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Multilayer Neural Network in Machine Learning

Limitations of Perceptrons
  • Perceptrons have a monotinicity property: -   If a link has positive weight, activation can only increase as the corresponding input value increase (irrespective of other input values)
  • Can't represent functions where input interactions can cancel one another's effect (e.g. XOR)
  • Can represent only linearly separable functions.
A Solution : multiple layers


Power / Expressiveness of Multi-layer Networks
  • Can represent interactions among inputs
  • Two layer networks can represent any Boolean function, and continuous functions (within a tolerance) as long as the number of hidden units is sufficient and appropriate activation functions used
  • Learning algorithms exist, but weaker guarantees than perceptron learning algorithms


Multilayer Network


Two-layer back-propagation neural network


The back-propagation training algorithm
  • Step 1: Installation
  • Set all the weights and threshold levels of the network to random numbers uniformly distributed inside a small range

Backprop

Initialization
  - Set all the weights and threshold levels of the network to random numbers uniformly distributed inside a small range
 
Forward computing :
  - Apply an input vector x to input units
  - Compute activation / output vector z on hidden layer
             zj = Φ (∑i 𝒱ij𝒳i)
  - Compute the output vector y on output layer
            yk = Φ (∑i 𝒲ik𝒳j)
           y is the result of the computation

Learning for BP Nets
  • Update of weights in W (between output and hidden layers):  - delta rule
  • Not applicable to updating V (between input and hidden )  - don't know the target values for hidden units z1, z2, ..., zp
  • Solution : Propagate errors at output units to hidden units to drive the update of weight in V (again by delta rule)  (error BACK-PROPAGATION learning)
  • Error back propagation can be continued downward if the net has more than one hidden layer.
  • How to compute errors on hidden units?
Derivation


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