ScoreQuant

Result · SOFT-HARD-ATOMLESS-EQUIVALENCE

Randomized and hard population values can coincide

Derived from the literaturepublication: derived from published

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Meaning

Atomless purification equates optimal values for criteria depending on the preserved cell moments.

Scientific context

Randomized rules reduce to deterministic ones when the score law has no atoms. This is a known result, by the Dvoretzky–Wald–Wolfowitz purification theorem, and is cited here rather than claimed.

Statement

Because the objectives depend only on (W_b,m_b), atomless purification implies randomized and deterministic quantizers have the same attainable population objective values.

Relationships

Rests on

Enables

Raises

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Local graph
Local map of Randomized and deterministic population optima coincide for moment-based criteria under atomlessness2 prerequisites to the left, 1 results it enables or questions it raises to the right, 0 boundary cases below.Atomless purification preserves (W_b,m_b)Atomless purification…Quantized Fisher-information identityQuantized Fisher-info…Atomic randomization gapAtomic randomization…Randomized and deterministic population optima coincide for moment-based criteria under atomlessnessRandomized and determ…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

S5. Atomless purification

For an atomless score law, Dvoretzky–Wald–Wolfowitz purification preserves all (Wb,mb)(W_b,m_b) of a randomized finite-action quantizer. Hence randomized and deterministic quantizers have the same population optimum value for every criterion depending only on these moments.

This is a classical theorem applied to the present Fisher-score moment representation.

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