Conference Agenda

Uncertainty quantification is a key issue when considering the application of deep neural network methods in science and engineering. To this end, numerous Bayesian neural network approaches have been introduced. The main challenge is to construct an algorithm which is applicable to the large sample sizes and parameter dimensions of modern applications on the one hand and which admits statistical guarantees on the other hand. A stochastic Metropolis-Hastings step saves computational costs by calculating the acceptance probabilities only on random (mini-)batches, but reduces the effective sample size leading to less accurate estimates. We demonstrate that this drawback can be fixed with a simple correction term. Focusing on deep neural network regression, we prove a PAC-Bayes oracle inequality which yields optimal contraction rates and we analyze the diameter and show high coverage probability of the resulting credible sets. The method is illustrated with a simulation example.

We extend the Monte Carlo method of Oechsler 2024 to the setting of a target distribution with heavy tails: We choose a regularly varying distribution and prove the convergence of a solution of a stochastic differential equation to this target. Hereby, the stochastic differential equation is driven by a general Lévy process - unlike in the case of a classical Langevin diffusion. This method is justified, apart from the possibility to sample from non-smooth targets, by the fact that an exponential convergence to the invariant distribution holds, which in general cannot be guaranteed by the classical Langevin diffusion in presence of heavy tails. Advantageous compared to other Langevin Monte Carlo methods is the option of an easy implementation of the method by only using a compound Poisson process as noise term and a numerically manageable drift term.