论文标题
关于Langevin型算法的高维后措施的多项式计算
On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms
论文作者
论文摘要
考虑了产生高维后分布的随机样品的问题。主要结果包括langevin型MCMC算法的非反应计算保证，该算法以关键数量（例如模型的尺寸，所需的精度水平和可用统计测量的数量）进行多项式扩展。直接结果，结果表明，后均值向量以及基于优化的最大后验（MAP）估计值在多项式时间内可计算，并且在数据的分布下具有很高的概率。这些结果是通过恢复基础真相参数生成数据的统计保证的补充。 我们的结果是在一般的高维非线性回归设置（带有高斯工艺先验的）中得出的，其中后验措施不一定是对数孔的，并采用了一组局部“几何”假设，并假设算法的良好初始分析器可用。该理论应用于涉及稳态Schrödinger方程的PDE的代表性非线性示例。
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 Schrödinger equation.
