  Recently, there has been a significant focus on exploring the theoretical
aspects of deep learning, especially regarding its performance in
classification tasks. Bayesian deep learning has emerged as a unified
probabilistic framework, seeking to integrate deep learning with Bayesian
methodologies seamlessly. However, there exists a gap in the theoretical
understanding of Bayesian approaches in deep learning for classification. This
study presents an attempt to bridge that gap. By leveraging PAC-Bayes bounds
techniques, we present theoretical results on the prediction or
misclassification error of a probabilistic approach utilizing Spike-and-Slab
priors for sparse deep learning in classification. We establish non-asymptotic
results for the prediction error. Additionally, we demonstrate that, by
considering different architectures, our results can achieve minimax optimal
rates in both low and high-dimensional settings, up to a logarithmic factor.
Moreover, our additional logarithmic term yields slight improvements over
previous works. Additionally, we propose and analyze an automated model
selection approach aimed at optimally choosing a network architecture with
guaranteed optimality.
