Perceptron

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# Perceptron - PowerPoint PPT Presentation

Perceptron. Inner-product scalar Perceptron Perceptron learning rule XOR problem linear separable patterns Gradient descent Stochastic Approximation to gradient descent LMS Adaline. Inner-product. A measure of the projection of one vector onto another. Example. Activation function.

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## PowerPoint Slideshow about ' Perceptron' - nuri

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### Perceptron

Inner-product scalar
• Perceptron
• Perceptron learning rule
• XOR problem
• linear separable patterns
• Stochastic Approximation to gradient descent
Inner-product
• A measure of the projection of one vector onto another
1040 Neurons
• 104-5 connections

per neuron

x1

x2

xn

Perceptron
• Linear treshold unit (LTU)

x0=1

w1

w0

w2

o

.

.

.

wn

McCulloch-Pitts model of a neuron

The goal of a perceptron is to correctly classify the set of pattern D={x1,x2,..xm} into one of the classes C1 and C2
• The output for class C1 is o=1 and fo C2 is o=-1
• For n=2 
Perceptron learning rule
• Consider linearly separable problems
• How to find appropriate weights
• Look if the output pattern o belongs to the desired class, has the desired value d
•  is called the learning rate
• 0 <  ≤ 1
In supervised learning the network has ist output compared with known correct answers
• Supervised learning
• Learning with a teacher
• (d-o) plays the role of the error signal
Perceptron
• The algorithm converges to the correct classification
• if the training data is linearly separable
• and  is sufficiently small
• When assigning a value to  we must keep in mind two conflicting requirements
• Averaging of past inputs to provide stable weights estimates, which requires small 
• Fast adaptation with respect to real changes in the underlying distribution of the process responsible for the generation of the input vector x, which requires large 
Rosenblatt\'s bitter rival and professional nemesis was Marvin Minsky of Carnegie Mellon University
• Minsky despised Rosenblatt, hated the concept of the perceptron, and wrote several polemics against him
• For years Minsky crusaded against Rosenblatt on a very nasty and personal level, including contacting every group who funded Rosenblatt\'s research to denounce him as a charlatan, hoping to ruin Rosenblatt professionally and to cut off all funding for his research in neural nets
XOR problem and Perceptron
• By Minsky and Papert in mid 1960
• To understand, consider simpler linear unit, where
• Let\'s learn wi that minimize the squared error, D={(x1,t1),(x2,t2), . .,(xd,td),..,(xm,tm)}
• (t for target)

wi=wi+wi w=w+w

Each training example is a pair of the form <(x1,…xn),t> where (x1,…,xn) is the vector of input values, and t is the target output value,  is the learning rate (e.g. 0.1)

• Initialize each wi to some small random value
• Until the termination condition is met,
• Do
• Initialize each wi to zero
• For each <(x1,…xn),t> in training_examples
• Do
• Input the instance (x1,…,xn) to the linear unit and compute the output o
• For each linear unit weight wi
• Do
• wi= wi +  (t-o) xi
• For each linear unit weight wi
• Do
• wi=wi+wi
• The gradient decent training rule updates summing over all the training examples D
• Calculate error for each example
• Known as delta-rule or LMS (last mean-square) weight update
LMS
• Estimate of the weight vector
• No steepest decent
• No well defined trajectory in the weight space
• Converge only asymptotically toward the minimum error
• Can approximate gradient descent arbitrarily closely if made small enough 
Summary
• Perceptron training rule guaranteed to succeed if
• Training examples are linearly separable
• Sufficiently small learning rate 
• Linear unit training rule uses gradient descent or LMS guaranteed to converge to hypothesis with minimum squared error
• Given sufficiently small learning rate 
• Even when training data contains noise
• Even when training data not separable by H
Inner-product scalar
• Perceptron
• Perceptron learning rule
• XOR problem
• linear separable patterns