  One way to avoid overfitting in machine learning is to use model parameters
distributed according to a Bayesian posterior given the data, rather than the
maximum likelihood estimator. Stochastic gradient Langevin dynamics (SGLD) is
one algorithm to approximate such Bayesian posteriors for large models and
datasets. SGLD is a standard stochastic gradient descent to which is added a
controlled amount of noise, specifically scaled so that the parameter converges
in law to the posterior distribution [WT11, TTV16]. The posterior predictive
distribution can be approximated by an ensemble of samples from the trajectory.
  Choice of the variance of the noise is known to impact the practical behavior
of SGLD: for instance, noise should be smaller for sensitive parameter
directions. Theoretically, it has been suggested to use the inverse Fisher
information matrix of the model as the variance of the noise, since it is also
the variance of the Bayesian posterior [PT13, AKW12, GC11]. But the Fisher
matrix is costly to compute for large- dimensional models.
  Here we use the easily computed Fisher matrix approximations for deep neural
networks from [MO16, Oll15]. The resulting natural Langevin dynamics combines
the advantages of Amari's natural gradient descent and Fisher-preconditioned
Langevin dynamics for large neural networks.
  Small-scale experiments on MNIST show that Fisher matrix preconditioning
brings SGLD close to dropout as a regularizing technique.
