The problem of generating random samples of high-dimensional posterior distributions is considered. The main results consist of non-asymptotic computational guarantees for Langevin-type MCMC algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. As a direct consequence, it is shown that posterior mean vectors as well as optimisation based maximum a posteriori (MAP) estimates are computable in polynomial time, with high probability under the distribution of the data. These results are complemented by statistical guarantees for recovery of the ground truth parameter generating the data. Our results are derived in a general high-dimensional non-linear regression setting (with Gaussian process priors) where posterior measures are not necessarily log-concave, employing a set of local `geometric' assumptions on the parameter space, and assuming that a good initialiser of the algorithm is available. The theory is applied to a representative non-linear example from PDEs involving a steady-state Schr\"odinger equation.

中文翻译：

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朗之万型算法对高维后验测度的多项式时间计算

考虑了生成高维后验分布的随机样本的问题。主要结果包括对 Langevin 型 MCMC 算法的非渐近计算保证，这些算法在关键数量（例如模型的维度、所需的精度水平和可用统计测量的数量）中进行多项式缩放。直接结果表明，后验平均向量以及基于优化的最大后验 (MAP) 估计在多项式时间内是可计算的，在数据​​分布下具有很高的概率。这些结果由用于恢复生成数据的地面实况参数的统计保证补充。我们的结果是在一般的高维非线性回归设置（具有高斯过程先验）中得出的，其中后验度量不一定是对数凹的，在参数空间上采用一组局部“几何”假设，并假设一个好的算法的初始化程序可用。该理论被应用于 PDE 中一个具有代表性的非线性示例，该示例涉及稳态 Schr\"odinger 方程。