Influence Is the Target-Response Jacobian Times Residuals

Paul Geertsema, Simon Geertsema

We introduce the target-response Jacobian $J = [\partial \hat y_i/\partial y_j]$, which measures how model predictions respond to changes in individual training targets. $J$ complements training-data attribution (TDA) methods, which characterize sensitivity to the weights placed on training observations in the loss rather than to their targets, and can be computed matrix-free at the same cost as influence-based TDA methods. We show that sample-weight influence $\mathcal{I}$ and $J$ follow from the same implicit differentiation argument and differ only by a per-example multiplier: for pointwise losses, $\mathcal{I} = J\operatorname{diag}(m)$ exactly at any stationary point with an invertible Hessian. For canonical-link losses (squared error, logistic, Poisson) the multiplier is the training residual, $\mathcal{I} = J\operatorname{diag}(r)$, and softmax cross-entropy obeys a multiclass version. The identity predicts three ways in which influence fails when used in place of $J$: small residuals shrink its predicted shift towards zero (interpolation collapse), residuals of mixed sign wash out its direction (direction washout), and under softmax cross-entropy it collapses a training…