Proposition: Finite-design conditional certificate

Let $D=\{(x_i,y_i){\}}_{i=1}^n$ be fixed and possibly reused by the controller. Given $D$ and the preceding history, fix the current checkpoint and draw new independent noise for each example. Define

Then

This does not bound $\text{abs}{Q_t}^D-{Q_t}^{\mathrm{population}}$ without an additional generalization argument.

Proof. Conditional correctness indicators are independent and bounded, although their means can differ by example. Hoeffding applies to their average around the average of their conditional means. The theorem contains no comparison between a data-dependent checkpoint's empirical-design and population targets.