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CS344: Introduction to Artificial Intelligence

This lecture delves into the intricacies of sigmoid neurons and multilayer feedforward neural networks (MLPs) while providing a thorough overview of their functioning, particularly in solving the XOR problem. It discusses the process of calculating weights, training with gradient descent, and the significance of non-linearity for computation. The course highlights the limitations of linear neurons in computing non-linear functions like XOR, emphasizing the crucial role of activation functions with derivatives. Gain insights into the mechanics of MLP training and its applications in artificial intelligence.

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CS344: Introduction to Artificial Intelligence

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  1. CS344: Introduction to Artificial Intelligence Pushpak BhattacharyyaCSE Dept., IIT Bombay Lecture 18,19: Sigmoid neuron 13thand 14th Feb, 2012

  2. FFNN Example- XOR • = 0.5 w1=1 w2=1  Calculation of XOR x1x2 x1x2 Calculation of x1x2 • = 1 w1=-1 w2=1.5 x2 x1

  3. Example - XOR • = 0.5 w1=1 w2=1 x1x2 1 1 x1x2 1.5 -1 -1 1.5 x2 x1

  4. Some Terminology • A multilayer feedforward neural network has • Input layer • Output layer • Hidden layer (asserts computation) Output units and hidden units are called computation units.

  5. 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”

  6. 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

  7. 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

  8. 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.

  9. Derivative of Sigmoid function

  10. 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)

  11. Calculation of ∆wi

  12. 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

  13. Can Linear Neurons Work? h2 h1 x2 x1

  14. Note: The whole structure shown in earlier slide is reducible to a single neuron with given behavior Claim: A neuron with linear I-O behavior can’t compute X-OR. Proof: Considering all possible cases: [assuming 0.1 and 0.9 as the lower and upper thresholds] For (0,0), Zero class: For (0,1), One class:

  15. For (1,0), One class: For (1,1), Zero class: These equations are inconsistent. Hence X-OR can’t be computed. Observations: • A linear neuron can’t compute X-OR. • A multilayer FFN with linear neurons is collapsible to a single linear neuron, hence no a additional power due to hidden layer. • Non-linearity is essential for power.

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