AdamMCMC: Combining Metropolis Adjusted Langevin with Momentum-based Optimization

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.