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Closer look into Learning Algorithm of a Neuron

ML Neural Networks | 23 Jan 2017
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In perceptron model weights are changed to get a better set of weights. In multi-layer neural network this algorithm won't work and we don't use perceptron learning algorithm.

For multi-layer nets the objective is the actual outputs values reaching target output values.

Linear Neurons

The neuron has a real-valued output which is weighted sum of its inputs:

\begin{equation}  y = \sum_{i}{x_i w_i} = W^T X  \end{equation}

$W$ - Weight vector
$X$ - input vector

The aim will be to minimize the error. Error in this case is the squared difference between the desired output and the actual output.

Delta Rule & Iterative Algorithm

\begin{equation} \Delta w_i = \varepsilon x_i (t - y) \end{equation}

where:

$\varepsilon$ - learning rate

$t - y$ - (difference between the target and the estimate, i.e. the residual error)

Based on iterative algorithm delta rule adjustments will be done in the system to ensure that weights are adjusted by $ \Delta w_i $ with each iteration which should lead with to better result in case of Linear Neuron.

Note: (Deriving Delta Rule is explained separately)

Error Surface of a Linear Neuron

Assume the space where "all the horizontal dimensions" correspond to the weights and a vertical dimension corresponds to the error.

In this case errors made on each set of weights would define the error surface, which is a quadratic bowl.

 

You can get the same on octave, using the source code:

vx = [-5:0.5:5];
vy = vx;
[x, y] = meshgrid(vx, vy);
z = x.^2 + y.^2;
surfc(x, y, z)
xlabel('w1')
ylabel('w2')
zlabel('E')
title('Error Surface of a Linear Neuron with Two Input Weights')

Logistic Neurons

\begin{equation} z = b + \sum_{i}{x_i w_i} \end{equation}

\begin{equation} y = \dfrac{1}{1+e^{-z}} \end{equation}

Deducing the delta rule

In order to find the derivatives needed for learning the weights of a logistic unit we need to find the quation for the following:

\begin{equation} \Delta w_i = \dfrac{\partial E}{\partial w_i} \end{equation}

using the chain rule, we would get:

\begin{equation} \dfrac{\partial E}{\partial w_i} = \sum_{n}{\dfrac{\partial y^n}{\partial w_i} \dfrac{\partial E}{\partial y^n}} \end{equation}

Another usage of chain rule and we would get:

\begin{equation} \dfrac{\partial y}{\partial w_i} = \dfrac{\partial z}{\partial w_i} \dfrac{d y}{d z} \end{equation}

By definition of $y$ and $z$ we would obtain that:

\begin{equation} \dfrac{\partial z}{\partial w_i} = x_i \end{equation}

\begin{equation} \dfrac{d y}{d z} = y (1 - y) \end{equation}

which means that:

\begin{equation} \dfrac{\partial y}{\partial w_i} = \dfrac{\partial z}{\partial w_i} \dfrac{d y}{d z} = x_i y (1 - y) \end{equation}

wrapping this up to the main question:

\begin{equation} \Delta w_i = \dfrac{\partial E}{\partial w_i} = - \sum_{n}{x_i ^n y^n (1 - y^n)(t^n - y^n) } \end{equation}

By original delta rule, applied on linear neuron we can interpret this result as a delta rule, adjusted by $y^n (1 - y^n)$ - the slope of the logistic function.

Learning by Randomly perturbing weights

One way to adjust the weights would be to do a random change on one of the weight and investigate if this improves the performance of the network. If so, save the weight and repeat again.

This is a form of reinforcement learning.

It is not efficient: we need to run the network on the whole training data to determine whether the performance was improved or not by the weight change.

Finite difference approximation

This method is considered more efficient and also relies on perturbing weights. Here we don't perturb the weights randomly, however we adjust the weights one by one and measure the impact.

for each weight:

The Backpropagation algorithm

In  contrast to the previously mentioned algorithms, the back propagation algorithm is the algorithm for taking one training case, and computing efficiently for every weight in the network, how the error will change as, on that particular training case, as you change the weight.

Main topics will affect the performance of BP algorithm:

Frequency of weight update:

How much to update:

Overfitting:

Training data contains useful relations which we want to model, and a lot of regularities which come in form of noise. In any set of chosen cases there is a sampling error, due to the choice of these particular cases. When fitting a model we may overfit: i.e. we can be modeling the sampling error along other regularities. This can lead to really poor network design.

Overfitting can be reduced by using the following methods:

Backpropagation cannot be used with binary threshold neurons: Backpropagation works with derivatives. In a binary threshold neuron the derivatives of the output function are zero so the error signal will not be able to propagate through it.