# Learning rule demonstration

## Hypothesys

This page demonstrates the learning rule for updating weights in a single layer artificial neural network. Since the learning rule is the same for each perceptron, we will focus on a single one. In this demonstration, we will assume we want to update the weights with respect to the gradient descent algorithm.

## Transfert function

Let's consider the following perceptron:

The transfert function is given by:

$$\begin{equation} y= f(w_1.x_1 + w_2.x_2 + ... + w_N.x_N) = f(\sum\limits_{i=1}^N w_i.x_i) \label{eq:transfert-function} \end{equation}$$

Let's define the sum $$S$$:

$$\begin{equation} S(w_i,x_i)= \sum\limits_{i=1}^N w_i.x_i \label{eq:sum} \end{equation}$$

Let's rewrite $$y$$ as a function of $$S$$ by merging equations $$\eqref{eq:sum}$$ and $$\eqref{eq:transfert-function}$$:

$$y(S)= f(\sum\limits_{i=1}^N w_i.x_i)=f(S(w_i,x_i))$$

## Error (or loss)

In artificial neural networks, the error we want to minimize is:

$$E=(y'-y)^2$$

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:

$$E=\frac{1}{2}(y'-y)^2$$

The algorithm (gradient descent) used to train the network (i.e. updating the weights) is given by:

$$\begin{equation} w_i'=w_i-\eta.\frac{dE}{dw_i} \label{eq:gradient-descent} \end{equation}$$

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:

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

Thanks to the chain rule

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

the equation $$\eqref{eq:error}$$ can be rewritten:

$$\frac{dE}{dw_i} = \frac{2}{2}(y'-y)\frac{d}{dw_i} (y'-y) = -(y'-y)\frac{dy}{dw_i}$$

Let's now calculate the derivative of $$y$$:

$$\begin{equation} \frac{dy}{dw_i} = \frac{df(S(w_i,x_i))}{dw_i} \label{eq:dy-dwi} \end{equation}$$

Once again, we use the chain rule to rewrite equation $$\eqref{eq:dy-dwi}$$ :

$$\frac{df(S)}{dw_i} = \frac{df(S)}{dS}\frac{dS}{dw_i} = x_i\frac{df(S)}{dS}$$

The derivative of the error becomes:

$$\begin{equation} \frac{dE}{dw_i} = -x_i(y'-y)\frac{df(S)}{dS} \label{eq:derror} \end{equation}$$

## Updating the weights

By merging equations $$\eqref{eq:gradient-descent}$$ and $$\eqref{eq:derror}$$ the weights can be updated with the following formula:

$$w_i'=w_i-\eta.\frac{dE}{dw_i} = w_i + \eta. x_i.(y'-y).\frac{df(S)}{dS}$$

In conclusion:

$$w_i'= w_i + \eta.x_i.(y'-y).\frac{df(S)}{dS}$$