Result · CONSISTENCY-RESTRICTED-AFFINE

Consistency within a restricted rule family

Proved herepublication: internal

Meaning

Uniform convergence supports value consistency in a compact finite-capacity family with mass and conditioning margins.

Scientific context

For a compact, finite-capacity family of nearest-centre rules with uniformly positive cell masses and bounded conditioning, cell moments and objective values converge uniformly, and approximate empirical maximizers are value-consistent. This is the statement that fitting a rule on a sample and deploying it on new data is asymptotically sound, but only inside that restricted class.

This is an adaptation of the standard empirical-process route to consistency; its registry statement adds uniform conditioning margins and an isolated population optimum as extra hypotheses beyond that standard route. The unrestricted version, without those margins, is open (OPEN-D-UNRESTRICTED-CONSISTENCY).

Statement

For a compact finite-capacity affine-max class with uniform cell-mass and conditioning margins and appropriate score integrability, cell moments converge uniformly; objective values converge uniformly; approximate empirical maximizers are value-consistent, and isolated population optima give decision/parameter consistency up to labels.

Assumptions

compact affine-max family
uniform positive cell masses
required information eigenvalues bounded away from zero
bounded scores or suitable uniform integrability

Where it stops

The unrestricted/global consistency problem remains open.

Relationships

Enables

Raises

Unrestricted global D consistencyOPEN-D-UNRESTRICTED-CONSISTENCY

Open next door

Local graph

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

Proof

C1. Restricted affine-class consistency

Let $\mathcal Q$ be a compact parameterized class of $K$ -cell affine-max quantizers. Assume bounded scores or suitable uniform integrability, cell masses uniformly bounded below, and the required information matrices uniformly bounded away from singularity.

Then empirical $W_b$ and $m_b$ converge uniformly over $\mathcal Q$ . Consequently D, regular $D_s$ , and E objective values converge uniformly on that regular subset. Approximate empirical maximizers are value-consistent; with an isolated population optimum, standard argmax theory gives parameter/decision consistency up to label permutations.

The proof uses finite capacity of multiclass affine decision regions + ULLN + continuity.

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