In the delta rule for error minimization, weights are adjusted with respect to:

Concept

The delta rule is a supervised gradient-descent learning rule. It minimizes the squared error between a desired output d and the actual output y of a unit.

For a linear unit, E = ½(d − y)2 and the update is Δw = η(d − y)x. The residual d − y supplies the error signal, while the input x scales the change in the weight.

Application

1. Identify d as the desired output and y as the actual output.

2. Compute the error signal e = d − y.

3. Substitute it into Δw = ηex; therefore, the weight adjustment is governed by the difference between the desired and actual outputs.

Contrast

• The change in the network output measures how y moves between observations; it is not the target residual in the squared-error objective.

• The input-output difference x − y compares quantities with different roles; it is not the supervised error signal.

• The current weight value describes the parameter before the update; by itself it does not define the supervised error signal.

Cross-check

If d = y, the residual is zero and the delta rule makes no weight change. If d and y differ, the sign and magnitude of their residual drive the correction, exactly as gradient descent requires.

Result

Therefore, weights are adjusted with respect to the difference between the desired output and the actual output.