Uncertainty Quantification for Sparse Deep Learning
This work addresses uncertainty quantification for deep learning practitioners, offering foundational theoretical justifications, though it is incremental as it builds on existing Bayesian non-parametric tools.
The paper tackles the lack of uncertainty quantification in deep learning by adopting a Bayesian approach, providing theoretical guarantees that Bayesian credible regions have valid frequentist coverage for sparse deep ReLU architectures in non-parametric regression.
Deep learning methods continue to have a decided impact on machine learning, both in theory and in practice. Statistical theoretical developments have been mostly concerned with approximability or rates of estimation when recovering infinite dimensional objects (curves or densities). Despite the impressive array of available theoretical results, the literature has been largely silent about uncertainty quantification for deep learning. This paper takes a step forward in this important direction by taking a Bayesian point of view. We study Gaussian approximability of certain aspects of posterior distributions of sparse deep ReLU architectures in non-parametric regression. Building on tools from Bayesian non-parametrics, we provide semi-parametric Bernstein-von Mises theorems for linear and quadratic functionals, which guarantee that implied Bayesian credible regions have valid frequentist coverage. Our results provide new theoretical justifications for (Bayesian) deep learning with ReLU activation functions, highlighting their inferential potential.