Single-Layer Perceptron Training Example

Learn an `OR` function using the following perceptron:

graph LR
x1 --> w1
x2 --> w2
x0 --> w0
w0 --> n[neuron]
w1 --> n
w2 --> n
n --> ϕ
ϕ --> y

where:

$w_0$ is a bias and $w_1$ and $w_2$ are the two inputs to the neuron.

We want to adjust the weights until all the input patterns on $x_1$ and $x_2$ produce the correct outputs.

The general method is as follows:

Calculate the induced local field by calculating the dot product of the inputs and weights:
\[\mathbf x \cdot \mathbf w =\text{ILF}\]
For this example:
\[\begin{bmatrix}-1 & 0 & 1\end{bmatrix} \cdot\begin{bmatrix}0 & 1 & 0\end{bmatrix}=0\]
Calculate $y$ by using the activation function:
\[y=\phi(\text{ILF})\]
The activation function in this case has a positive bias so pulls values $\geq0$ to $1$ and drops values $<0$ to $-1$:
\[y=\phi(0) =1\]
For an input of $x_1=0, x_2=1$ we expect an output of $d=1$. We calculate the error $e$ by taking the difference between $d$ and $y$:
\[\begin{aligned} e&=d-y\\ &=1-1\\ &=0 \end{aligned}\]
If the error is non-zero we adjust the weights by the following function:
\[\mathbf w(n+1)=\mathbf w(n)+\eta e(n)\mathbf x(n)\]
In this case where no adjustment is needed and $\eta=\frac12$:
\[\begin{aligned} \mathbf w(n+1)&=\mathbf w(n)+\eta e(n)\mathbf x(n)\\ &=\begin{bmatrix}0\\1\\0\end{bmatrix}+\frac12\times 0\times\begin{bmatrix}-1\\0\\1\end{bmatrix}\\ &=\begin{bmatrix}0\\1\\0\end{bmatrix} \end{aligned}\]