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Perceptron Learning Rule

The Perceptron Learning Rule is a method that converges in a finite time for linearly separable problems. This rule adjusts weights based on input-output products to correctly classify new patterns. Inspired by neuroscience, the rule is akin to modifying synapses in neurons. The update process involves correcting misclassifications by adjusting weights proportional to the input pattern and desired output. An example demonstrates how the weights evolve through iterations until all points are correctly classified. Despite its simplicity, the process may require multiple updates for optimal accuracy.

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Perceptron Learning Rule

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  1. Perceptron Learning Rule Assuming the problem is linearly separable, there is a learning rule that converges in a finite time Motivation A new (unseen) input pattern that is similar to an old (seen) input pattern is likely to be classified correctly

  2. Learning Rule, Ctd • Basic Idea – go over all existing data patterns, whose labeling is known, and check their classification with a current weight vector • If correct, continue • If not, add to the weights a quantity that is proportional to the product of the input pattern with the desired output Z (1 or –1)

  3. Weight Update Rule

  4. Biological Motivation • Learning means changing the weights (“synapses”) between the neurons • the product between input and output is important in computational neuroscience

  5. Hebb Rule • In 1949, Hebb postulated that the changes in a synapse are proportional to the correlation between firing of the neurons that are connected through the synapse (the pre- and post- synaptic neurons) • Neurons that fire together, wire together

  6. Example: a simple problem 4 points linearly separable 2 1.5 (1/2, 1) 1 (-1,1/2) (1,1/2) 0.5 0 Z = 1 Z = - 1 -0.5 (-1,1) -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

  7. Updating Weights • Upper left point is wrongly classified • eta = 1/3 , W(0) = (0,1) • W ==>W + eta * Z * X • W_x = 0 + 1/3 *(-1) * (-1) = 1/3 • W_y = 1 + 1/3 * (-1) * (1/2) = 5/6 • W(1) = (1/3,5/6)

  8. first correction 2 1.5 W(1) = (1/3,5/6) 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

  9. Updating Weights, Ctd • Upper left point is still wrongly classified • W ==>W + eta * Z * X • W_x = 1/3 + 1/3 *(-1) * (-1) = 2/3 • W_y = 5/6 + 1/3 * (-1) * (1/2) = 4/6 = 2/3 • W(2) = (2/3,2/3)

  10. Example, Ctd • All 4 points are classified correctly • Toy problem – only 2 updates required • Correction of weights was simply a rotation of the separating hyper plane • Rotation can be applied to the right direction, but may require many updates

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