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Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay

CS621: Artificial Intelligence Lecture 22-23: Sigmoid neuron, Backpropagation (Lecture 20 and 21 taken by Anup on Graphical Models). Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay. Training of the MLP. Multilayer Perceptron (MLP)

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Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay

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  1. CS621: Artificial IntelligenceLecture 22-23: Sigmoid neuron, Backpropagation(Lecture 20 and 21 taken by Anup on Graphical Models) Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay

  2. Training of the MLP • Multilayer Perceptron (MLP) • Question:- How to find weights for the hidden layers when no target output is available? • Credit assignment problem – to be solved by “Gradient Descent”

  3. Gradient Descent Technique • Let E be the error at the output layer • ti = target output; oi = observed output • i is the index going over n neurons in the outermost layer • j is the index going over the p patterns (1 to p) • Ex: XOR:– p=4 and n=1

  4. Weights in a ff NN • wmn is the weight of the connection from the nth neuron to the mth neuron • E vs surface is a complex surface in the space defined by the weights wij • gives the direction in which a movement of the operating point in the wmn co-ordinate space will result in maximum decrease in error m wmn n

  5. Sigmoid neurons • Gradient Descent needs a derivative computation - not possible in perceptron due to the discontinuous step function used!  Sigmoid neurons with easy-to-compute derivatives used! • Computing power comes from non-linearity of sigmoid function.

  6. Derivative of Sigmoid function

  7. Training algorithm • Initialize weights to random values. • For input x = <xn,xn-1,…,x0>, modify weights as follows Target output = t, Observed output = o • Iterate until E <  (threshold)

  8. Calculation of ∆wi

  9. Observations Does the training technique support our intuition? • The larger the xi, larger is ∆wi • Error burden is borne by the weight values corresponding to large input values

  10. Backpropagation on feedforward network

  11. Backpropagation algorithm …. Output layer (m o/p neurons) j • Fully connected feed forward network • Pure FF network (no jumping of connections over layers) wji …. i Hidden layers …. …. Input layer (n i/p neurons)

  12. Gradient Descent Equations

  13. Backpropagation – for outermost layer

  14. Backpropagation for hidden layers …. Output layer (m o/p neurons) k …. j Hidden layers …. i …. Input layer (n i/p neurons) kis propagated backwards to find value of j

  15. Backpropagation – for hidden layers

  16. General Backpropagation Rule • General weight updating rule: • Where for outermost layer for hidden layers

  17. = 0.5 w1=1 w2=1 x1x2 1 1 x1x2 1.5 -1 -1 1.5 x2 x1 How does it work? • Input propagation forward and error propagation backward (e.g. XOR)

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