Learning rule demonstration


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

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