We study the complexity of heavy-tailed sampling and present a separation result in terms of obtaining high-accuracy versus low-accuracy guarantees i.e., samplers that require only $O(\log(1/\varepsilon))$ versus $\Omega(\text{poly}(1/\varepsilon))$ iterations to output a sample which is $\varepsilon$-close to the target in $\chi^2$-divergence. Our results are presented for proximal samplers that are based on Gaussian versus stable oracles. We show that proximal samplers based on the Gaussian oracle have a fundamental barrier in that they necessarily achieve only low-accuracy guarantees when sampling from a class of heavy-tailed targets. In contrast, proximal samplers based on the stable oracle exhibit high-accuracy guarantees, thereby overcoming the aforementioned limitation. We also prove lower bounds for samplers under the stable oracle and show that our upper bounds cannot be fundamentally improved.

翻译：我们研究了重尾分布采样的复杂度，并针对高精度与低精度保证的获取提出了分离性结果，即分别需要$O(\log(1/\varepsilon))$次迭代与$\Omega(\text{poly}(1/\varepsilon))$次迭代才能输出一个在$\chi^2$散度下与目标分布$\varepsilon$接近的样本的采样器。我们的结果针对基于高斯预言机与稳定预言机的邻近采样器进行阐述。我们证明，基于高斯预言机的邻近采样器存在一个根本性局限：在对一类重尾目标分布进行采样时，它们必然只能实现低精度保证。相比之下，基于稳定预言机的邻近采样器则展现出高精度保证，从而克服了上述限制。我们还证明了在稳定预言机下采样器的下界，并表明我们的上界无法在根本上进一步改进。