This paper provides a comprehensive error analysis of learning with vector-valued random features (RF). The theory is developed for RF ridge regression in a fully general infinite-dimensional input-output setting, but nonetheless applies to and improves existing finite-dimensional analyses. In contrast to comparable work in the literature, the approach proposed here relies on a direct analysis of the underlying risk functional and completely avoids the explicit RF ridge regression solution formula in terms of random matrices. This removes the need for concentration results in random matrix theory or their generalizations to random operators. The main results established in this paper include strong consistency of vector-valued RF estimators under model misspecification and minimax optimal convergence rates in the well-specified setting. The parameter complexity (number of random features) and sample complexity (number of labeled data) required to achieve such rates are comparable with Monte Carlo intuition and free from logarithmic factors.

Several studies have compared the in-distribution (ID) and out-ofdistribution (OOD) performance of models in computer vision and NLP. They report a frequent positive correlation, but surprisingly, almost never an inverse correlation that would be indicative of a necessary trade-off. Such inverse patterns are possible theoretically, and their occurrence in practice is important to determine whether ID performance can serve as a proxy for OOD generalization.

A.1 Creation of the Multimodal Web Document Dataset A.1.1 Collecting of a Large Number of HTMLFiles Our data collection process begins by considering the 25 most recent Common Crawl6 dumps available at the time of dataset creation. It contains webpages spanning from February 2020 to January/February 2023. We use a modified version of readability-lxml7 to extract the main text from the pages, discarding any pages that contain text of excessively high perplexity. This process yields a total of 41.2 billion documents. Selection of English content To identify non-English content, we apply the FastText classifier (Joulin et al., 2017) to the extracted text, e ectively filtering out 63.6% of the documents. Early text deduplication Often, a set of URLs is crawled repeatedly across di erent Common Crawl snapshots. However, the content of these websites may vary as web administrators make changes over time. Hence, at this stage, we refrain from deduplicating documents based on their URLs. Instead, we perform MinHash (Broder, 1997) deduplication with 16 hashes calculated over 5-grams. To further refine the data, we eliminate documents containing substantial proportions of repeated paragraphs and n-grams, employing the methodology described in MassiveText (Rae et al., 2022).

Invariant graph representation learning aims to learn the invariance among data from different environments for out-of-distribution generalization on graphs. As the graph environment partitions are usually expensive to obtain, augmenting the environment information has become the de facto approach. However, the usefulness of the augmented environment information has never been verified. In this work, we find that it is fundamentally impossible to learn invariant graph representations via environment augmentation without additional assumptions. Therefore, we develop a set of minimal assumptions, including variation sufficiency and variation consistency, for feasible invariant graph learning.

Invariant graph representation learning aims to learn the invariance among data from different environments for out-of-distribution generalization on graphs. As the graph environment partitions are usually expensive to obtain, augmenting the environment information has become the de facto approach. However, the usefulness of the augmented environment information has never been verified. In this work, we find that it is fundamentally impossible to learn invariant graph representations via environment augmentation without additional assumptions. Therefore, we develop a set of minimal assumptions, including variation sufficiency and variation consistency, for feasible invariant graph learning.