- By
**nuri** - Follow User

- 382 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Perceptron' - nuri

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

- 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

- 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

Lineary separable patterns pattern

Perceptron learning rule pattern

- 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 with known correct answers

- 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

Several nodes with known correct answers

Constructions with known correct answers

Frank Rosenblatt with known correct answers

- 1928-1969

- Rosenblatt's bitter rival and professional with known correct answersnemesis 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 with known correct answers

- By Minsky and Papert in mid 1960

Gradient Descent with known correct answers

- 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)

Error for different hypothesis, for with known correct answersw0 and w1 (dim 2)

- We want to move the weigth vector in the direction that decrease E
wi=wi+wi w=w+w

- The gradient decrease

Differentiating decrease E

Update rule for gradient decent decrease

Gradient Descent decrease

Gradient-Descent(training_examples, )

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

Stochastic Approximation to gradient descent decrease

- The gradient decent training rule updates summing over all the training examples D
- Stochastic gradient approximates gradient decent by updating weights incrementally
- Calculate error for each example
- Known as delta-rule or LMS (last mean-square) weight update
- Adaline rule, used for adaptive filters Widroff and Hoff (1960)

LMS decrease

- Estimate of the weight vector
- No steepest decent
- No well defined trajectory in the weight space
- Instead a random trajectory (stochastic gradient descent)
- Converge only asymptotically toward the minimum error
- Can approximate gradient descent arbitrarily closely if made small enough

Summary decrease

- 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 decrease
- Perceptron
- Perceptron learning rule

- XOR problem
- linear separable patterns

- Gradient descent
- Stochastic Approximation to gradient descent
- LMS Adaline

Download Presentation

Connecting to Server..