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：本文通过解析优化期望平方跃迁距离，定义了Langevin扩散的最优预条件。这一最优预条件为逆Fisher信息协方差矩阵，其中协方差矩阵通过目标分布下对数目标梯度外积的期望计算得出。我们将该结果应用于Metropolis调整Langevin算法（MALA），并推导出一种计算高效的自适应MCMC方案，该方案在算法运行过程中从梯度历史中学习预条件。多项实验表明，所提算法在高维空间中具有极强鲁棒性，且显著优于其他方法，包括采用标准自适应MCMC学习预条件的紧密相关自适应MALA方案，以及依赖于位置的黎曼流形MALA采样器。