Optimal Preconditioning and Fisher Adaptive Langevin Sampling

TL;DR

This paper introduces an optimal preconditioning method for Langevin diffusion based on Fisher information, leading to an adaptive MCMC algorithm that is robust and efficient in high-dimensional sampling tasks.

Contribution

It derives an optimal preconditioning matrix as the inverse Fisher information and develops an adaptive MCMC scheme that learns this preconditioning during sampling.

Findings

The proposed method outperforms existing adaptive MALA schemes.

It is highly robust in high-dimensional settings.

The algorithm significantly improves sampling efficiency.

Abstract

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.