Characterizing the Generalization Error of Random Feature Regression with Arbitrary Data-Augmentation

Data augmentation (DA) is now a standard ingredient in modern machine learning pipelines, with extensive empirical evidence reporting improvements in generalization across modalities and tasks Mumuni and Mumuni (2022); Wang et al. (2025). It is often used to encode task-relevant symmetries directly into the training procedure, for instance by encouraging invariance to image rotations or other transformations of the input Shorten and Khoshgoftaar (2019); Chen et al. (2020). It has also been identified as one of the most effective regularization techniques across both supervised learning settings Bishop (1995); Cubuk et al. (2019); Mumuni and Mumuni (2022); Wang et al. (2025) and self-supervised/unsupervised learning Feng et al. (2021); Van Assel et al. (2025). Domain-specific augmentation pipelines have been central to progress in computer vision Shorten and Khoshgoftaar (2019); Kumar et al. (2024), natural language processing Feng et al. (2021); Shorten et al. (2021); Bayer et al. (2022), and time-series or audio applications Wen et al. (2020); Iwana and Uchida (2021); Iglesias et al. (2023). Despite these empirical successes, the benefits of DA remain highly task-and data-dependent, and augmentation schemes are often engineered in an ad hoc manner Fawzi et al. (2016); Cubuk et al. (2019); Lim et al. (2019); Hataya et al. (2020). In contrast with this rich empirical literature, comprehensive theoretical analyses of DA remain relatively scarce. Two classical starting points are, first, the interpretation of additive Gaussian noise as a form of explicit (ridge-like) regularization Bishop (1995); Lin et al. (2024), and second, the idea that leveraging distributional invariances and group structure in the learning objective helps decrease the variance of the model without increasing its bias Chen et al. (2020). Yet, when applied to modern and complex augmentation schemes, these works either provide only upper bounds on the generalization error Lin et al. (2024), or require very strong assumptions on the data distribution (e.g.