AdamMCMC: Combining Metropolis Adjusted Langevin with Momentum-based
  Optimization
Sebastian Bieringer Gregor Kasieczka Maximilian F. Steffen Mathias Trabs
Uncertainty estimation is a key issue when considering the application of
deep neural network methods in science and engineering. In this work, we
introduce a novel algorithm that quantifies epistemic uncertainty via Monte
Carlo sampling from a tempered posterior distribution. It combines the well
established Metropolis Adjusted Langevin Algorithm (MALA) with momentum-based
optimization using Adam and leverages a prolate proposal distribution, to
efficiently draw from the posterior. We prove that the constructed chain admits
the Gibbs posterior as an invariant distribution and converges to this Gibbs
posterior in total variation distance. Numerical evaluations are postponed to a
first revision.