# Posterior Concentration for Sparse Deep Learning

## TL;DR

This paper introduces Spike-and-Slab Deep Learning (SS-DL), a Bayesian regularization method that adaptively recovers smooth input-output maps with unknown smoothness levels, providing theoretical guarantees for deep ReLU networks.

## Contribution

It presents a Bayesian approach that achieves near minimax rates for smoothness-adaptive deep learning, guiding architecture design without prior smoothness knowledge.

## Key findings

- Posterior concentrates at near minimax rate for smooth maps
- SS-DL does not overfit, favoring smaller networks
- Provides theoretical justification for deep ReLU networks

## Abstract

Spike-and-Slab Deep Learning (SS-DL) is a fully Bayesian alternative to Dropout for improving generalizability of deep ReLU networks. This new type of regularization enables provable recovery of smooth input-output maps with unknown levels of smoothness. Indeed, we show that the posterior distribution concentrates at the near minimax rate for $\alpha$-H\"older smooth maps, performing as well as if we knew the smoothness level $\alpha$ ahead of time. Our result sheds light on architecture design for deep neural networks, namely the choice of depth, width and sparsity level. These network attributes typically depend on unknown smoothness in order to be optimal. We obviate this constraint with the fully Bayes construction. As an aside, we show that SS-DL does not overfit in the sense that the posterior concentrates on smaller networks with fewer (up to the optimal number of) nodes and links. Our results provide new theoretical justifications for deep ReLU networks from a Bayesian point of view.