拓展对称性理论，证明非紧凑对称模型在真实数据上更优。
Symmetries in PAC-Bayesian Learning
- 基于PAC-Bayes框架，将对称性理论扩展到平移等非紧凑对称情形。
- 在多个非均匀数据集上验证，新边界比以往结果更紧且仍成立。
- 为对称模型在真实场景中的优势提供理论支撑，适合理论研究者。
对称性被证实能提升机器学习模型的实证性能，但解释这些增益的理论保障仍有限。以往研究主要聚焦于紧致群对称性，并常假设数据分布本身不变，这一假设在真实应用中很少成立。本文将泛化保证拓展至更广泛的非紧凑对称情形，如平移变换，且适用于非不变的数据分布。基于PAC-Bayes框架，我们改进并收紧了现有界，以McAllester的PAC-Bayes界为例，证明该方法可推广至多种典型PAC-Bayes界。实验在多个具有非均匀、非紧凑变换的数据集上进行，所导出的保证不仅成立，还优于先前结果。这些发现为对称数据下选择对称模型提供了理论依据，突破了传统紧致群与分布不变性的限制，推动了对机器学习中对称性更普遍的理解。
原文摘要 · Abstract (English)
Symmetries are known to improve the empirical performance of machine learning models, yet theoretical guarantees explaining these gains remain limited. Prior work has focused mainly on compact group symmetries and often assumes that the data distribution itself is invariant, an assumption rarely satisfied in real-world applications. In this work, we extend generalization guarantees to the broader setting of non-compact symmetries, such as translations and to non-invariant data distributions. Building on the PAC-Bayes framework, we adapt and tighten existing bounds, demonstrating the approach on McAllester's PAC-Bayes bound while showing that it applies to a wide range of PAC-Bayes bounds. We validate our theory with experiments on several datasets with non-uniform and non-compact transformations, where the derived guarantees not only hold but also improve upon prior results. These findings provide theoretical evidence that, for symmetric data, symmetric models are preferable beyond the narrow setting of compact groups and invariant distributions, opening the way to a more general understanding of symmetries in machine learning.
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