DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Loss discrepancy between domains

Definition

Let $(X, Σ_{X})$ and $(Y, Σ_{Y})$ be measurable spaces, let $P$ and $Q$ be probability measures on $X$ (in particular, source and target input marginals), and let $H$ be a nonempty class of measurable functions $h : X \to Y$ . Let $L : Y \times Y \to [0, + \infty]$ be product-measurable. For $h, h^{'} \in H$ , the map $x \mapsto (h^{'} (x), h (x))$ is measurable because the preimage of each measurable rectangle $A \times B$ is $h^{' - 1} (A) \cap h^{- 1} (B)$ and such rectangles generate the product sigma-algebra; composing with $L$ shows that $x \mapsto L (h^{'} (x), h (x))$ is measurable. Assume that, for every $h, h^{'} \in H$ , the two integrals below are finite, and define $L_{P} (h^{'}, h) : = \int_{X} L (h^{'} (x), h (x)) P (d x), L_{Q} (h^{'}, h) : = \int_{X} L (h^{'} (x), h (x)) Q (d x) .$ The $L$ -discrepancy between $P$ and $Q$ over $H$ is the extended nonnegative number ${disc}_{L} (P, Q) : = \sup_{h, h^{'} \in H} ∣ L_{P} (h^{'}, h) - L_{Q} (h^{'}, h) ∣ \in [0, + \infty] .$ The supremum need not be attained.

Definition references

Source and target joint laws, risks, and covariate shift: supplies the source and target input marginals and their common measurable input space.

A measurable function between measurable spaces: a map is measurable when preimages of measurable sets are measurable.

The product sigma-algebra and its finite iterates: the product sigma-algebra is generated by measurable rectangles.

Mansour, Mohri, and Rostamizadeh, Definition 4 and Theorem 8, arXiv version 3, pp. 4–6: the discrepancy distance for a general loss and its domain-adaptation bound.

Sugiyama, §2 and §3.1, book pp. 928–931: generalization risk under distinct input laws and loss reweighting under a shared conditional label law.

Depends on

Used by

Loss-discrepancy domain-adaptation bound Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh, Domain Adaptation: Learning Bounds and Algorithms, arXiv:0902.3430v3, Definition 4 (standard reference, not scraped)
Masashi Sugiyama, Learning under Non-stationarity: Covariate Shift Adaptation by Importance Weighting, Handbook of Computational Statistics, 2nd ed., Chapter 31 (2012) (standard reference, not scraped)