Neural Network

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Transcript Neural Network

Machine Learning
Neural Networks
Introduction

Artificial Neural Network is based on the
biological nervous system as Brain

It is composed of interconnected computing
units called neurons

ANN like human, learn by examples
Why Artificial Neural Networks?
There are two basic reasons why we are interested in
building artificial neural networks (ANNs):

Technical viewpoint: Some problems such as
character recognition or the prediction of future
states of a system require massively parallel and
adaptive processing.

Biological viewpoint: ANNs can be used to
replicate and simulate components of the human
(or animal) brain, thereby giving us insight into
natural information processing.
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Science: Model how biological neural
systems, like human brain, work?
 How do we see?
 How is information stored
in/retrieved from memory?
 How do you learn to not to touch
fire?
 How do your eyes adapt to the
amount of light in the environment?
 Related fields: Neuroscience,
Computational Neuroscience,
Psychology, Psychophysiology,
Cognitive Science, Medicine, Math,
Physics.
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Brief History
Old Ages:
 Association (William James; 1890)
 McCulloch-Pitts Neuron (1943,1947)
 Perceptrons (Rosenblatt; 1958,1962)
 Adaline/LMS (Widrow and Hoff; 1960)
 Perceptrons book (Minsky and Papert; 1969)
Dark Ages:
 Self-organization in visual cortex (von der Malsburg; 1973)
 Backpropagation (Werbos, 1974)
 Foundations of Adaptive Resonance Theory (Grossberg; 1976)
 Neural Theory of Association (Amari; 1977)
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History
Modern Ages:
 Adaptive Resonance Theory (Grossberg; 1980)
 Hopfield model (Hopfield; 1982, 1984)
 Self-organizing maps (Kohonen; 1982)
 Reinforcement learning (Sutton and Barto; 1983)
 Simulated Annealing (Kirkpatrick et al.; 1983)
 Boltzmann machines (Ackley, Hinton, Terrence; 1985)
 Backpropagation (Rumelhart, Hinton, Williams; 1986)
 ART-networks (Carpenter, Grossberg; 1992)
 Support Vector Machines
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Hebb’s Learning Law

•
•
•
In 1949, Donald Hebb formulated William James’
principle of association into a mathematical form.
If the activation of the neurons, y1 and y2 , are
both on (+1) then the weight between the two
neurons grow. (Off: 0)
Else the weight between remains the same.
However, when bipolar activation {-1,+1}
scheme is used, then the weights can also
decrease when the activation of two neurons
does not match.
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Real Neural Learning
Synapses change size and strength with
experience.
 Hebbian learning: When two connected
neurons are firing at the same time, the
strength of the synapse between them
increases.
 “Neurons that fire together, wire together.”

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Biological Neurons



Human brain = tens of thousands
of neurons
Each neuron is connected to
thousands other neurons
A neuron is made of:
– The soma: body of the neuron
– Dendrites: filaments that provide
input to the neuron
– The axon: sends an output signal
– Synapses: connection with other
neurons – releases certain
quantities of chemicals called
neurotransmitters to other
neurons
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Modeling of Brain Functions
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The biological neuron


The pulses
generated by the
neuron travels
along the axon
as an electrical
wave.
Once these
pulses reach the
synapses at the
end of the axon
open up
chemical
vesicles exciting
the other neuron.
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How do NNs and ANNs work?

Information is transmitted as a series of
electric impulses, so-called spikes.

The frequency and phase of these spikes
encodes the information.

In biological systems, one neuron can be
connected to as many as 10,000 other neurons.

Usually, a neuron receives its information
from other neurons in a confined area
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Navigation of a car

Done by Pomerlau. The network takes inputs from a 34X36 video image
and a 7X36 range finder. Output units represent “drive straight”, “turn
left” or “turn right”. After training about 40 times on 1200 road images,
the car drove around CMU campus at 5 km/h (using a small workstation
on the car). This was almost twice the speed of any other non-NN
algorithm at the time.
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Automated driving at 70 mph on a
public highway
Camera
image
30 outputs
for steering
4 hidden
units
30x32 pixels
as inputs
30x32 weights
into one out of
four hidden
unit
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Computers vs. Neural Networks
“Standard” Computers
Neural Networks
one CPU
highly parallel
processing
fast processing units
units
slow processing
reliable units
unreliable units
static infrastructure
infrastructure
dynamic
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Neural Network
Neural Network Application
•Pattern recognition can be implemented using NN
•The figure can be T or H character, the network should
identify each class of T or H.
Simple Neuron
X1
Inputs
X2
Output
Xn
b
An Artificial Neuron
synapses
neuron i
x1
x2
Wi,1
Wi,2
…
Wi,n
…
xi
xn
n
net input signal
net i (t )   wi , j (t ) x j (t )
j 1
output
x i (t )  f i ( net i (t ))
Neural Network
Input Layer
Hidden 1
Hidden 2
Output Layer
Network Layers
The common type of ANN consists of three
layers of neurons: a layer of input neurons
connected to the layer of hidden neuron
which is connected to a layer of output
neurons.
Architecture of ANN
Feed-Forward networks
Allow the signals to travel one way from input
to output
 Feed-Back Networks
The signals travel as loops in the network, the
output is connected to the input of the
network

How do NNs and ANNs Learn?

NNs are able to learn by adapting their
connectivity patterns so that the
organism improves its behavior in terms
of reaching certain (evolutionary) goals.

The NN achieves learning by
appropriately adapting the states of its
synapses.
Learning Rule

The learning rule modifies the weights of
the connections.

The learning process is divided into
Supervised and Unsupervised learning
Supervised Network

Which means there exists an external
teacher. The target is to minimization of the
error between the desired and computed
output
Unsupervised Network
Uses no external teacher and is based upon
only local information.
Perceptron

It is a network of one neuron and hard limit
transfer function
X1
W1
Inputs
X2
W2
Wn
Xn

f
Output
Perceptron

The perceptron is given first a randomly
weights vectors

Perceptron is given chosen data pairs (input
and desired output)

Preceptron learning rule changes the
weights according to the error in output
Perceptron Learning Rule
W new = W old + (t-a) X
Where W new is the new weight
W old is the old value of weight
X is the input value
t is the desired value of output
a is the actual value of output
Example

Let
–
–
–
–

X1 = [0
X2 = [0
X3 = [1
X4 = [1
0]
1]
0]
1]
and
and
and
and
W = [2 2] and b = -3
t =0
t=0
t=0
t=1
AND Network

This example means we construct a network
for AND operation. The network draw a
line to separate the classes which is called
Classification
Perceptron Geometric View
The equation below describes a (hyper-)plane in the input
space consisting of real valued m-dimensional vectors. The
plane splits the input space into two regions, each of them
describing one class.
decision
region for C1
x2 w x + w x + w >= 0
m
1 1
2 2
0
w x  w
i 1
i i
0
 0 decision
C1
boundary
C2
x1
w1x1 + w2x2 + w0 = 0
Problems
Four one-dimensional data belonging to two
classes are
X = [1
-0.5 3
-2]
T = [1
-1 1
-1]
W = [-2.5 1.75]

Boolean Functions
Take in two inputs (-1 or +1)
 Produce one output (-1 or +1)
 In other contexts, use 0 and 1
 Example: AND function

– Produces +1 only if both inputs are +1

Example: OR function
– Produces +1 if either inputs are +1

Related to the logical connectives from
F.O.L.
The First Neural Neural
Networks
X1
1
Y
X2
1
AND Function
Threshold(Y) = 2
AND
X1
1
1
0
0
X2
1
0
1
0
Y
1
0
0
0
Simple Networks
-1
W = 1.5
x
t = 0.0
W=1
y
Exercises
Design a neural network to recognize the
problem of
 X1=[2
2] , t1=0
 X=[1
-2], t2=1
 X3=[-2 2], t3=0
 X4=[-1 1], t4=1
Start with initial weights w=[0 0] and bias =0

Perceptron: Limitations






The perceptron can only model linearly separable
classes, like (those described by) the following
Boolean functions:
AND
OR
COMPLEMENT
It cannot model the XOR.
You can experiment with these functions in the
Matlab practical lessons.
Types of decision regions
w0  w1x1  w2 x2  0
1
Network
with a single
node
w0
x1 w1
w0  w1x1  w2 x2  0
L1
L2
w2
1
1
1
Convex
region
L3
x2
x1
L4
x2
1
-3.5
1
1
One-hidden layer
network that realizes
the convex region
Gaussian Neurons
Another type of neurons overcomes this problem by using
a Gaussian activation function:
f i ( net i (t ))  e
net i ( t ) 1
2
fi(neti(t))
1
0
-1
1 neti(t)
Gaussian Neurons
Gaussian neurons are able to realize non-linear
functions.
Therefore, networks of Gaussian units are in
principle unrestricted with regard to the functions
that they can realize.
The drawback of Gaussian neurons is that we
have to make sure that their net input does not
exceed 1.
This adds some difficulty to the learning in
Gaussian networks.
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Sigmoidal Neurons
Sigmoidal neurons accept any vectors of real
numbers as input, and they output a real number
between 0 and 1.
Sigmoidal neurons are the most common type of
artificial neuron, especially in learning networks.
A network of sigmoidal units with m input
neurons and n output neurons realizes a network
function
f: Rm  (0,1)n
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Sigmoidal Neurons
f i ( net i (t )) 
1
1  e  ( net i ( t )  ) / 
fi(neti(t))
1
=
1
0
-1
1 neti(t)
The parameter  controls the slope of the sigmoid function, while the
parameter  controls the horizontal offset of the function in a way
similar to the threshold neurons.
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Sigmoidal Neurons
This leads to a simplified form of the sigmoid function:
1
S ( net ) 
1  e (  net )
We do not need a modifiable threshold , because we will use
“dummy” inputs as we did for perceptrons.
The choice  = 1 works well in most situations and results in a very
simple derivative of S(net).
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Sigmoidal Neurons
1
S ( x) 
x
1 e
dS ( x )
e x
S ' ( x) 

dx
(1  e  x ) 2
1  e x 1
1
1



x 2
x
(1  e )
1 e
(1  e  x ) 2
 S ( x )(1  S ( x ))
This result will be very useful when we develop the backpropagation
algorithm.
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Multi-layers Network

Let the network of 3 layers
– Input layer
– Hidden layer
– Output layer
Each layer has different number of neurons
 The famous example to need the multi-layer
network is XOR unction

Learning rule

The perceptron learning rule can not be
applied to multi-layer network

We use BackPropagation Algorithm in
learning process
Feed-forward +
Backpropagation

Feed-forward:
– input from the features is fed forward in the network from input
layer towards the output layer

Backpropagation:
– Method to asses the blame of errors to weights
– error rate flows backwards from the output layer to the input
layer (to adjust the weight in order to minimize the output error)
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Backprop

Back-propagation training algorithm illustrated:
Network activation
Error computation
Forward Step
Error propagation
Backward Step

Backprop adjusts the weights of the NN in order to
minimize the network total mean squared error.
Correlation Learning
Hebbian Learning (1949):
“When an axon of cell A is near enough to excite a cell B
and repeatedly or persistently takes place in firing it, some
growth process or metabolic change takes place in one or
both cells such that A’s efficiency, as one of the cells firing
B, is increased.”
Weight modification rule:
wi,j = cxixj
Eventually, the connection strength will reflect the
correlation between the neurons’ outputs.
Competitive Learning
•
Nodes compete for inputs
•
Node with highest activation is the winner
•
Winner neuron adapts its tuning (pattern of
weights) even further towards the current input
•
Individual nodes specialize to win competition
for a set of similar inputs
•
Process leads to most efficient neural
representation of input space
•
Typical for unsupervised learning
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Backpropagation Learning
Similar to the Adaline, the goal of the Backpropagation
learning algorithm is to modify the network’s weights so
that its output vector
op = (op,1, op,2, …, op,K)
is as close as possible to the desired output vector
dp = (dp,1, dp,2, …, dp,K)
for K output neurons and input patterns p = 1, …, P.
The set of input-output pairs (exemplars)
{(xp, dp) | p = 1, …, P} constitutes the training set.
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Bp Algorithm

The weight change rule is
ijnew  ijold  .error. f ' (inputi )
Where  is the learning factor <1
 Error is the error between actual and trained
value
 f’ is is the derivative of sigmoid function =
f(1-f)

Delta Rule




Each observation contributes a variable amount to
the output
The scale of the contribution depends on the input
Output errors can be blamed on the weights
A least mean square (LSM) error function can be
defined (ideally it should be zero)
E = ½ (t – y)2
Example
For the network with one neuron in input layer
and one neuron in hidden layer the following
values are given
X=1, w1 =1, b1=-2, w2=1, b2 =1, =1 and t=1
Where X is the input value
W1 is the weight connect input to hidden
W2 is the weight connect hidden to output
B1 and b2 are bias
T is the training value

Exercises
Design a neural network to recognize the
problem of
 X1=[2
2] , t1=0
 X=[1
-2], t2=1
 X3=[-2 2], t3=0
 X4=[-1 1], t4=1
Start with initial weights w=[0 0] and bias =0

Exercises
Perform one iteration of backprpgation to
network of two layers. First layer has one
neuron with weight 1 and bias –2. The
transfer function in first layer is f=n2
 The second layer has only one neuron with
weight 1 and bias 1. The f in second layer is
1/n.
 The input to the network is x=1 and t=1

Neural Network
Construct a neural network to solve the problem
X1
1.0
9.4
2.5
8.0
0.5
7.9
7.0
2.8
1.2
7.8
X2
1.0
6.4
2.1
7.7
2.2
8.4
7.0
0.8
3.0
6.1
Output
1
-1
1
-1
1
-1
-1
1
1
-1
Initialize the
weights 0.75 , 0.5,
and –0.6
Neural Network
Construct a neural network to solve the XOR problem
X1
1
0
1
0
X2
1
0
0
1
Output
0
0
1
1
Initialize the weights –7.0 , -7.0, -5.0 and –4.0
-0.5
The transfer function is linear
function.
-2
1
1
1
-1
-1
1
3
0.5
-0.5
Consider a transfer function as f(n) = n2. Perform
one iteration of BackPropagation with a= 0.9 for
neural network of two neurons in input layer and
one neuron in output layer. The input values are
X=[1 -1] and t = 8, the weight values between
input and hidden layer are w11 = 1, w12 = - 2,
w21 = 0.2, and w22 = 0.1. The weight between
input and output layers are w1 = 2 and w2= -2.
The bias in input layers are b1 = -1, and b2= 3.
W11
W1
X1
W12
W21
W2
X2
W22
Some variations


True gradient descent assumes infinitesmall learning rate
(). If  is too small then learning is very slow. If large,
then the system's learning may never converge.
Some of the possible solutions to this problem are:
– Add a momentum term to allow a large learning rate.
– Use a different activation function
– Use a different error function
– Use an adaptive learning rate
– Use a good weight initialization procedure.
– Use a different minimization procedure
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Problems with Local Minima

Backpropagation is gradient descent search
– Where the height of the hills is determined by error
– But there are many dimensions to the space
• One for each weight in the network

Therefore backpropagation
– Can find its ways into local minima

One partial solution:
– Random re-start: learn lots of networks
• Starting with different random weight settings
– Can take best network
– Or can set up a “committee” of networks to categorise examples

Another partial solution: Momentum
Adding Momentum

Imagine rolling a ball down a hill
Gets stuck
here
Without Momentum
With Momentum
Momentum in Backpropagation

For each weight
– Remember what was added in the previous epoch

In the current epoch
– Add on a small amount of the previous Δ

The amount is determined by
– The momentum parameter, denoted α
– α is taken to be between 0 and 1
How Momentum Works


If direction of the weight doesn’t change
– Then the movement of search gets bigger
– The amount of additional extra is compounded in each
epoch
– May mean that narrow local minima are avoided
– May also mean that the convergence rate speeds up
Caution:
– May not have enough momentum to get out of local
minima
– Also, too much momentum might carry search
• Back out of the global minimum, into a local
minimum
Momentum
Weight update becomes:
 wij (n+1) =  (pj opi) +   wij(n)
 The momentum parameter  is chosen
between 0 and 1, typically 0.9. This
allows one to use higher learning rates.
The momentum term filters out high
frequency oscillations on the error
surface.
What would the learning rate be in a deep
valley?

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Problems with Overfitting

Plot training example error versus test example error:

Test set error is increasing!
– Network is overfitting the data
– Learning idiosyncrasies in data, not general principles
– Big problem in Machine Learning (ANNs in particular)
Avoiding Overfitting



Bad idea to use training set accuracy to terminate
One alternative: Use a validation set
– Hold back some of the training set during training
– Like a miniature test set (not used to train weights at
all)
– If the validation set error stops decreasing, but the
training set error continues decreasing
• Then it’s likely that overfitting has started to
occur, so stop
Another alternative: use a weight decay factor
– Take a small amount off every weight after each
epoch
– Networks with smaller weights aren’t as highly fine
tuned (overfit)