Ouestion No $04$:

Consider a single perceptron unit with a bias term and $n$ inputs $x0,x1,$dotsxn.

The perceptron uses weights $w0,w1,$dotswn and the given sigmoid activation

function to compute the

output.

$($i$.$e$.\{$:$f(x)=\frac{1}{1+e^{-x}})$

Derive the weight update equation for such a perceptron using the following

definition of

where yj is the target

Error $={\displaystyle \underset{j}{\overset{m}{}}}(y_j-y_j^{**}{)}^2+{\displaystyle \underset{i=0}{\overset{n}{}}}{w}_i^2$

output and $y{j}^{**}$

is the output computed using weights and the activation function of neuron.

Hints:

Derivative of

function can be

$f^{\text{'}}(x)=f(x)*(1-f(x))$

sigmoid activation

written as

Weights are updated

$W_i=W_i-E_i$

using the equation

Therefore, to derive the final equation you need to compute the derivative of

Error with respect to each of the weights components wi and then use the weight

update equation given above to derive the final equation