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Misclassification bounds for PAC-Bayesian sparse deep learning

The Tien Mai

TL;DR

The paper develops a PAC-Bayesian analysis for sparse deep classifiers using Spike-and-Slab priors and hinge-loss based risk, yielding non-asymptotic misclassification bounds via an EWA (Gibbs) posterior. It proves slow- and fast-rate oracle inequalities and shows minimax-optimal rates in both low- and high-dimensional settings up to logarithmic factors, with explicit architectural regimes. An automatic architecture selection procedure is proposed, achieving adaptivity and guaranteeing near-optimal rates by balancing expected hinge risk and posterior–prior complexity. The results bridge Bayesian DNN theory with classical minimax theory, providing practical generalization guarantees and principled model-architecture selection for sparse networks.

Abstract

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.

Misclassification bounds for PAC-Bayesian sparse deep learning

TL;DR

The paper develops a PAC-Bayesian analysis for sparse deep classifiers using Spike-and-Slab priors and hinge-loss based risk, yielding non-asymptotic misclassification bounds via an EWA (Gibbs) posterior. It proves slow- and fast-rate oracle inequalities and shows minimax-optimal rates in both low- and high-dimensional settings up to logarithmic factors, with explicit architectural regimes. An automatic architecture selection procedure is proposed, achieving adaptivity and guaranteeing near-optimal rates by balancing expected hinge risk and posterior–prior complexity. The results bridge Bayesian DNN theory with classical minimax theory, providing practical generalization guarantees and principled model-architecture selection for sparse networks.

Abstract

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.
Paper Structure (12 sections, 14 theorems, 50 equations)

This paper contains 12 sections, 14 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Problem and method
  3. Problem statement
  4. Deep neural networks
  5. EWA procedure using empirical hinge loss
  6. Sparsity prior
  7. Main results
  8. Assumptions
  9. General results
  10. Rates in specific settings
  11. Architecture design via model selection
  12. Proof

Key Result

Theorem 1

Given Assumption asm1 and Assumption assume_bound_on_thetruebayes, for $\lambda = \sqrt{n}$, we find that with a probability of at least $1-2\epsilon$, where $\epsilon \in (0,1)$, the following holds: where $c$ depends only ong $C_B, C'$.

Theorems & Definitions (29)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Example 1
  • Proposition 6
  • Remark 2
  • Example 2
  • ...and 19 more