We show that high-accuracy guarantees for log-concave sampling -- that is, iteration and query complexities which scale as $\mathrm{poly}\log(1/δ)$, where $δ$ is the desired target accuracy -- are achievable using stochastic gradients with subexponential tails. Notably, this exhibits a separation with the problem of convex optimization, where stochasticity (even additive Gaussian noise) in the gradient oracle incurs $\mathrm{poly}(1/δ)$ queries. We also give an information-theoretic argument that light-tailed stochastic gradients are necessary for high accuracy: for example, in the bounded variance case, we show that the minimax-optimal query complexity scales as $Θ(1/δ)$. Our framework also provides similar high accuracy guarantees under stochastic zeroth order (value) queries, and an improved complexity result for sampling from finite-sum potentials.

翻译：我们证明，对于对数凹采样问题，若使用具有亚指数尾部的随机梯度，则可实现高精度保证——即迭代次数与查询复杂度均按 $\mathrm{poly}\log(1/δ)$ 缩放（其中 $δ$ 为期望的目标精度）。值得注意的是，这一结论与凸优化问题形成显著分离：在梯度黑箱中引入随机性（甚至加性高斯噪声）时，后者的查询复杂度需达到 $\mathrm{poly}(1/δ)$。我们进一步通过信息论论证表明，轻尾随机梯度是实现高精度的必要条件。例如，在方差有界情形下，极小极大最优查询复杂度为 $Θ(1/δ)$。此外，本文框架同样适用于基于随机零阶（函数值）查询的场景，并为有限和势能采样问题提供了改进的复杂度结果。