backpropagation in neural networks

BackpropagationBackpropagation
NetworksNetworks

Introduction toIntroduction to
- In 1969 a method for learning in multi-layer network,
BackpropagationBackpropagation, was invented by Bryson and Ho.
- The Backpropagation algorithm is a sensible approach
for dividing the contribution of each weight.
- Works basically the same as perceptrons

Backpropagation Learning Principles:
Hidden Layers and Gradients
There are two differences for the updating rule :differences for the updating rule :
1) The activation of the hidden unit is used instead ofinstead of
activation of the input value.activation of the input value.
2) The rule contains a term for the gradient of the activation
function.

Backpropagation NetworkBackpropagation Network
trainingtraining
• 1. Initialize network with random weights
• 2. For all training cases (called examples):
– a. Present training inputs to network and
calculate output
– b.b. For all layers (starting with output layer,
back to input layer):
• i. Compare network output with correct output
(error function)
• ii. Adapt weights in current layer
This is
what
you
want

Backpropagation LearningBackpropagation Learning
DetailsDetails
• Method for learning weights in feed-forward (FF)
nets
• Can’t use Perceptron Learning Rule
– no teacher values are possible for hidden units
• Use gradient descent to minimize the error
– propagate deltas to adjust for errors
backward from outputs
to hidden layers
to inputs
forward
backward

Backpropagation Algorithm – Main
Idea – error in hidden layers
The ideas of the algorithm can be summarized as follows :
1. Computes the error term for the output units using the
observed error.
2. From output layer, repeat
- propagating the error term back to the previous layer
and
- updating the weights between the two layers
until the earliest hidden layer is reached.

Backpropagation Algorithm
• Initialize weights (typically random!)
• Keep doing epochs
– For each example ee in training set do
• forward passforward pass to compute
– O = neural-net-output(network,e)
– miss = (T-O) at each output unit
• backward pass to calculate deltas to weights
• update all weights
– end
• until tuning set error stops improving
Backward pass explained in next slideForward pass explained
earlier

Backward Pass
• Compute deltas to weights
– from hidden layer
– to output layer
• Without changing any weights (yet),
compute the actual contributions
– within the hidden layer(s)
– and compute deltas

Gradient DescentGradient Descent
• Think of the N weights as a point in an N-
dimensional space
• Add a dimension for the observed error
• Try to minimize your position on the “error
surface”

Error Surface
Error as function of
weights in
multidimensional space
error
weights

Gradient
• Trying to make error decrease the fastest
• Compute:
• GradE = [dE/dw1, dE/dw2, . . ., dE/dwn]
• Change i-th weight by
• deltawi = -alphaalpha * dE/dwi
• We need a derivative!
• Activation function must be continuouscontinuous,
differentiable, non-decreasing, and easy to
compute
Derivatives of error for weights
Compute
deltas

Can’t use LTU
• To effectively assign credit / blame to units
in hidden layers, we want to look at the
first derivative of the activation function
• Sigmoid function is easy to differentiate
and easy to compute forward
Linear Threshold Units Sigmoid function

Updating hidden-to-output
• We have teacher supplied desired values
• deltawji = α * aj * (Ti - Oi) * g’(ini)
= α * aj * (Ti - Oi) * Oi * (1 - Oi)
– for sigmoid the derivative is, g’(x) = g(x) * (1 - g(x))
alphaalpha
derivative
miss
Here we have
general formula with
derivative, next we
use for sigmoid

Updating interior weights
• Layer k units provide values to all layer
k+1 units
• “miss” is sum of misses from all units on k+1
• missj = Σ [ ai(1- ai) (Ti - ai) wji ]
• weights coming into this unit are adjusted
based on their contribution
deltakj = α * Ik * aj * (1 - aj) * missj
For layer k+1
Compute deltas

How do we pick αα?
1. Tuning set, or
2. Cross validation, or
3. Small for slow, conservative learning

How many hidden layers?
• Usually just one (i.e., a 2-layer net)
• How many hidden units in the layer?
– Too few ==> can’t learn
– Too many ==> poor generalization

How big a training set?
• Determine your target error rate, e
• Success rate is 1- e
• Typical training set approx. n/e, where n is the
number of weights in the net
• Example:
– e = 0.1, n = 80 weights
– training set size 800
trained until 95% correct training set classification
should produce 90% correct classification
on testing set (typical)

ExamplesExamples of Backpropagation
Learning
In the restaurant
problem NN was
worse than the
decision tree
Error
decreases
with number
of epochs
Decision tree still
better for
restaurant
example

Examples of Backpropagation Learning
Majority example,
perceptron better
Restaurant
example, DT
better

Learning MathLearning Math
See next
slide for
explanation

Visualization ofVisualization of
learninglearning
Backprop output layer

bias neuron in input layer
Bias Neurons inBias Neurons in BackpropBackpropagationagation
LearningLearning

Training pairs
Software for Backpropagation LearningSoftware for Backpropagation Learning
Calculate difference to
desired output
Calculate total error
Run network forward.
Was explained earlier
This routine
calculate error for
backpropagation

Update output weights
Software for Backpropagation LearningSoftware for Backpropagation Learning
continuationcontinuation
Calculate hidden difference values
Update input weights
Return total error
Here we do not use
alpha, the learning rate

The generalgeneral Backpropagation Algorithm for updating weights in a multilayermultilayer network
Run network to
calculate its
output for this
example
Go through all
examples
Compute the
error in output
Update weights
to output layer
Compute error in
each hidden layer
Update weights in
each hidden layer
Repeat until
convergent
Return learned network
Here we use alpha, the
learning rate

Examples andExamples and
ApplicationsApplications
of ANNof ANN

Neural Network in Practice
NNs are used for classification and function approximation
or mapping problems which are:
- Tolerant of some imprecision.
- Have lots of training data available.
- Hard and fast rules cannot easily be applied.

NETalk (1987)
• Mapping character strings into phonemes so they
can be pronounced by a computer
• Neural network trained how to pronounce each
letter in a word in a sentence, given the three
letters before and three letters after it in a window
• Output was the correct phoneme
• Results
– 95% accuracy on the training data
– 78% accuracy on the test set

Other Examples
• Neurogammon (Tesauro & Sejnowski, 1989)
– Backgammon learning program
• Speech Recognition (Waibel, 1989)
• Character Recognition (LeCun et al., 1989)
• Face Recognition (Mitchell)

ALVINNALVINN
• Steer a van down the road
– 2-layer feedforward
• using backpropagation for learning
– Raw input is 480 x 512 pixel image 15x per sec
– Color image preprocessed into 960 input units
– 4 hidden units
– 30 output units, each is a steering direction

Neural Network Approaches
ALVINN - Autonomous Land Vehicle In a Neural Network

Learning on-the-
fly
• ALVINN learned as the vehicle
traveled
– initially by observing a human
driving
– learns from its own driving by
watching for future corrections
– never saw bad driving
• didn’t know what was
dangerous, NOT correct
• computes alternate views of
the road (rotations, shifts, and
fill-ins) to use as “bad”
examples
– keeps a buffer pool of 200 pretty
old examples to avoid overfitting
to only the most recent images

Feed-forward vs. Interactive
Nets
• Feed-forward
– activation propagates in one direction
– We usually focus on this
• Interactive
– activation propagates forward & backwards
– propagation continues until equilibrium is reached in
the network
– We do not discuss these networks here, complex
training. May be unstable.

Ways of learning with an ANN
• Add nodes & connections
• Subtract nodes & connections
• Modify connection weights
– current focus
– can simulate first two
• I/O pairs:
– given the inputs, what should the output be?
[“typical” learning problem]

More Neural NetworkMore Neural Network
ApplicationsApplications
- May provide a model for massive parallel computation.
- More successful approach of “parallelizing” traditional
serial algorithms.
- Can compute any computable function.
- Can do everything a normal digital computer can do.
- Can do even more under some impractical assumptions.

Neural Network Approaches to driving
- Developed in 1993.
- Performs driving with
Neural Networks.
- An intelligent VLSI image
sensor for road following.
- Learns to filter out image
details not relevant to
driving.
Hidden layer
Output units
Input units
•Use special hardware
•ASIC
•FPGA
•analog

Neural Network Approaches
Hidden Units Output unitsInput Array

Actual Products Available
ex1. Enterprise Miner:ex1. Enterprise Miner:
- Single multi-layered feed-forward neural networks.
- Provides business solutions for data mining.
ex2. Nestor:ex2. Nestor:
- Uses Nestor Learning System (NLS).
- Several multi-layered feed-forward neural networks.
- Intel has made such a chip - NE1000 in VLSI technology.

Ex1. Software tool - Enterprise Miner
- Based on SEMMA (Sample, Explore, Modify, Model,
Access) methodology.
- Statistical tools include :
Clustering, decision trees, linear and logistic
regression and neural networks.
- Data preparation tools include :
Outliner detection, variable transformation, random
sampling, and partition of data sets (into training,
testing and validation data sets).

Ex 2. Hardware Tool - Nestor
- With low connectivity within each layer.
- Minimized connectivity within each layer results in rapid
training and efficient memory utilization, ideal for VLSI.
- Composed of multiple neural networks, each specializing
in a subset of information about the input patterns.
- Real time operation without the need of special computers
or custom hardware DSP platforms
•Software exists.

Problems with using ANNs
1. Insufficiently characterized development
process compared with conventional software
– What are the steps to create a neural network?
1. How do we create neural networks in a
repeatable and predictable manner?
2. Absence of quality assurance methods for
neural network models and implementations
– How do I verify my implementation?

Solving Problem 1 – The Steps to createSolving Problem 1 – The Steps to create
a ANNa ANN
Define the process of developing neural networks:
1. Formally capture the specifics of the problem in
a document based on a template
2. Define the factors/parameters for creation
– Neural network creation parameters
– Performance requirements
1. Create the neural network
2. Get feedback on performance

Neural Network Development Process

Problem Specification Phase
• Some factors to define in problem specification:
– Type of neural networks (based on experience or
published results)
– How to collect and transform problem data
– Potential input/output representations
– Training & testing method and data selection
– Performance targets (accuracy and precision)
• Most important output is the ranked collection of
factors/parameters

Problem 2 –Problem 2 –
How to create a Neural NetworkHow to create a Neural Network
• Predictability (with regard to resources)
– Depending on creation approach used, record time
for one iteration
– Use timing to predict maximum and minimum times
for all of the combinations specified
• Repeatability
– Relevant information must be captured in problem
specification and combinations of parameters

Problem 3 - Quality Assurance
• Specification of generic neural network software
(models and learning)
• Prototype of specification
• Comparison of a given implementation with
specification prototype
• Allows practitioners to create arbitrary neural
networks verified against models

Two Methods for Comparison
• Direct comparison of outputs:
• Verification of weights generated by learning
algorithm:
20-10-5 (with particular connections and input)
Prototype <0.123892, 0.567442, 0.981194, 0.321438, 0.699115>
Implementation <0.123892, 0.567442, 0.981194, 0.321438, 0.699115>
20-10-5 Iteration 100 Iteration 200 ………. Iteration n
Prototype Weight state 1 Weight state 2 ………. Weight state n
Implementation Weight state 1 Weight state 2 ………. Weight state n

Further Work on
improvements
• Practitioners to use the development process or
at least document in problem specification
• Feedback from neural network development
community on the content of the problem
specification template
• Collect problem specifications and analyse to
look for commonalities in problem domains and
improve predictability (eg. control)
• More verification of specification prototype

Further Work (2)
• Translation methods for formal specification
• Extend formal specification to new types
• Fully prove aspects of the specification
• Cross discipline data analysis methods (eg. ICA,
statistical analysis)
• Implementation of learning on distributed
systems
– Peer-to-peer network systems (farm each
combination of parameters to a peer)
• Remain unfashionable

SummarySummary
- Neural network is a computational model that simulate
some properties of the human brain.
- The connections and nature of units determine the
behavior of a neural network.
- Perceptrons are feed-forward networks that can only
represent linearly separable functions.

Summary
- Given enough units, any function can be represented
by Multi-layer feed-forward networks.
- Backpropagation learning works on multi-layer
feed-forward networks.
- Neural Networks are widely used in developing
artificial learning systems.

ReferencesReferences
- Russel, S. and P. Norvig (1995). Artificial Intelligence - A
Modern Approach. Upper Saddle River, NJ, Prentice
Hall.
- Sarle, W.S., ed. (1997), Neural Network FAQ, part 1 of 7:
Introduction, periodic posting to the Usenet newsgroup
comp.ai.neural-nets,
URL: ftp://ftp.sas.com/pub/neural/FAQ.html

Eddy Li
Eric Wong
Martin Ho
Kitty Wong
Sources

backpropagation in neural networks

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