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.

翻译：我们证明，利用具有次指数尾部的随机梯度，可以实现对数凹分布采样的高精度保证——即迭代和查询复杂度按 $\mathrm{poly}\log(1/δ)$ 缩放，其中 $δ$ 为期望的目标精度。值得注意的是，这揭示了与凸优化问题的分离：在凸优化中，梯度预言机中的随机性（即使是加性高斯噪声）会导致 $\mathrm{poly}(1/δ)$ 的查询复杂度。我们还通过信息论论证表明，实现高精度必须使用轻尾随机梯度：例如，在有界方差情形下，我们证明极小极大最优查询复杂度按 $Θ(1/δ)$ 缩放。我们的框架同样能在随机零阶（函数值）查询下提供类似的高精度保证。