We define an optimal preconditioning for the Langevin diffusion by analytically optimizing the expected squared jumped distance. This yields as the optimal preconditioning an inverse Fisher information covariance matrix, where the covariance matrix is computed as the outer product of log target gradients averaged under the target. We apply this result to the Metropolis adjusted Langevin algorithm (MALA) and derive a computationally efficient adaptive MCMC scheme that learns the preconditioning from the history of gradients produced as the algorithm runs. We show in several experiments that the proposed algorithm is very robust in high dimensions and significantly outperforms other methods, including a closely related adaptive MALA scheme that learns the preconditioning with standard adaptive MCMC as well as the position-dependent Riemannian manifold MALA sampler.

翻译：我们通过解析优化期望平方跳跃距离，为Langevin扩散定义了最优预条件。该优化结果得到的最优预条件为逆Fisher信息协方差矩阵，其中协方差矩阵由目标分布下对数目标梯度外积的期望计算得出。我们将此结果应用于Metropolis调整Langevin算法（MALA），并推导出一种计算高效的自适应MCMC方案，该方案通过算法运行过程中产生的梯度历史学习预条件。多项实验表明，所提出的算法在高维空间中具有极强的鲁棒性，且显著优于其他方法，包括一种密切相关的自适应MALA方案（该方案采用标准自适应MCMC学习预条件）以及位置依赖的黎曼流形MALA采样器。