Parameter-State Evaluation Graphs: Structure and Prediction in Finite-Horizon Adaptation

Learning procedures can differ not only in which updates they apply, but also in which historical parameter state each update reads. We study this choice in finite-horizon adaptation and represent it as a parameter-state evaluation graph. Common summaries describe the amount of staleness, but discard the update identity associated with each stale read. We show that this missing identity information matters. Graphs with the same updates, order, and aggregate delay statistics can still reach different endpoints because the same delays attach to different updates. An exact affine analysis explains this difference. From a single fully sequential reference trajectory, we predict the endpoints of unexecuted graphs by measuring the defect introduced by each stale read and propagating it through the remaining updates. In the primary supervised setting, this pathwise prediction improves graph ranking and captures matched-delay differences missed by aggregate summaries. Our results indicate that the parameter state read by each update should be treated as part of the finite-horizon learning algorithm, not merely as an implementation detail, and that part of its effect can be predicted from a single reference trajectory.