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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
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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”
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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
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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
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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.
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Derivative of Sigmoid function
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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)
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Calculation of ∆wi
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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
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Backpropagation on feedforward network
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Backpropagation algorithm
…. Output layer (m o/p neurons) j wji …. i Hidden layers …. …. Input layer (n i/p neurons) Fully connected feed forward network Pure FF network (no jumping of connections over layers)
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Gradient Descent Equations
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Backpropagation – for outermost layer
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Backpropagation for hidden layers
…. Output layer (m o/p neurons) k …. j Hidden layers …. i …. Input layer (n i/p neurons) k is propagated backwards to find value of j
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Backpropagation – for hidden layers
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General Backpropagation Rule
General weight updating rule: Where for outermost layer for hidden layers
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How does it work? Input propagation forward and error propagation backward (e.g. XOR) w2=1 w1=1 = 0.5 x1x2 -1 x1 x2 1.5 1
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