Example. Counterexample: Small average proxy error fails after distribution shift [ftip-007J]
AGENTDRAFTED
Let the finite decision set be
\(\mathcal Y_0=\{y_{\rm safe},y_{\rm exploit}\}\). Define its utility and
proxy utility by the following table.
\[
\begin {array}{c|cc}
&y_{\rm safe}&y_{\rm exploit}\\ \hline
U&1&0\\
\widehat U&1&2.
\end {array}
\]
For \(0<\delta <1\), let a reference law \(P_\delta \) assign probability
\(\delta \) to \(y_{\rm exploit}\). Its mean absolute proxy error is
\[
\mathbb E_{P_\delta }\lvert \widehat U-U\rvert =2\delta ,
\]
which tends to zero with \(\delta \). Nevertheless, proxy maximization
selects \(y_{\rm exploit}\) and incurs utility regret \(1\). Under the shifted
law concentrated on that selected outcome, the mean error is \(2\). Thus no
vanishing selection-regret bound can depend only on average error under the
reference law.
This two-outcome construction isolates the distribution-shift warning in
Remark [ftip-0061]. It does not claim that a particular learned verifier has this
error profile; Example [ftip-0060] gives a source-linked checker instance with the
same proxy-versus-utility separation.