# How Many Samples Are Enough for Learning Across Domains?

arXiv: https://arxiv.org/abs/2609.39336 · HTML: https://mlsanity.com/paper/2609.39336/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Domain Generalization via Invariant Feature Representation](https://mlsanity.com/paper/1301.2115/) (2013)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [The Risks of Invariant Risk Minimization](https://mlsanity.com/paper/2010.05761/) (2020)
- [Accuracy on the Line: On the Strong Correlation Between Out-of-Distribution and In-Distribution Generalization](https://mlsanity.com/paper/2107.04649/) (2021)
- [Invariant Risk Minimization Games](https://mlsanity.com/paper/2002.04692/) (2020)
- [A causal framework for distribution generalization](https://mlsanity.com/paper/2006.07433/) (2020)
- [Provable Domain Generalization via Invariant-Feature Subspace Recovery](https://mlsanity.com/paper/2201.12919/) (2022)
- [Causal Balancing for Domain Generalization](https://mlsanity.com/paper/2206.05263/) (2022)
- [(Almost) Provable Error Bounds Under Distribution Shift via Disagreement Discrepancy](https://mlsanity.com/paper/2306.00312/) (2023)
- [Generalization Error Bounds for Multiclass Sparse Linear Classifiers](https://mlsanity.com/paper/2204.06264/) (2022)
- [Learning Optimal Features via Partial Invariance](https://mlsanity.com/paper/2301.12067/) (2023)
- [How Many Domains Suffice for Domain Generalization? A Tight Characterization via the Domain Shattering Dimension](https://mlsanity.com/paper/2506.16704/) (2025)
- [Distribution Shift Is Key to Learning Invariant Prediction](https://mlsanity.com/paper/2601.12296/) (2026)

## Abstract (arXiv)

Understanding the fundamental mechanisms of learning is essential for designing systems with strong generalization. Recent studies have shown that increasing the number of training domains, or enlarging the distribution shift among them, improves generalization when each domain contains sufficiently many data samples. However, the conditions under which the data samples can be considered sufficient remain unexplored. In this work, we fill this gap by establishing criteria for per-domain sample requirements based on the presented learning bounds. These criteria not only reveal an inverse linear scaling law between the number of training domains and the number of samples required per domain, but also explain the fundamental rationale behind the assumption of data sufficiency, thereby providing theoretical guidance for assessing the adequacy of existing datasets and constructing datasets. This differs from classical learning theory, as the number of samples required is highly dependent on the number of training domains. Additionally, we prove the close relationship between in-domain learning and out-of-domain generalization through the presented generalization bounds, and lastly discuss some key arguments.