ScoreQuant

Result · HARD-GEOMETRIC-EMPIRICAL-PIECEWISE-CONSTANT

Hard empirical objectives have flat regions

Proved herepublication: internal

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Meaning

Moving a boundary without crossing a sample point leaves its labels and hard objective unchanged.

Statement

For hard affine/Voronoi parameterizations, finite empirical labels and cell moments remain unchanged until a training row crosses a boundary; ordinary gradient is therefore zero almost everywhere.

optimizer design guardrail

Relationships

Enables

Raises

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In the library

Carried by SoftVoronoiConfig. Reference.

Every object and refusal

Local graph
Local map of Hard finite geometric objective is piecewise constant in boundary parameters0 prerequisites to the left, 1 results it enables or questions it raises to the right, 0 boundary cases below.Soft-to-hard zero-temperature limitSoft-to-hard zero-tem…Hard finite geometric objective is piecewise constant in boundary parametersHard finite geometric…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

S3. Hard empirical geometric objective is piecewise constant

For a hard affine/Voronoi quantizer parameterized by continuous generators/hyperplanes, the finite empirical objective is piecewise constant in those parameters: until a training row crosses a boundary, labels and cell moments do not change.

Therefore ordinary gradient descent on the hard finite objective is zero almost everywhere and is not a useful generic solver.

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