Definition. Uniform proxy approximation [ftip-007G]
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Let \(\mathcal Y_0\) be a nonempty finite set of candidate outcomes. Let \(U:\mathcal Y_0\to \mathbb R\) be an independently declared utility and \(\widehat U:\mathcal Y_0\to \mathbb R\) a proxy score. The proxy is a uniform approximation to \(U\) on \(\mathcal Y_0\) with error \(\varepsilon \) when \(\varepsilon \geq 0\) and

\[ \sup _{y\in \mathcal Y_0}\lvert \widehat U(y)-U(y)\rvert \leq \varepsilon . \]

The candidate set is part of the claim. Approximation on training outcomes, on one policy's support, or in expectation is not uniform approximation on a larger set reached after optimization.

Definition. Uniform proxy approximation [ftip-007G]
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Proxy reward and verifier fidelity [ftip-007F]

Definition. Uniform proxy approximation [ftip-007G]
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Theorem. Uniform proxy error gives two-epsilon selection regret [ftip-007H]

Proof.

Remark. The two-epsilon theorem is a decision bound [ftip-007I]

Example. Counterexample: Small average proxy error fails after distribution shift [ftip-007J]

Definition. Repeated verifier audit [ftip-007K]

Theorem. Independent false accepts amplify with candidate count [ftip-007L]

Proof.

Theorem. A false-accept union bound without independence [ftip-007M]

Proof.

Example. Weight updates and candidate selection are different proxy operators [ftip-007N]

Remark. What verifier error rates do not identify [ftip-007O]

Theorem. Uniform proxy error and objective regret [ftip-00IA]

Theorem. Uniform proxy error gives two-epsilon selection regret [ftip-007H]

Proof.