DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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.

One-sided empirical Rademacher complexity

Definition

Let $m \geq 1$ , let $S = (z_{1}, \dots, z_{m})$ be a fixed sample in a measurable space $X$ , and let $F$ be a nonempty finite or countable class of measurable real functions such that the displayed supremum is finite and measurable on the finite sign space. Let $σ_{1}, \dots, σ_{m}$ be independent uniform signs. The one-sided empirical Rademacher complexity is

${\hat{R}}_{S} (F) : = \frac{1}{m} E_{σ} \sup_{f \in F} \sum_{i = 1}^{m} σ_{i} f (z_{i}) .$

There is no absolute value in this definition and $0$ need not belong to $F$ . It differs from the absolute-value complexity $m^{- 1} E_{σ} \sup_{f \in F} ∣ \sum_{i} σ_{i} f (z_{i}) ∣$ .

Remarks

This convention is the one used in One-sided coordinate Rademacher contraction.

Depends on

Empirical, centered, and Rademacher processes

Used by

A two-function class on a Bernoulli sample Example
Absolute-value Rademacher contraction Theorem
One-sided coordinate Rademacher contraction Theorem

Dependency tree · two levels

6 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

Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., Definition 3.1 (standard reference, not scraped)
David Pollard, Empirical Processes: Theory and Applications, Chapter 5 (standard reference, not scraped)