Counterexample · E-FIRSTORDER-NOT-FINITE

A positive E margin need not improve a finite move

Counterexamplepublication: internal

Meaning

An exact counterexample separates infinitesimal screening information from the gain of a whole-observation relocation.

Scientific context

For the smallest-eigenvalue criterion, a positive first-order margin can come with a negative exact gain — the same kind of failure documented for the determinant criterion's profiled case, but here affecting the base criterion itself rather than only a profiled version of it. As with E-GLOBAL-GEOMETRY-FAILS, no prior-art search is recorded for this example.

Statement

A positive E subgradient-rule margin can coexist with a strictly negative exact finite E gain.

Where it stops

Refuted by

CE-E-GEOMETRY-001
At a finite global E optimum with a simple minimum eigenvalue, the rank-one metric vv^T built from the minimum eigenvector assigns every training row to its own cell (nearest-cell rule).

Relationships

Enables

Quantitative E necessity bound under spectral gapOPEN-E-SPECTRAL-GAP-BOUND

Raises

Quantitative E necessity bound under spectral gapOPEN-E-SPECTRAL-GAP-BOUND

Open next door

Local graph

Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The example

E4. Positive first-order E margin need not imply exact improvement

Reported suite: 2,167/8,965 subgradient-rule-improving moves had strictly negative exact E gain; one example had margin 2.27 and exact change $-0.240$ .

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