Definition. First positive sparse-reward time [ftip-00A4]
Definition. First positive sparse-reward time [ftip-00A4]
Fix an executable sparse-reward rule and let \(r_j^{\mathrm {sp}}\) be the reward returned on rollout \(j\). The first positive sparse-reward time is the extended-valued index
\[ J_+=\inf \{j\geq 1:r_j^{\mathrm {sp}}>0\}, \qquad \inf \varnothing :=\infty . \]This definition concerns the reward implementation named by an experiment cell. The event \(r_j^{\mathrm {sp}}>0\) coincides with a target-response event only when that equality is part of the declared reward semantics. In particular, a length penalty or partial-credit rule can separate positive reward from exact target match.