Editing Perceptron (section)

=== Definitions ===

We first define some variables:
*<math>r</math> is the [[learning rate]] of the perceptron. Learning rate is a positive number usually chosen to be less than 1. The larger the value, the greater the chance for volatility in the weight changes.
*<math>y = f(\mathbf{z}) </math> denotes the ''output'' from the perceptron for an input vector <math>\mathbf{z}</math>.
*<math>D = \{(\mathbf{x}_1,d_1),\dots,(\mathbf{x}_s,d_s)\} </math> is the ''training set'' of <math>s</math> samples, where:
** <math>\mathbf{x}_j</math> is the <math>n</math>-dimensional input vector.
** <math>d_j </math> is the desired output value of the perceptron for that input.
We show the values of the features as follows:
*<math>x_{j,i} </math> is the value of the <math>i</math>th feature of the <math>j</math>th training ''input vector''.
*<math>x_{j,0} = 1 </math>.
To represent the weights: 
*<math>w_i </math> is the <math>i</math>th value in the ''weight vector'', to be multiplied by the value of the <math>i</math>th input feature.
*Because <math>x_{j,0} = 1 </math>, the <math>w_0 </math> is effectively a bias that we use instead of the bias constant <math>b</math>.
To show the time-dependence of <math>\mathbf{w}</math>, we use:
*<math>w_i(t) </math> is the weight <math>i</math> at time <math>t</math>.