# Simplest perceptron

### Simple perceptron

Let's consider the following simple perceptron with a transfert function given by $$f(x)=x$$ to keep the maths simple: ### Transfert function

The perceptron global transfert function is given by the following equation: \begin{equation} y= w_1.x_1 + w_2.x_2 + ... + w_N.x_N = \sum\limits_{i=1}^N w_i.x_i \label{eq:transfert-function} \end{equation}

### Error (or loss)

In artificial neural networks, the error we want to minimize is: \begin{equation} E=(y'-y)^2 \label{eq:error} \end{equation} with:
• $$E$$ the error
• $$y'$$ the expected output (from training data set)
• $$y$$ the real output of the network (from network)
In practice and to simplify the maths, this error is divided by two: $$E=\frac{1}{2}(y'-y)^2$$

### Gradient descent

The algorithm (gradient descent) used to train the network (i.e. updating the weights) is given by: $$w_i'=w_i-\eta.\frac{dE}{dw_i}$$ where:
• $$w_i$$ the weight before update
• $$w_i'$$ the weight after update
• $$\eta$$ the learning rate

### Derivating the error

Let's derivate the error: \begin{equation} \frac{dE}{dw_i} = \frac{1}{2}\frac{d}{dw_i}(y'-y)^2 \label{eq:eq-error} \end{equation}

Thanks to the chain rule

$$(f \circ g)'=(f' \circ g).g'$$

equation \eqref{eq:eq-error} can be rewritten as:

$$\frac{dE}{dw_i} = \frac{2}{2}(y'-y)\frac{d}{dw_i} (y'-y) = -(y'-y)\frac{dy}{dw_i}$$ As $$y= w_1.x_1 + w_2.x_2 + ... + w_N.x_N$$ : $$\frac{dE}{dw_i} = -(y'-y)\frac{d}{dw_i}(w_1.x_1 + w_2.x_2 + ... + w_N.x_N) = -(y'-y)x_i$$

### Updating the weights

The weights can be updated with the following formula: $$w_i'=w_i-\eta.\frac{dE}{dw_i} = w_i+\eta(y'-y)x_i$$ In conclusion : $$w_i'= w_i + \eta(y'-y)x_i$$