====== Lecture 3 ======

===== 3.1 - Learning the weights of a linear neuron =====

  * For multi-layers neural networks the average of two good solutions may be a bad solution => we cannot use the perceptron learning procedure.
  * Here instead of making the weights get closer to a set of good weights, we just try to make the output get closer to the expected output.

  * For linear neurons, the output is \(y = w^T x\)
  * To measure the error we use the squared error between Y and t.
  * We then use the "delta-rule" for learning: \(\Delta w_i = \epsilon x_i (t-y)\)

==== Deriving the delta rule ====

  * We start with \(E = \frac 12 \sum\limits_n (t^n - y^n)^2\)
  * Then we differentiate to get error derivatives for weights: \(\frac{\partial E}{\partial w_i} = \frac 12 \sum\limits_n \frac {\partial y^n}{\partial w_i} \frac{d E^n}{d y^n} = - \sum\limits_n x_i^n (t^n - y^n)\)

  * Then we use \(\Delta w_i = - \epsilon \frac{\partial E}{\partial w_i} = \sum\limits_n \epsilon x_i^n (t^n - y^n)\)

===== 3.2 - The error surface for a linear neuron =====

  * For linear neuron, the error surface is a quadratic bowl

===== 3.3 - Learning the weights of a logistic output neuron =====

==== Derivatives of a logistic neuron ====

  * \(z = b + \sum\limits_i x_i w_i\)
  * \(\frac{\partial z}{\partial w_i} = x_i\)
  * \(\frac{\partial z}{\partial x_i} = w_i\)

  * Then \(y=\frac{1}{1+e^{-z}}\), so \(\frac{dy}{dz} = y(1-y)\)

  * So we get \(\frac{\partial y}{\partial w_i} = \frac {\partial z}{\partial w_i} \frac{d y}{d z} = x_i y (1-y)\)

  * And \(\frac{\partial E}{\partial w_i} = \sum\limits_n \frac {\partial y^n}{\partial w_i} \frac{d E^n}{d y^n} = - \sum\limits_n x_i^n y^n (1-y^n) (t^n - y^n)\)

===== 3.4 - The backpropagation algorithm =====

  * Randomly pertubing one weight and then checking if it improves the performances may work, but it is very inefficient compared to backpropagation.

FIXME : add the formulas for the backpropagation here (after understanding them!)


===== 3.5 - Using the derivatives computed by backpropagation =====