Subpolynomial-Query Proximal Sampling Beyond Log-Concavity

Abstract

We study the gradient query complexity of sampling from a density $\pi(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ has a $\beta$-Lipschitz gradient but need not be convex. Our approach uses the proximal sampler, which alternates Gaussian perturbation with sampling a position conditional on the noisy observation. We implement its observation chain using the high-order noisy conditional sampler of [OpenAI 2026], obtaining improved dimension dependence under functional inequalities. If $\pi$ satisfies a log-Sobolev inequality (LSI) with constant $\alpha$, then for every fixed $\nu>0$, there is a finite constant $C_\nu$ depending only on $\nu$ such that the algorithm returns a sample within total-variation distance $\varepsilon$ using at most $C_\nu\kappa(d+\kappa+\varepsilon^{-1}+\|\nabla V(x_0)\|^2/\beta)^\nu$ queries, where $\kappa=\beta/\alpha$ and $x_0$ is an arbitrary reference point for initialization. The analysis controls center-dependent local errors along exact proximal histories and removes the final observation noise by conditional decoding. Under a Poincar\'e inequality (PI), we obtain a corresponding guarantee from a warm start with a known bound on its R\'enyi divergence. An oracle lower bound shows that a stationary point alone, which suffices under LSI, is insufficient under PI.