AlgorithmAlgorithm: AI-adaptedProof: 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.

Finite-list empirical risk minimization

Definition

Given a sample $S$ of length $m \geq 1$ , an ordered finite nonempty list of measurable predictors $(f_{1}, \dots, f_{n})$ , and a finite tolerance $η \geq 0$ , compute $r_{j} : = {\hat{R}}_{S} (f_{j})$ for each $j \in {1, \dots, n}$ . If every $r_{j} = + \infty$ , return index $1$ . Otherwise set $r_{*} : = \min_{1 \leq j \leq n} r_{j}$ and return the least index $j$ for which $r_{j} < + \infty$ and $r_{j} \leq r_{*} + η$ . The output index denotes the selected predictor from the supplied list. When $η = 0$ , the output is a first exact minimizer, including the all-infinite case. The algorithm evaluates only a finite number of empirical risks and makes no population-risk guarantee.

Definition references

Supervised prediction problems: fixes the measurable observation and predictor spaces.

Population and empirical risk: defines each empirical risk evaluated by the procedure.

Empirical risk minimizers: defines exact and approximate ERM, including the all-infinite branch.

Population and empirical risks are measurable: the sample objective for each listed predictor is measurable.

Depends on

Used by

Duplicate predictors do not enlarge the finite class Example
Zero training error can coexist with positive population risk Example
The finite-list ERM selector is measurable Lemma
Finite-class agnostic ERM bound Theorem
Finite-class realizable PAC bound Theorem

Dependency tree · two levels

14 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

Shai Shalev-Shwartz and Shai Ben-David, Understanding Machine Learning: From Theory to Algorithms, Chapter 2 §2.2 (standard reference, not scraped)
Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., Chapter 2 §2.2 (standard reference, not scraped)