Learning with the perceptron means to adjust the weights automatically according to some rule so that the perceptron will produce a specific output for a given input. The rule that has been proposed by Frank Rosenblatt uses the error, which is the deviation of the perceptron’s output from the expected output as correctiv signal. For the AND function the expected output \(y\) is [0,0,0,1] given the inputs A [0,0,1,1] and B [0,1,0,1]. The error \(e\) is thus formulated as the difference of the expected output and the output computed by the perceptron. $$ e = y -output $$
This error term is then multiplied with a learning factor and with the specific input combination that produced the error to produce an update term \( \Delta w_{i}\)
$$\Delta w_{i}= \lambda \times e \times x_{i} $$
The corresponding weight \(w_{i}\) is then updated by integrating this update term \(\Delta w_{i}\)
$$w_{i,new}=w_{i,old} + \Delta w_{i}$$