooth. (2) The second moment $\mathbf{E}_{X\sim μ}[|X|^2]\leq M$. Recently, He and Zhang (COLT'25) showed that the query complexity of this problem is at least $\left(\frac{LM}{dε}\right)^{Ω(d)}$ where $ε$ is the desired accuracy in total variation distance, and the Poincaré constant can be unbounded. Meanwhile, another common assumption in the study of diffusion based samplers (see e.g., the work of Chen, Chewi, Li, Li, Salim and Zhang (ICLR'23)) strengthens (1) to the following: (1*) The potential function of every distribution along the Ornstein-Uhlenbeck process starting from $μ$ is $L$-smooth. We show that under the assumptions (1*) and (2), the query complexity of sampling from $μ$ can be $\mathrm{poly}(L,d)\cdot \left(\frac{Ld+M}{ε^2}\right)^{\mathcal{O}(L+1)}$, which is polynomial in $d$ and $\frac{1}ε$ when $L=\mathcal{O}(1)$ and $M=\mathrm{poly}(d)$. This improves the algorithm with quasi-polynomial query complexity developed by Huang et al. (COLT'24). Our results imply that the seemingly moderate strengthening from (1) to (1*) yields an exponential gap in the query complexity. Furthermore, we show that together with the assumption (1*) and the stronger moment assumption that $|X|$ is $λ$-sub-Gaussian for $X\simμ$, the Poincaré constant of $μ$ is at most $\mathcal{O}(λ)^{2(L+1)}$. We also establish a modified log-Sobolev inequality for $μ$ under these conditions. As an application of our technique, we obtain a new estimate of the modified log-Sobolev constant for a specific class of mixtures of strongly log-concave distributions.

💡 Research Summary

The paper investigates the computational task of sampling from a probability distribution μ on ℝⁿ whose density is proportional to e^{‑V(x)}. The authors assume only two minimal conditions: (1) the potential V is L‑smooth (i.e., its gradient is L‑Lipschitz) and (2) the second moment of μ is bounded by a constant M. Under these assumptions alone, recent lower‑bound work (He and Zhang, COLT 2025) shows that any sampling algorithm must make at least (LM/(d ε))^{Ω(d)} oracle queries to achieve total‑variation error ε, implying an exponential dependence on the dimension d.

A common strengthening in diffusion‑based sampling literature is that the potential remains L‑smooth not only for the original distribution but for every intermediate distribution along the Ornstein‑Uhlenbeck (OU) flow that starts from μ. The authors denote this stronger smoothness assumption as (1*). While (1*) looks modest, it captures a structural regularity of the whole OU trajectory that is absent under the plain smoothness assumption (1).

The main contributions are twofold. First, the authors design a variant of the restricted Gaussian dynamics (also known as the proximal sampler) that works under assumptions (1*) and (2). The algorithm proceeds in discrete time: at each step a Gaussian perturbation is added to the current state, and then a sample is drawn from the conditional distribution ν_T(· | \hat Y) obtained by exponentially tilting μ with the observed perturbation. The analysis hinges on bounding the Poincaré constant of the induced Markov chain. By interpreting the OU process through the lens of stochastic localization, the authors translate (1*) into a uniform covariance bound (Condition 1) for all exponentially tilted measures. This bound guarantees an “approximate conservation of variance” along the localization path, which in turn yields a controllable Poincaré constant for the chain.

A technical obstacle is that the integral ∫₀ᵀ (1+L_s)/(s(1+s)) ds, which appears in the variance‑conservation estimate, diverges near zero. To overcome this, the authors introduce a three‑phase “late‑initialization” scheme. They start the process at a tiny time s₀>0 where the law is close to a Gaussian, run the dynamics up to a moderate time T where the covariance bound is finite, and finally exploit the fact that for T>L the tilted measure ν_T becomes (T‑L)‑strongly log‑concave. This concatenation of phases eliminates the singularity and yields a total query complexity

 N = poly(L,d)·(Ld+M)^{O(L+1)}/ε².

When L=O(1) and M=poly(d), the algorithm runs in polynomial time, dramatically improving upon the quasi‑polynomial bound of Huang et al. (COLT 2024) and matching the lower bound only when (1*) is absent.

Second, the paper shows that adding a stronger moment condition—namely that ‖X‖ is λ‑sub‑Gaussian for X∼μ—together with (1*) yields explicit functional inequalities for μ. The authors prove that μ satisfies a Poincaré inequality with constant C_{PI}(μ) ≤ O(λ)^{2(L+1)} and a modified log‑Sobolev inequality (mLSI) with a comparable constant. The proof again relies on the variance‑conservation property of the stochastic localization process, now combined with sub‑Gaussian tail control to bound the Dirichlet form. The authors argue that without (1*), such inequalities cannot hold for the hard instances constructed in He‑Zhang (2025), because they would imply a polynomial‑time sampler contradicting the lower bound.

As an application, the authors analyze mixtures of a strongly log‑concave, L‑smooth component ρ (e.g., a Gaussian) and a bounded‑support component ν. They prove that the mixture μ = ν * ρ satisfies an mLSI with constant C_{mLSI}(μ) ≤ (1/(2m))·e^{L R²}, where m is the strong convexity parameter of ρ and R bounds the support of ν. This recovers and generalizes recent results on Gaussian mixtures (Ma‑Sly, 2023).

Overall, the paper demonstrates that the seemingly modest assumption (1*)—smoothness of all OU intermediates—has a profound impact: it reduces the sampling complexity from exponential to polynomial and enables sharp functional inequalities for non‑log‑concave distributions. The work bridges algorithmic sampling theory with geometric functional analysis, introducing a novel three‑phase concatenation technique that may be useful for studying other complex, multimodal distributions beyond the current setting.