Polynomial time sampling from log-smooth distributions in fixed dimension under semi-log-concavity of the forward diffusion with application to strongly dissipative distributions

Abstract: In this article, we provide a stochastic sampling algorithm with polynomial complexity in fixed dimension that leverages the recent advances on diffusion models where it is shown that under mild conditions, sampling can be achieved via an accurate estimation of intermediate scores across the marginals $(p_t)_{t\ge 0}$ of the standard Ornstein-Uhlenbeck process started at $\mu$, the density we wish to sample from. The heart of our method consists into approaching these scores via a computationally cheap estimator and relating the variance of this estimator to the smoothness properties of the forward process. Under the assumption that the density to sample from is $L$-log-smooth and that the forward process is semi-log-concave: $-\nabla^2 \log(p_t) \succeq -\beta I_d$ for some $\beta \geq 0$, we prove that our algorithm achieves an expected $\epsilon$ error in $\text{KL}$ divergence in $O(d^7(L+\beta)^2L^{d+2}\epsilon^{-2(d+3)}(d+m_2(\mu))^{2(d+1)})$ time with $m_2(\mu)$ the second order moment of $\mu$. In particular, our result allows to fully transfer the problem of sampling from a log-smooth distribution into a regularity estimate problem. As an application, we derive an exponential complexity improvement for the problem of sampling from an $L$-log-smooth distribution that is $\alpha$-strongly log-concave outside some ball of radius $R$: after proving that such distributions verify the semi-log-concavity assumption, a result which might be of independent interest, we recover a $poly(R, L, \alpha^{-1}, \epsilon^{-1})$ complexity in fixed dimension which exponentially improves upon the previously known $poly(e^{LR^2}, L,\alpha^{-1}, \log(\epsilon^{-1}))$ complexity in the low precision regime.