Existing guarantees for algorithms sampling from nonlogconcave measures on $\mathbb{R}^d$ are generally inexplicit or unscalable. Even for the class of measures with logdensities that have bounded Hessians and are strongly concave outside a Euclidean ball of radius $R$, no available theory is comprehensively satisfactory with respect to both $R$ and $d$. In this paper, it is shown that complete polynomial complexity can in fact be achieved if $R\leq c\sqrt{d}$, whilst an exponential number of point evaluations is generally necessary for any algorithm as soon as $R\geq C\sqrt{d}$ for constants $C>c>0$. A simple importance sampler with tail-matching proposal achieves the former, owing to a blessing of dimensionality. On the other hand, if strong concavity outside a ball is replaced by a distant dissipativity condition, then sampling guarantees must generally scale exponentially with $d$ in all parameter regimes.

翻译：现有关于从 $\mathbb{R}^d$ 上非对数凹测度采样的算法保证通常是不显式或不可扩展的。即使对于对数密度具有有界 Hessian 矩阵且在半径为 $R$ 的欧几里得球外强凹的测度类，现有理论在 $R$ 和 $d$ 两方面均未提供全面令人满意的结果。本文证明，若 $R\leq c\sqrt{d}$，则实际上可实现完全多项式复杂度；而一旦 $R\geq C\sqrt{d}$（其中常数 $C>c>0$），任何算法通常都需要指数级的点评估次数。一个具有尾部匹配建议分布的简单重要性采样器凭借维数祝福实现了前者。另一方面，若将球外强凹性替换为远距离耗散条件，则采样保证在所有参数范围内通常必须随 $d$ 呈指数级缩放。