As yet another example we define now a simple perceptron model with all quantities given by scalars. We consider only one input variable \( x \) and one target value \( y \). We define an activation function \( \sigma_1 \) which takes as input
$$ z_1 = w_1x+b_1, $$where \( w_1 \) is the weight and \( b_1 \) is the bias. These are the parameters we want to optimize. The output is \( a_1=\sigma(z_1) \) (see graph from whiteboard notes). This output is then fed into the cost/loss function, which we here for the sake of simplicity just define as the squared error
$$ C(x;w_1,b_1)=\frac{1}{2}(a_1-y)^2. $$