Widrow and Hoff's 1960 LMS gradient is a closed-form product of measured error and input for one pattern, with no backward sweep and no dynamics model

widrowhofflmsadalinegradientmadalinebackpropagation-origins

In "Adaptive Switching Circuits" (Widrow and Hoff, 1960) the single-pattern error is e(k) = d(k) − a₀ − a₁s₁(k) − … − a_n s_n(k) (Eq. 9). For that pattern the partials of the squared error reduce to −2·e·s_j (Eqs. 12–13). The paper says that "all partial derivatives have the same magnitude, and have signs determined by the error sign and the respective input signal signs," and that the procedure has the advantage "that derivatives are very easy to measure and that no storage is required within the adaptive machinery except for the gain values."

Mechanically this is a one-stage gradient available in closed form from the forward error alone. Nothing lies between the weights and the output error, so there is no analogue of the adjoint system in claim-kelley-bryson-optimal-control-precursor, and no model of system dynamics is needed. The paper's only multi-unit scheme (its section F) assigns "responsibility" for a desired +1 output to whichever of two Adalines has the smaller measured error and adapts only that one; when the desired output is −1, both machines adapt in the usual manner (precision corrected 2026-09-27, cross-model audit). No error signal is propagated through one neuron to another. This is consistent with the vault's account of Widrow's multilayer work (claim-madaline-rule-i-first-layer-trainable-second-fixed, claim-widrow-abandoned-multilayer-training-until-1985-backprop).

The single-sample character of the step is covered in claim-widrow-hoff-1960-original-paper-describes-lms-as-stochastic-steepest-descent, and its unbiasedness in claim-widrow-lehr-1990-lms-instantaneous-gradient-unbiased-estimate. The contrast with Kelley's computation is drawn in claim-kelley-and-widrow-hoff-1960-share-steepest-descent-root-not-backward-sweep-mechanism.