Subpolynomial-Query Proximal Sampling Beyond Log-Concavity

We study the gradient query complexity of sampling from a density π(x)-V(x) on Rᵈ, where V has a $β$-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 $π$ satisfies a log-Sobolev inequality (LSI) with constant $α$, then for every fixed $ν>0$, there is a finite constant C_ν depending only on $ν$ such that the algorithm returns a sample within total-variation distance ε using at most C_νκ(d+κ+ε⁻¹+\|∇ V(x₀)\|²/β)^ν queries, where $κ=β/α$ and x₀ 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 Poincaré inequality (PI), we obtain a corresponding guarantee from a warm start with a known bound on its Rényi divergence. An oracle lower bound shows that a stationary point alone, which suffices under LSI, is insufficient under PI.