Supervised learning in a single-layer neural network

Let's consider a single-layer neural network with b inputs and c outputs:
 

Supervised learning

We have seen that different weights of a neural network produce different functions of the input. To train a network, we can present some sample inputs and compare the actual output to the desired results.  The difference is called the error.
[an error term is computed and fed back]
The different learning rules tell us which way to adjust the weights to reduce this error.  We say that training has converged when this error reaches some small, acceptable level.

Often the learning rule takes the following form:
   Wij  (t+1) = Wij  (t) + eta  . err (p)
where 0 <= eta < 1 is a parameter that controls the learning rate, and err(p) is the error when input pattern p is presented.

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Adaline learning

ADALINE is an acronym for ADAptive LINear Element (or ADAptive LInear NEuron).  It was developed by Bernard Widrow and Marcian Hoff (1960).

The adaline learning rule (also known as the least-mean-squares rule, the delta rule, and the Widrow-Hoff rule) is a training rule that minimises the output error using (approximate) gradient descent. After each training pattern Ip  is presented, the correction to apply to the weights is proportional to the error.  The correction is calculated before the thresholding step, using errij (p)=Tp-Wij Ip:

error=(inner product - target value)
 

Thus, the weights are adjusted by

    Wij  (t+1) = Wij  (t) + eta  (Tp-Wij Ip)  (Ip)
This corresponds to gradient descent on the quadratic error surface, Ej=Sump [Tp - Wj . Ip] 2

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Perceptron learning

In perceptron learning, the weights are adjusted only when a pattern is misclassified.    The correction to the weights after applying the training pattern p is
    Wij  (t+1) = Wij  (t)  + eta (Tp - Ap)  (Ip)
This corresponds to gradient descent on the error surface  E (Wij )= Summisclassified [Wij (Ap)(Ip)].

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Pocket algorithm

The perceptron learning algorithm does not terminate if the learning set is not linearly separable.  In many real-world cases, however,  we want to find the "best" linear separation even when the learning sets are not ideal. The pocket algorithm is a modification of the perceptron rule proposed by S. I. Gallant (1990). It stores the best weight vector so far in a "pocket" while continuing to learn.  The weights are actually modified only if a better weight vector is found.
 

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Backpropagation

The backpropagation algorithm was developed for training multilayer perceptron networks. In this applet, we will study how it works for a single-layer network.  It was popularized by Rumelhart, Hinton and Williams (1986), although similar ideas had been developed previously by others (Werbos, 1974; Parker, 1985).  The idea is to train a network by propagating the output errors backward through the layers. The errors serve to evaluate the derivatives of the error function with respect to the weights, which can then be adjusted.

The backpropagation algorithm for a single-layer network using the sum-of-squares error function consists of two phases:

  1. Feedforward - apply an input; evaluate the activations aj and store the error deltaj at each node j

  2.     aj = Sum i(Wij  (t)   Ipi)
        Ap = g (aj )
        delta = Ap -Ipj
     
  3. Backpropagation - compute the adjustments and update the weights.  Since there is just one layer, the output layer, we compute

  4.      Wij  (t+1) = Wij  (t) - eta  delta Ipj
    (This is called "on-line" learning, because the weights are adjusted each time a new input is presented.  In "batch" learning, the weights are adjusted after summing over all the patterns in the training set.)
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Optimal Perceptron learning

In the case of linear separable problems a perceptron can find different solutions:


 It would now be interesting to find the hyperplane that assures the maximal safety tolerance:


The margins of  that hyperplane touches a limited number of special points which define the hyperplane and which are called the Support Vectors.

The perceptron has to determine the samples for which . The remaining samples with are the Support Vectors sv

Represents the distance between a sample and. z- and z+ represent the projection of the critical points on the axis defined by.

Algorithm of the Optimal Perceptron:



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Further reading


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