Jeff Howbert Introduction to Machine Learning Winter 2012 1

Classification

Neural Networks 1

Jeff Howbert Introduction to Machine Learning Winter 2012 2

Neural networks

Topics– Perceptrons

structure training expressiveness

– Multilayer networks possible structures

– activation functions training with gradient descent and backpropagation expressiveness

Jeff Howbert Introduction to Machine Learning Winter 2012 3

Connectionist models

Consider humans:– Neuron switching time ~ 0.001 second– Number of neurons ~ 1010

– Connections per neuron ~ 104-5

– Scene recognition time ~ 0.1 second– 100 inference steps doesn’t seem like enough

Massively parallel computation

Jeff Howbert Introduction to Machine Learning Winter 2012 4

Neural networks

Properties:– Many neuron-like threshold switching units– Many weighted interconnections among units– Highly parallel, distributed process– Emphasis on tuning weights automatically

Jeff Howbert Introduction to Machine Learning Winter 2012 5

Neural network application

ALVINN: An Autonomous Land Vehicle In a Neural Network

(Carnegie Mellon University Robotics Institute, 1989-1997)

ALVINN is a perception system which learns to control the NAVLAB vehicles by watching a person drive. ALVINN's architecture consists of a single hidden layer back-propagation network. The input layer of the network is a 30x32 unit two dimensional "retina" which receives input from the vehicles video camera. Each input unit is fully connected to a layer of five hidden units which are in turn fully connected to a layer of 30 output units. The output layer is a linear representation of the direction the vehicle should travel in order to keep the vehicle on the road.

Jeff Howbert Introduction to Machine Learning Winter 2012 6

Neural network application

ALVINN drives 70 mph on highways!

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X2

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Output node

Input nodes

Perceptron structure

Model is an assembly of nodes connected by weighted links

Output node sums up its input values according to the weights of their links

Output node sum then compared against some threshold t

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Output

Input

Example: modeling a Boolean function

Output Y is 1 if at least two of the three inputs are equal to 1.

Jeff Howbert Introduction to Machine Learning Winter 2012 9

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Perceptron model

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Jeff Howbert Introduction to Machine Learning Winter 2012 10

Example: decision surface for Boolean function on preceding slides

Perceptron decision boundary

Perceptron decision boundaries are linear(hyperplanes in higher dimensions)

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Can model any function where positive and negative examples are linearly separable

– Examples: Boolean AND, OR, NAND, NOR

Cannot (fully) model functions which are not linearly separable.

– Example: Boolean XOR

Expressiveness of perceptrons

Jeff Howbert Introduction to Machine Learning Winter 2012 12

1. Initialize weights with random values.

2. Do

a. Apply perceptron to each training example.

b. If example is misclassified, modify weights.

3. Until all examples are correctly classified, or process has converged.

Perceptron training process

Jeff Howbert Introduction to Machine Learning Winter 2012 13

Two rules for modifying weights during training:– Perceptron training rule

train on thresholded outputs driven by binary differences between correct and

predicted outputs modify weights with incremental updates

– Delta rule train on unthresholded outputs driven by continuous differences between correct

and predicted outputs modify weights via gradient descent

Perceptron training process

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1. Initialize weights with random values.

2. Do

a. Apply perceptron to each training sample i.

b. If sample i is misclassified, modify all weights j.

3. Until all samples are correctly classified.

Perceptron training rule

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Jeff Howbert Introduction to Machine Learning Winter 2012 15

a. If sample i is misclassified, modify all weights j.

Examples:

Perceptron training rule

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Jeff Howbert Introduction to Machine Learning Winter 2012 16

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Example of processing one sample

Perceptron training rule

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Based on squared error function for weight vector:

Note that error is difference between correct output and unthresholded sum of inputs, a continuous quantity (rather than binary difference between correct output and thresholded output).

Weights are modified by descending gradient of error function.

Delta training rule

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Squared error function for weight vector w

Jeff Howbert Introduction to Machine Learning Winter 2012 19

Gradient of error function

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Jeff Howbert Introduction to Machine Learning Winter 2012 20

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Gradient of squared error function

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1. Initialize weights with random values.

2. Do

a. Apply perceptron to each training sample i.

b. If sample i is misclassified, modify all weights j.

3. Until all samples are correctly classified, or process converges.

Delta training rule

Jeff Howbert Introduction to Machine Learning Winter 2012 22

Gradient descent: batch vs. incremental

Incremental mode (illustrated on preceding slides)

– Compute error and weight updates for a single sample.

– Apply updates to weights before processing next sample.

Batch mode– Compute errors and weight updates for a block of

samples (maybe all samples).

– Apply all updates simultaneously to weights.

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Perceptron training rule vs. delta rule

Perceptron training rule guaranteed to correctly classify all training samples if:

– Samples are linearly separable.

– Learning rate is sufficiently small.

Delta rule uses gradient descent. Guaranteed to converge to hypothesis with minimum squared error if:

– Learning rate is sufficiently small.

Even when:

– Training data contains noise.

– Training data not linearly separable.

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Equivalence of perceptron and linear models

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Output node

Input nodes

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