AlgorithmAlgorithm: 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.

Finite-sample AdaBoost.M1 with half-log coefficients

Definition

Fix a finite training sample $(x_{i}, y_{i})_{i = 1}^{N}$ with $N \geq 1$ , $x_{i} \in X$ , and $y_{i} \in {- 1, + 1}$ , a finite round limit $M \geq 0$ , and a weak-learning routine which at each requested round returns a classifier $h_{m} : X \to {- 1, + 1}$ for the current observation weights. Initialize $w_{i}^{(1)} = 1 / N$ and $F_{0} (x) = 0$ . At round $m$ , compute the weighted error $e_{m} = \sum_{i = 1}^{N} w_{i}^{(m)} 1 {h_{m} (x_{i}) \neq y_{i}}$ . If $e_{m} = 0$ , halt and return $h_{m}$ itself as a perfect classifier with zero training error; do not form an infinite coefficient. If $e_{m} \geq 1 / 2$ , halt and report that the weak-learning condition failed. For $0 < e_{m} < 1 / 2$ , set

$a_{m} = \frac{1}{2} \log \frac{1 - e_{m}}{e_{m}}, Z_{m} = 2 \sqrt{e_{m} (1 - e_{m})},$

update $w_{i}^{(m + 1)} = w_{i}^{(m)} \exp (- a_{m} y_{i} h_{m} (x_{i})) / Z_{m}$ , and set $F_{m} (x) = F_{m - 1} (x) + a_{m} h_{m} (x)$ . After $M$ completed weak rounds, return the classifier ${\hat{y}}_{M} (x) = {sign}_{+} (F_{M} (x))$ , where ${sign}_{+} (u) = + 1$ for $u \geq 0$ and $- 1$ for $u < 0$ . Thus ties are assigned to $+ 1$ . The coefficient convention is the half-log convention; multiplying every coefficient by the same positive factor does not change the sign classifier. The definition makes no claim about test error or generalization.

Definition references

Trevor Hastie, Robert Tibshirani, and Jerome Friedman, The Elements of Statistical Learning, 2nd ed., §10.1, Algorithm 10.1, printed pp. 337–338

Robert E. Schapire, “The Boosting Approach to Machine Learning: An Overview,” §2, PDF pp. 2–4

Used by

AdaBoost stops at a perfect round Example
Two AdaBoost rounds with normalized weights Example
AdaBoost's exponential training-loss and nonpositive-margin bound Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Trevor Hastie, Robert Tibshirani, and Jerome Friedman, The Elements of Statistical Learning, 2nd ed., 12th printing (standard reference, not scraped)
Robert E. Schapire, "The Boosting Approach to Machine Learning: An Overview," MSRI Workshop on Nonlinear Estimation and Classification, 2002 (standard reference, not scraped)