Transcript Document

Artificial Neural Networks
Notes based on Nilsson and Mitchell’s
Machine learning
Outline
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Perceptrons (LTU)
Gradient descent
Multi-layer networks
Backpropagation
Biological Neural Systems
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Neuron switching time : > 10-3 secs
Number of neurons in the human brain:
~1010
Connections (synapses) per neuron : ~104–
105
Face recognition : 0.1 secs
High degree of parallel computation
Distributed representations
Properties of Artificial Neural Nets (ANNs)
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Many simple neuron-like threshold switching
units
Many weighted interconnections among units
Highly parallel, distributed processing
Learning by adaptation of the connection
weights
Appropriate Problem Domains for Neural Network Learning
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Input is high-dimensional discrete or realvalued (e.g. raw sensor input)
Output is discrete or real valued
Output is a vector of values
Form of target function is unknown
Humans do not need to interpret the results
(black box model)
General Idea
• A network of neurons. Each neuron is characterized
by:
• number of input/output wires
• weights on each wire
• threshold value
• These values are not explicitly programmed, but they
evolve through a training process.
• During training phase, labeled samples are presented.
If the network classifies correctly, no weight changes.
Otherwise, the weights are adjusted.
• backpropagation algorithm used to adjust weights.
ALVINN (Carnegie Mellon Univ)
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
Another Example
NETtalk – Program that learns to pronounce English text.
(Sejnowski and Rosenberg 1987).
-A difficult task using conventional programming models.
- Rule-based approaches are too complex since
pronunciations are very irregular.
- NETtalk takes as input a sentence and produces a
sequence of phonemes and an associated stress for each
letter.
NETtalk
A phoneme is a basic unit of sound in a language.
Stress – relative loudness of that sound.
Because the pronunciation of a single letter depends upon its
context and the letters around it, NETtalk is given a seven
character window.
Each position is encoded by one of 29 symbols, (26 letters
and 3 punctuations.)
Letter in each position activates the corresponding unit.
NETtalk
The output units encode phonemes using 21 different
features of human articulation.
Remaining five units encode stress and syllable boundaries.
NETtalk also has a middle layer (hidden layer) that has 80
hidden units and nearly 18000 connections (edges).
NETtalk is trained by giving it a 7 character window so that
it learns the pronounce the middle character.
It learns by comparing the computed pronunciation to the
correct pronunciation.
Handwritten character recognition
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This is another area in which neural networks have been
successful.
In fact, all the successful programs have a neural
network component.
Threshold Logic Unit (TLU)
inputs
x1
x2
.
.
.
xn
w1
w2
wn
weights

output
activation
a=i=1n wi xi
y=
q
1 if a  q
{ 0 if a < q
y
Activation Functions
y
threshold
linear
y
a
y
piece-wise linear
a
a
y
sigmoid
a
Decision Surface of a TLU
1
w
1
x2
Decision line
w1 x1 + w2 x2 = q
1
0
0
0
1
0
0
x1
Scalar Products & Projections
v
v
v
j
w•v>0
w
j
j
w
w•v=0
w•v<0
v
j
w
w • v = |w||v| cos j
w
Geometric Interpretation
The relation w•x=q defines the decision line
x2
Decision line
w•x=q
w
y=1
xw
|xw|=q/|w|
x1
x
y=0
Geometric Interpretation
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In n dimensions the relation w • x=q defines a n1 dimensional hyper-plane, which is
perpendicular to the weight vector w.
On one side of the hyper-plane (w • x > q) all
patterns are classified by the TLU as “1”, while
those that get classified as “0” lie on the other
side of the hyper-plane.
If patterns can be not separated by a hyper-plane
then they cannot be correctly classified with a
TLU.
Linear Separability
w1=1
w2=1
q=1.5
x2
0
0
w1=?
w2=?
q= ?
1
x1
0
0
1
1
0
x1
Logical XOR
Logical AND
x1
x2
a
0
0
0
y
0
x1
0
x2
0
y
0
0
1
1
0
0
1
0
1
1
1
0
1
1
1
0
1
0
1
1
1
2
Threshold as Weight
q=wn+1
x1
x2
.
.
.
xn
w1
w2
wn
xn+1=-1
wn+1

y
a= i=1n+1 wi xi
y=
1 if a  0
{ 0 if a <0
Geometric Interpretation
The relation w • x=0 defines the decision line
x2
w
Decision line
w•x=0
y=1
x1
y=0
x
Training ANNs
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Training set S of examples {x, t}
 x is an input vector and
 t the desired target vector
 Example: Logical And
S = {(0,0),0}, {(0,1),0}, {(1,0),0}, {(1,1),1}
Iterative process
 Present a training example x , compute network output y,
compare output y with target t, adjust weights and thresholds
Learning rule
 Specifies how to change the weights w and thresholds q of the
network as a function of the inputs x, output y and target t.
Adjusting the Weight Vector
x
x
w’ = w + ax
ax
j>90
Target t=1
Output y=0
w
w
Move w in the direction of x
x
x
w
-ax
j<90
Target t=0
Output y=1
w
w’ = w - ax
Move w away from the direction of x
Perceptron Learning Rule
w’= w + a (t-y) x
Or in components
 w’i = wi + Dwi = wi + a (t-y) xi (i=1..n+1)
With wn+1 = q and xn+1= –1
 The parameter a is called the learning rate. It
determines the magnitude of weight updates Dwi .
 If the output is correct (t = y) the weights are not
changed (Dwi =0).
 If the output is incorrect (t  y) the weights wi are
changed such that the output of the TLU for the
new weights w’i is closer/further to the input xi.
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Perceptron Training Algorithm
repeat
for each training vector pair (x,t)
evaluate the output y when x is the input
if yt then
form a new weight vector w’ according
to w’=w + a (t-y) x
else
do nothing
end if
end for
until y=t for all training vector pairs
Perceptron Convergence Theorem
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The algorithm converges to the correct classification
 if the training data is linearly separable
 and a is sufficiently small
If two classes of vectors X1 and X2 are linearly separable,
the application of the perceptron training algorithm will
eventually result in a weight vector w0, such that w0
defines a TLU whose decision hyper-plane separates X1
and X2 (Rosenblatt 1962).
Solution w0 is not unique, since if w0 x =0 defines a
hyper-plane, so does w’0 = k w0.
Example
x1
x2
1
9.4
2.5
8.0
0.5
7.9
7.0
2.8
1.2
7.8
1
6.4
2.1
7.7
2.2
8.4
7.0
0.8
3.0
6.1
Initial weights:
output
1
-1
1
-1
1
-1
-1
1
1
-1
(0.75, -0.5, -0.6)
Linear Unit
inputs
x1
x2
.
.
.
xn
w1
w2
wn
weights
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activation
output
y
a=i=1n wi xi
y= a = i=1n wi xi
Gradient Descent Learning Rule
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Consider linear unit without threshold and
continuous output o (not just –1,1)
 0 =w0 + w1 x1 + … + wn xn
Train the wi’s such that they minimize the squared
error
2
 e = aD (fa-da)
where D is the set of training examples
Here fa is the actual output, da is the desired
output.
Gradient Descent rule:
We want to choose the weights wi so that e is minimized. Recall that
e = aD (fa – da)2
Since our goal is to work this error function for one input at a time,
let us consider a fixed input x in D, and define
e = (fx – dx)2
We will drop the subscript and just write this as:
e = (f – d)2
Our goal is to find the weights that will minimize this expression.
e/W= [e/w1,… e/wn+1]
Since s, the threshold function, is given by s = X . W, we have:
e/W = e/s * s/W. However, s/W = X. Thus,
e/W = e/s * X
Recall from the previous slide that e = (f – d)2
So, we have: e/s = 2(f – d)* f /s (note: d is constant)
This gives the expression:
e/W = 2(f – d)* f /s * X
A problem arises when dealing with TLU, namely f is not a
continuous function of s.
For a fixed input x, suppose the desired output is d, and the actual
output is f, then the above expression becomes:
Dw = - 2(d – f) x
This is what is known as the Widrow-Hoff procedure, with 2
replaced by c:
W  W  c( d - f ) X
The key idea is to move the weight vector along the gradient.
When will this converge to the correct weights?
• We are assuming that the data is linearly separable.
• We are also assuming that the desired output from the linear
threshold gate is available for the training set.
• Under these conditions, perceptron convergence theorem shows
that the above procedure will converge to the correct weights after
a finite number of iterations.
Neuron with Sigmoid-Function
inputs
x1
x2
.
.
.
xn
w1
w2
wn
weights

activation
output
y
a=i=1n wi xi
y=s(a) =1/(1+e-a)
Sigmoid Unit
x1
x2
.
.
.
xn
w1
w2
wn
x0=-1

w0 a=i=0n wi xi
y=s(a)=1/(1+e-a)
y
s(x) is the sigmoid function: 1/(1+e-x)
ds(x)/dx= s(x) (1- s(x))
Derive gradient descent rules to train:
Sigmoid function
f
s
1
f (s) =
1  e-s
f
Thus,
= f (1 - f )
s
Gradient Descent Rule for Sigmoid Output Function
s
sigmoid
Ep[w1,…,wn] = (tp-yp)2
Ep/wi = /wi (tp-yp)2
a
s’
= /wi(tp- s(i wi xip))2
= (tp-yp) s’(i wi xip) (-xip)
for y=s(a) = 1/(1+e-a)
s’(a)= e-a/(1+e-a)2=s(a) (1-s(a))
a
w’i= wi + Dwi = wi + a y(1-y)(tp-yp) xip
Presentation of Training Examples
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Presenting all training examples once to the
ANN is called an epoch.
In incremental stochastic gradient descent
training examples can be presented in
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Fixed order (1,2,3…,M)
Randomly permutated order (5,2,7,…,3)
Completely random (4,1,7,1,5,4,……) (repetitions
allowed arbitrarily)
Capabilities of Threshold Neurons
The threshold neuron can realize any linearly
separable function Rn  {0, 1}.
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Although we only looked at two-dimensional
input, our findings apply to any dimensionality
n.
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For example, for n = 3, our neuron can realize
any function that divides the three-dimensional
input space along a two-dimension plane.
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Capabilities of Threshold Neurons
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What do we do if we need a more complex function?
We can combine multiple artificial neurons to form
networks with increased capabilities.
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For example, we can build a two-layer network with any
number of neurons in the first layer giving input to a single
neuron in the second layer.
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The neuron in the second layer could, for example,
implement an AND function.
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Capabilities of Threshold Neurons
o
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o
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o
 1
o
 2
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.
.
.
o
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o
 1
o
 2
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What kind of function can such a network realize?
Capabilities of Threshold Neurons
Assume that the dotted lines in the diagram represent the inputdividing lines implemented by the neurons in the first layer:
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2nd comp.
 st
1 comp.
Then, for example, the second-layer neuron could output 1 if the
input is within a polygon, and 0 otherwise.
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Capabilities of Threshold Neurons
However, we still may want to implement functions that
are more complex than that.
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An obvious idea is to extend our network even further.
Let us build a network that has three layers, with
arbitrary numbers of neurons in the first and second layers
and one neuron in the third layer.
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The first and second layers are completely connected,
that is, each neuron in the first layer sends its output to
every neuron in the second layer.
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Capabilities of Threshold Neurons
o
 1
o
 2
o
 1
o
 2
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.
.
.
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.
.
.
o
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o
 1
o
 2
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What type of function can a three-layer network realize?
Capabilities of Threshold Neurons
Assume that the polygons in the diagram indicate the
input regions for which each of the second-layer neurons
yields output 1:
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2nd comp.
 st
1 comp.
Then, for example, the third-layer neuron could output 1 if the
input is within any of the polygons, and 0 otherwise.
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Capabilities of Threshold Neurons
The more neurons there are in the first layer, the more
vertices can the polygons have.
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With a sufficient number of first-layer neurons, the
polygons can approximate any given shape.
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The more neurons there are in the second layer, the more
of these polygons can be combined to form the output
function of the network.
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With a sufficient number of neurons and appropriate
weight vectors wi, a three-layer network of threshold neurons
can realize any function Rn  {0, 1}.
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Terminology
Usually, we draw neural networks in such a way that the
input enters at the bottom and the output is generated at
the top.
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Arrows indicate the direction of data flow.
The first layer, termed input layer, just contains the
input vector and does not perform any computations.
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The second layer, termed hidden layer, receives input
from the input layer and sends its output to the output
layer.
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After applying their activation function, the neurons in
the output layer contain the output vector.
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Terminology
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Example: Network function f: R3  {0, 1}2
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output vector
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input vector
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output layer
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hidden layer
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input layer
Multi-Layer Networks
output layer
hidden layer
input layer
Training-Rule for Weights to the Output Layer
yj
wji
xi
Ep[wij] = ½ j (tjp-yjp)2
Ep/wij = /wij ½ j (tjp-yjp)2
=…
= - yjp(1-ypj)(tpj-ypj) xip
Dwij = a yjp(1-yjp) (tpj-yjp) xip
= a djp xip
with djp := yjp(1-yjp) (tpj-yjp)
Training-Rule for Weights to the Hidden Layer
dj
wjk
xk
dk
wki
xi
yj
Credit assignment problem:
No target values t for hidden
layer units.
Error for hidden units?
dk = j wjk dj yj (1-yj)
Dwki = a xkp(1-xkp) dkp xip
Training-Rule for Weights to the Hidden Layer
dj
wj
k
xk
dk
wk
i
xi
yj
Ep[wki] = ½ j (tjp-yjp)2
Ep/wki = /wki ½ j (tjp-yjp)2
=/wki ½j (tjp-s(kwjk xkp))2
=/wki ½j (tjp-s(kwjk s(iwki xip)))2
= -j (tjp-yjp) s’j(a) wjk s’k(a) xip
= -j dj wjk s’k(a) xip
= -j dj wjk xk (1-xk) xip
Dwki = a dk xip
with dk = j dj wjk xk(1-xk)
Backpropagation
dj
wjk
yj
Backward step:
propagate errors from
output to hidden layer
xk
dk
wki
xi
Forward step:
Propagate activation
from input to output layer
Backpropagation Algorithm
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Initialize weights wij with a small random value
repeat
for each training pair {(x1,…xn)p,(t1,...,tm)p} Do
p
 Present (x1,…,xn) to the network and compute the
outputs yj (forward step)
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Compute the errors dj in the output layer and
propagate them to the hidden layer (backward step)
Update the weights in both layers according to
Dwki = a dk xi
end for loop
until overall error E becomes acceptably low
Backpropagation Algorithm
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Initialize each wi to some small random value
Until the termination condition is met, Do
 For each training example <(x1,…xn),t> Do
 Input the instance (x1,…,xn) to the network and compute the
network outputs yk
 For each output unit k
 dk=yk(1-yk)(tk-yk)
 For each hidden unit h
dh=yh(1-yh) k wh,k dk
For each network weight wi,j Do
wi,j=wi,j+Dwi,j where
Dwi,j=  dj xi,j
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Backpropagation
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Gradient descent over entire network weight vector
Easily generalized to arbitrary directed graphs
Will find a local, not necessarily global error minimum
-in practice often works well (can be invoked multiple times
with different initial weights)
Often include weight momentum term
Dwi,j(n)=  dj xi,j + a Dwi,j (n-1)
Minimizes error training examples
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Will it generalize well to unseen instances (over-fitting)?
Training can be slow typical 1000-10000 iterations
(Using network after training is fast)
Backpropagation
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Easily generalized to arbitrary directed graphs without clear
layers.
BP finds a local, not necessarily global error minimum
- in practice often works well (can be invoked multiple times
with different initial weights)
Minimizes error over training examples
 How does it generalize to unseen instances ?
Training can be slow typical 1000-10000 iterations
(use more efficient optimization methods than gradient descent)
Using network after training is fast
Convergence of Backprop
Gradient descent to some local minimum perhaps not global
minimum
Add momentum term: Dwki(n)
 Dwki(n) = a dk(n) xi (n) + l Dwki(n-1)
with l  [0,1]
 Stochastic gradient descent
 Train multiple nets with different initial weights
Nature of convergence
 Initialize weights near zero
 Therefore, initial networks near-linear
 Increasingly non-linear functions possible as training progresses
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Expressive Capabilities of ANN
Boolean functions
 Every boolean function can be represented by network with
single hidden layer
 But might require exponential (in number of inputs) hidden
units
Continuous functions
 Every bounded continuous function can be approximated
with arbitrarily small error, by network with one hidden layer
[Cybenko 1989, Hornik 1989]
 Any function can be approximated to arbitrary accuracy by a
network with two hidden layers [Cybenko 1988]