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 a rotated MNIST dataset with a non-uniform rotation group, 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.

翻译：已知对称性能够提升机器学习模型的实证性能，然而解释这些增益的理论保证仍然有限。先前的工作主要集中于紧群对称性，并且通常假设数据分布本身是不变的，这一假设在现实应用中很少得到满足。在本工作中，我们将泛化保证扩展到更广泛的非紧对称性（例如平移）以及非不变数据分布的场景。基于PAC-Bayes框架，我们改进并收紧现有界，以McAllester的PAC-Bayes界为例展示了该方法，同时表明其适用于广泛的PAC-Bayes界。我们通过在具有非均匀旋转群的旋转MNIST数据集上的实验验证了我们的理论，其中推导出的保证不仅成立，而且优于先前的结果。这些发现提供了理论证据，表明对于对称数据，对称模型在紧群与不变分布这一狭窄设定之外同样更可取，从而为更广泛地理解机器学习中的对称性开辟了道路。