  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.
