Smoothness-Based Derandomization of PAC-Bayes Bounds

2026-06-17 · Source: Takara TLDR - Daily AI Papers · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, medium

Summary

Alexandre Lemire Paquin, Brahim Chaib-Draa, and Philippe Giguère introduce a framework for PAC-Bayes derandomization tailored for smooth loss functions. Their research aims to derive generalization bounds for deterministic predictors by leveraging the smoothness properties of both the loss function and the predictor class. They demonstrate that the transition from a Gibbs predictor to a deterministic predictor at the posterior mean incurs a precise cost, quantified by the generalization gap of the Jensen gap class. This class is controlled using Rademacher complexity, leading to bounds for deterministic predictors that incorporate flatness quantities, specifically parameter Jacobians and Hessians of the score map. The framework is applicable to both bounded and unbounded smooth loss functions, with specialized results for linear predictors and smooth neural networks. These theoretical Jacobian and Hessian quantities also inspire a practical regularizer, which was computed for BatchNorm networks using effective BatchNorm weights. Experiments on CIFAR-10 illustrate the regularizer's performance across varying batch sizes.

Key takeaway

For machine learning engineers developing robust models, understanding generalization for deterministic predictors is crucial. This research suggests you can achieve tighter PAC-Bayes bounds by explicitly accounting for the smoothness of your loss functions and predictor classes. Consider integrating regularizers derived from parameter Jacobians and Hessians into your neural network training, especially for BatchNorm architectures, to improve generalization and model stability. Your experiments on datasets like CIFAR-10 can validate the practical impact of these smoothness-based regularization techniques.

Key insights

PAC-Bayes derandomization for smooth loss functions quantifies the cost of deterministic predictors using flatness metrics.

Principles

• Smoothness of loss and predictor class enables tighter generalization bounds.
• Generalization gap of Jensen gap class quantifies Gibbs-to-deterministic cost.
• Flatness quantities (Jacobians, Hessians) are key to deterministic bounds.

Method

Control the Jensen gap class via Rademacher complexity to derive deterministic predictor bounds involving parameter Jacobians and Hessians of the score map.

In practice

• Apply the framework to linear predictors and smooth neural networks.
• Implement a regularizer based on parameter Jacobians and Hessians.
• Compute regularizer for BatchNorm networks by folding transformations.

Topics

• PAC-Bayes
• Generalization Bounds
• Smoothness
• Neural Networks
• Regularization
• BatchNorm

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Takara TLDR - Daily AI Papers.