Influence Is the Target-Response Jacobian Times Residuals

We introduce the target-response Jacobian $J=[\mathrm{\partial }{\widehat{y}}_i\mathrm{/}\mathrm{\partial }{y}_j]$, 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, $\mathcal{I}=J\mathrm{diag}(m)$ exactly at any stationary point with an invertible Hessian. For canonical-link losses (squared error, logistic, Poisson) the multiplier is the training residual, $\mathcal{I}=J\mathrm{diag}(r)$, 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 point’s per-class sensitivities towards its own true class (true-class collapse). We score both operators against refitting with a per-prediction $R^2$ across target perturbations, with no rescaling. On tanh networks trained with squared error (Friedman #1) and logistic loss (MAGIC, HIGGS), with $N$ from 2,000 to ${10}^6$, $J$ reaches $R^2$ between 0.99 and 1.00, whereas influence’s $R^2$ is negative on every dataset. On an MNIST softmax classifier, influence’s $R^2$ is again negative, while $J$ reaches 0.92. On HIGGS, one row of $J$, holding all ${10}^6$ sensitivities of one prediction, takes under 0.2 seconds on an RTX 3090 after a one-time setup of under five minutes, whereas refitting would take ${10}^6$ refits.