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.

Bias-free linear margins and the hard-margin SVM

Definition

A finite binary sample is a list $(x_{i}, y_{i})_{i = 1}^{n}$ with $n \geq 1$ and $d \geq 1$ , features $x_{i} \in R^{d}$ , and labels $y_{i} \in {- 1, + 1}$ . The bias-free linear score with parameter $w \in R^{d}$ is $x \mapsto ⟨ w, x ⟩$ , and its signed margin on example $i$ is $m_{i} (w) : = y_{i} ⟨ w, x_{i} ⟩$ . The sample is strictly linearly separable if there exists $u \in R^{d}$ with $\min_{1 \leq i \leq n} m_{i} (u) > 0$ . Its hard-margin feasible set is $F : = {w \in R^{d} : m_{i} (w) \geq 1 for all i}$ , and its hard-margin problem is minimizing $\frac{1}{2} ∥ w ∥_{2}^{2}$ over $F$ . No intercept or bias term is present in this score or objective.

Definition references

Daniel Soudry et al., The Implicit Bias of Gradient Descent on Separable Data, equations (1) and (4)

Matus Telgarsky, Deep Learning Theory lecture notes, §10.1

Used by

Dependency tree · 0 levels

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

Sources

Daniel Soudry, Elad Hoffer, Mor Shpigel Nacson, Suriya Gunasekar, and Nathan Srebro, The Implicit Bias of Gradient Descent on Separable Data, equations (1) and (4) (standard reference, not scraped)
Matus Telgarsky, Deep Learning Theory lecture notes, §10.1 (standard reference, not scraped)