cs.LGSep 30, 2026

How Many Samples Are Enough for Learning Across Domains?

Authors: Hong Zheng

Organizations: This work was completed during my Ph.D. period and was the result of a flash of inspiration. I decided to submit it to arXiv and leave its value to the readers.

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

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.

Figures & tables

Figure 1: Examples illustrating the advantage of pursuing tighter upper bounds. The solid lines corresponding to N1N_{1} and N2N_{2} are obtained from our criteria by considering the equality case, while the shaded regions denote the feasible regions defined by the inequality case. NminN_{min} denotes the limiting value of both N1N_{1} and N2N_{2} as E→∞E\to\infty . As illustrated, given ϵ\epsilon , N2N_{2} provides a more precise estimation of NN . Eventually, as EE increases, both N1N_{1} and N2N_{2} approach NminN_{min} .
Figure 2: Theoretical predictions for ε\varepsilon by controlling EE and NN . The left subfigure shows the results of ε(N)\varepsilon(N) with fixed EE , while the right subfigure shows the results of ε(E)\varepsilon(E) with fixed NN . The left subfigure demonstrates that, for different values of EE , ε\varepsilon eventually converges to similar asymptotic values as NN increases. The right subfigure demonstrates that, however, for different values of NN , ε\varepsilon converges to different asymptotic values as EE increases.

Explore similar work

CardsList

  1. Hierarchical Domain Generalization

    Jul 17, 2026Chenxiao Yang, Zhiyuan Li, Shai Ben-David +1Multimodal Domain GeneralizationHypothesis Class

    Toyota Technological Institute at Chicago · University of Waterloo

  2. Can Domain Generalization be Guaranteed in Small-Sample Learning?

    Sep 30, 2026Hong ZhengMultimodal Domain GeneralizationGeneralization Bounds

    This work was completed during my Ph.D. period and was the result of a flash of inspiration. I decided to submit it to arXiv and leave its value to the readers.

  3. Explaining Data Mixing Scaling Laws

    Jun 6, 2026Rui Dai, Shuran ZhengScaling LawsMixture Models

    Beijing Institute of Technology · IIIS, Tsinghua University