Regression and Classification with Neural Networks

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Sep 25th, 2001Copyright © 2001, 2003, Andrew W. Moore
Regression and
Classification with
Neural Networks
Andrew W. Moore
Professor
School of Computer Science
Carnegie Mellon University
www.cs.cmu.edu/~awm
awm@cs.cmu.edu
412-268-7599
Note to other teachers and users of
these slides. Andrew would be delighted
if you found this source material useful in
giving your own lectures. Feel free to use
these slides verbatim, or to modify them
to fit your own needs. PowerPoint
originals are available. If you make use
of a significant portion of these slides in
your own lecture, please include this
message, or the following link to the
source repository of Andrew’s tutorials:
http://www.cs.cmu.edu/~awm/tutorials
.
Comments and corrections gratefully
received.
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 2
Linear Regression
Linear regression assumes that the expected value of
the output given an input, E[y|x], is linear.
Simplest case: Out(x) = wxfor some unknown w.
Given the data, we can estimate w.
y
5
= 3.1x
5
= 4
y
4
= 1.9x
4
= 1.5
y
3
= 2x
3
= 2
y
2
= 2.2x
2
= 3
y
1
= 1
x
1
= 1
outputs
inputs
DATASET
←1 →

w

2
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 3
1-parameter linear regression
Assume that the data is formed by
y
i
= wx
i
+ noise
i
where…
• the noise signals are independent
• the noise has a normal distribution with mean 0
and unknown variance σ
2
P(y|w,x) has a normal distribution with
• mean wx
• variance σ
2
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 4
Bayesian Linear Regression
P(y|w,x) = Normal (mean wx, var σ
2
)
We have a set of datapoints (x
1
,y
1
) (x
2
,y
2
) … (x
n
,y
n
)
which are EVIDENCE about w.
We want to infer wfrom the data.
P(w|x
1
, x
2
, x
3
,…x
n
, y
1
, y
2
…y
n
)
•You can use BAYES rule to work out a posterior
distribution for wgiven the data.
•Or you could do Maximum Likelihood Estimation
3
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 5
Maximum likelihood estimation of w
Asks the question:
“For which value of wis this data most likely to have
happened?”
<=>
For what wis
P(y
1
, y
2
…y
n
|x
1
, x
2
, x
3
,…x
n
, w) maximized?
<=>
For what w is
maximized
?
),(
1
i
n
i
i
xwyP

=
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 6
For what w is
For what w is
For what w is
For what w is
maximized? ),(
1
i
n
i
i
xwyP

=
maximized? ))(
2
1
exp(
2
1
σ
ii
wxy
n
i

=


maximized?
2
1
2
1









=
σ
ii
n
i
wxy
( )
minimized?
2
1

=

n
i
ii
wxy
4
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 7
Linear Regression
The maximum
likelihood wis
the one that
minimizes sum-
of-squares of
residuals
We want to minimize a quadratic function of w.
(
)
( )
( )
2
22
2
2 wxwyxy
wxy
i
i
iii
i
ii
∑∑ ∑

+−=
−=Ε
E(w)
w
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 8
Linear Regression
Easy to show the sum of
squares is minimized
when
2


=
i
ii
x
yx
w
The maximum likelihood
model is
We can use it for
prediction
( )
wxx
=
Out
5
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 9
Linear Regression
Easy to show the sum of
squares is minimized
when
2


=
i
ii
x
yx
w
The maximum likelihood
model is
We can use it for
prediction
Note:
In Bayesian stats you’d have
ended up with a prob dist of w
And predictions would have given a prob
dist of expected output
Often useful to know your confidence.
Max likelihood can give some kinds of
confidence too.
p(w)
w
( )
wxx
=
Out
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 10
Multivariate Regression
What if the inputs are vectors?
Dataset has form
x
1
y
1
x
2
y
2
x
3
y
3
.: :
.
x
R
y
R
3
.
. 4
6 .
.
5
. 8
. 10
2-d input
example
x
1
x
2
6
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 11
Multivariate Regression
Write matrix X and Y thus:












=












=












=
RRmRR
m
m
R
y
y
y
xxx
xxx
xxx
M
M
M
2
1
21
22221
11211
2

...
...
...
..........
..........
..........
y
x
x
x
x
1
(there are R datapoints. Each input has mcomponents)
The linear regression model assumes a vector w such that
Out(x) = w
T
x = w
1
x[1] + w
2
x[2] + ….w
m
x[D]
The max. likelihood wis w = (X
T
X)
-1
(X
T
Y)
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 12
Multivariate Regression
Write matrix X and Y thus:












=












=












=
RRmRR
m
m
R
y
y
y
xxx
xxx
xxx
M
M
M
2
1
21
22221
11211
2

...
...
...
..........
..........
..........
y
x
x
x
x
1
(there are R datapoints. Each input has mcomponents)
The linear regression model assumes a vector w such that
Out(x) = w
T
x = w
1
x[1] + w
2
x[2] + ….w
m
x[D]
The max. likelihood wis w = (X
T
X)
-1
(X
T
Y)
IMPORTANT EXERCISE:
PROVE IT !!!!!
7
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 13
Multivariate Regression (con’t)
The max. likelihood w is w = (X
T
X)
-1
(X
T
Y)
X
T
X is an mx mmatrix: i,j’th elt is
X
T
Y is an m-element vector: i
’th
elt

=
R
k
kjki
xx
1

=
R
k
kki
yx
1
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 14
What about a constant term?
We may expect
linear data that does
not go through the
origin.
Statisticians and
Neural Net Folks all
agree on a simple
obvious hack.
Can you guess??
8
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 15
The constant term
• The trick is to create a fake input “X
0
” that
always takes the value 1
2055
1743
1642
YX
2
X
1
1
1
1
X
0
2055
1743
1642
YX
2
X
1
Before:
Y=w
1
X
1
+ w
2
X
2
…has to be a poor
model
After:
Y= w
0
X
0
+w
1
X
1
+ w
2
X
2
= w
0
+w
1
X
1
+ w
2
X
2
…has a fine constant
term
In this example,
You should be able
to see the MLE w
0
, w
1
andw
2
by
inspection
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 16
Regression with varying noise
• Suppose you know the variance of the noise that
was added to each datapoint.
x=0 x=3x=2x=1
y=0
y=3
y=2
y=1
σ=1/2
σ=2
σ=1
σ=1/2
σ=2
1/423
432
1/412
111
4½½
σ
i
2
y
i
x
i
),(~
2
iii
wxNy σ
Assume
W
h
a
t

s

t
h
e

M
L
E

e
s
t
i
m
a
t
e

o
f

w
?
9
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 17
MLE estimation with varying noise
=),,...,,,,...,,|,...,,(log
22
2
2
12121
argmax
wxxxyyyp
w
RRR
σσσ
=


=
R
i
i
ii
wxy
w
1
2
2
)(
argmin
σ
=








=


=
0
)(
such that
1
2
R
i
i
iii
wxyx
w
σ


















=
=
R
i
i
i
R
i
i
ii
x
yx
1
2
2
1
2
σ
σ
Assuming i.i.d. and
then plugging in
equation for Gaussian
and simplifying.
Setting dLL/dw
equal to zero
Trivial algebra
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 18
This is Weighted Regression
• We are asking to minimize the weighted sum of
squares
x=0 x=3x=2x=1
y=0
y=3
y=2
y=1
σ=1/2
σ=2
σ=1
σ=1/2
σ=2

=

R
i
i
ii
wxy
w
1
2
2
)(
argmin
σ
2
1
i
σ
where weight for i’th datapoint is
10
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 19
Weighted Multivariate Regression
The max. likelihood w is w = (WX
T
WX)
-1
(WX
T
WY)
(WX
T
WX) is an mx mmatrix: i,j’th elt is
(WX
T
WY) is an m-element vector: i
’th
elt

=
R
k
i
kjki
xx
1
2
σ

=
R
k
i
kki
yx
1
2
σ
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 20
Non-linear Regression
• Suppose you know that y is related to a function of x in
such a way that the predicted values have a non-linear
dependence on w, e.g:
x=0 x=3x=2x=1
y=0
y=3
y=2
y=1
33
23
32
2.51
½½
y
i
x
i
),(~
2
σ
ii
xwNy +
Assume
W
h
a
t

s

t
h
e

M
L
E

e
s
t
i
m
a
t
e

o
f

w
?
11
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 21
Non-linear MLE estimation
=
),,,...,,|,...,,(log
2121
argmax
wxxxyyyp
w
RR
σ
(
)
=+−

=
R
i
ii
xwy
w
1
2
argmin
=








=
+
+−

=
0such that
1
R
i
i
ii
xw
xwy
w
Assuming i.i.d. and
then plugging in
equation for Gaussian
and simplifying.
Setting dLL/dw
equal to zero
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 22
Non-linear MLE estimation
=
),,,...,,|,...,,(log
2121
argmax
wxxxyyyp
w
RR
σ
(
)
=+−

=
R
i
ii
xwy
w
1
2
argmin
=








=
+
+−

=
0such that
1
R
i
i
ii
xw
xwy
w
Assuming i.i.d. and
then plugging in
equation for Gaussian
and simplifying.
Setting dLL/dw
equal to zero
We’re down the
algebraic toilet
S
o

g
u
e
s
s

w
h
a
t

w
e

d
o
?
12
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 23
Non-linear MLE estimation
=
),,,...,,|,...,,(log
2121
argmax
wxxxyyyp
w
RR
σ
(
)
=+−

=
R
i
ii
xwy
w
1
2
argmin
=








=
+
+−

=
0such that
1
R
i
i
ii
xw
xwy
w
Assuming i.i.d. and
then plugging in
equation for Gaussian
and simplifying.
Setting dLL/dw
equal to zero
We’re down the
algebraic toilet
S
o

g
u
e
s
s

w
h
a
t

w
e

d
o
?
Common (but not only) approach:
Numerical Solutions:
• Line Search
• Simulated Annealing
• Gradient Descent
• Conjugate Gradient
• Levenberg Marquart
• Newton’s Method
Also, special purpose statistical-
optimization-specific tricks such as
E.M. (See Gaussian Mixtures lecture
for introduction)
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 24
GRADIENT DESCENT
Suppose we have a scalar function
We want to find a local minimum.
Assume our current weight is w
GRADIENT DESCENT RULE:
η is called the LEARNING RATE. A small positive
number, e.g. η = 0.05
ℜ→

:f(w)
( )
w
w
ww f


−← η
13
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 25
GRADIENT DESCENT
Suppose we have a scalar function
We want to find a local minimum.
Assume our current weight is w
GRADIENT DESCENT RULE:
η is called the LEARNING RATE. A small positive
number, e.g. η = 0.05
ℜ→

:f(w)
( )
w
w
ww f


−← η
QUESTION: Justify the Gradient Descent Rule
Recall Andrew’s favorite
default value for anything
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 26
Gradient Descent in “m” Dimensions
ℜ→ℜ
m
:)f(w
(
)
wf-ww


η
Given
points in direction of steepest ascent.
GRADIENT DESCENT RULE:
Equivalently
( )
wf-
j
jj
w
ηww



….where w
j
is the jth weight
“just like a linear feedback system”
( )
( )
( )


















=∇
wf
wf
wf
1
m
w
w
M
(
)
wf∇
is the gradient in that direction
14
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 27
What’s all this got to do with Neural
Nets, then, eh??
For supervised learning, neural nets are also models with
vectors of w parameters in them. They are now called
weights.
As before, we want to compute the weights to minimize sum-
of-squared residuals.
Which turns out, under “Gaussian i.i.d noise”
assumption to be max. likelihood.
Instead of explicitly solving for max. likelihood weights, we
use GRADIENT DESCENT to SEARCH for them.

W
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y
?


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.



A
h
a!
!


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e
p
l
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:


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e’
l
l

s
ee
l
at
e
r
.

Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 28
Linear Perceptrons
They are multivariate linear models:
Out(x) = w
T
x
And “training” consists of minimizing sum-of-squared residuals
by gradient descent.
QUESTION: Derive the perceptron training rule.
(
)
(
)
( )
2
2


−=
−=Ε
Τ
k
k
y
y
kk
kk
x
xOut
w
15
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 29
Linear Perceptron Training Rule

=
−=
R
k
k
T
k
yE
1
2
)( xw
Gradient descent tells us
we should update w
thusly if we wish to
minimize E:
j
jj
w
E
ηww


← -
So what’s
?
j
w
E


Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 30
Linear Perceptron Training Rule

=
−=
R
k
k
T
k
yE
1
2
)(
xw
Gradient descent tells us
we should update w
thusly if we wish to
minimize E:
j
jj
w
E
ηww


← -
So what’s
?
j
w
E



=



=


R
k
k
T
k
jj
y
ww
E
1
2
)( xw

=



−=
R
k
k
T
k
j
k
T
k
y
w
y
1
)()(2 xwxw

=


−=
R
k
k
T
j
k
w
δ
1
2 xw
k
T
kk
yδ xw−=
…where…
∑ ∑
= =


−=
R
k
m
i
kii
j
k
xw
w
δ
1 1
2

=
−=
R
k
kjk

1
2
16
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 31
Linear Perceptron Training Rule

=
−=
R
k
k
T
k
yE
1
2
)(
xw
Gradient descent tells us
we should update w
thusly if we wish to
minimize E:
j
jj
w
E
ηww


← -
…where…

=
−=


R
k
kjk
j

w
E
1
2

=
+←
R
k
kjkjj
xδηww
1
2
We frequently neglect the 2 (meaning
we halve the learning rate)
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 32
The “Batch” perceptron algorithm
1) Randomly initialize weights w
1
w
2
… w
m
2) Get your dataset (append 1’s to the inputs if
you don’t want to go through the origin).
3) for i = 1 to R
4) for j = 1 to m
5) if stops improving then stop. Else loop
back to 3.
iii
y xw
Τ
−=:δ

=
+←
R
i
ijijj
xww
1
δη

2
i
δ
17
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 33
ijijj
iii
xww
y
ηδ
δ
+←
−←
Τ
xw
A RULE KNOWN BY
MANY NAMES
T
h
e

L
M
S

R
u
l
e
T
h
e

d
e
l
t
a

r
u
l
e
T
h
e

W
i
d
r
o
w
H
o
f
f

r
u
l
e
C
l
a
s
s
i
c
a
l
c
o
n
d
i
t
i
o
n
i
n
g
T
h
e

a
d
a
l
i
n
e
r
u
l
e
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 34
If data is voluminous and arrives fast
Input-output pairs (x,y) come streaming in very
quickly. THEN
Don’t bother remembering old ones.
Just keep using new ones.
observe (x,y)
jjj
xδηwwj
y

xw
+←∀
−←
Τ
δ
18
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 35
GD Advantages (MI disadvantages):
• Biologically plausible
• With very very many attributes each iteration costs only O(mR). If
fewer than m iterations needed we’ve beaten Matrix Inversion
• More easily parallelizable (or implementable in wetware)?
GD Disadvantages (MI advantages):
• It’s moronic
• It’s essentially a slow implementation of a way to build the XTX matrix
and then solve a set of linear equations
• If m is small it’s especially outageous. If m is large then the direct
matrix inversion method gets fiddly but not impossible if you want to
be efficient.
• Hard to choose a good learning rate
• Matrix inversion takes predictable time. You can’t be sure when
gradient descent will stop.
Gradient Descent vs Matrix Inversion
for Linear Perceptrons
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 36
GD Advantages (MI disadvantages):
• Biologically plausible
• With very very many attributes each iteration costs only O(mR). If
fewer than m iterations needed we’ve beaten Matrix Inversion
• More easily parallelizable (or implementable in wetware)?
GD Disadvantages (MI advantages):
• It’s moronic
• It’s essentially a slow implementation of a way to build the XTX matrix
and then solve a set of linear equations
• If m is small it’s especially outageous. If m is large then the direct
matrix inversion method gets fiddly but not impossible if you want to
be efficient.
• Hard to choose a good learning rate
• Matrix inversion takes predictable time. You can’t be sure when
gradient descent will stop.
Gradient Descent vs Matrix Inversion
for Linear Perceptrons
19
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 37
GD Advantages (MI disadvantages):
• Biologically plausible
• With very very many attributes each iteration costs only O(mR). If
fewer than m iterations needed we’ve beaten Matrix Inversion
• More easily parallelizable (or implementable in wetware)?
GD Disadvantages (MI advantages):
• It’s moronic
• It’s essentially a slow implementation of a way to build the XTX matrix
and then solve a set of linear equations
• If m is small it’s especially outageous. If m is large then the direct
matrix inversion method gets fiddly but not impossible if you want to
be efficient.
• Hard to choose a good learning rate
• Matrix inversion takes predictable time. You can’t be sure when
gradient descent will stop.
Gradient Descent vs Matrix Inversion
for Linear Perceptrons
But we’ll
soon see that
GD
has an important extra
trick up its sleeve
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 38
Perceptrons for Classification
What if all outputs are 0’s or 1’s ?
or
We can do a linear fit.
Our prediction is 0 if out(x)≤1/2
1 if out(x)>1/2
WHAT’S THE BIG PROBLEM WITH THIS???
20
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 39
Perceptrons for Classification
What if all outputs are 0’s or 1’s ?
or
We can do a linear fit.
Our prediction is 0 if out(x)≤½
1 if out(x)>½
WHAT’S THE BIG PROBLEM WITH THIS???
Blue = Out(x)
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 40
Perceptrons for Classification
What if all outputs are 0’s or 1’s ?
or
We can do a linear fit.
Our prediction is 0 if out(x)≤½
1 if out(x)>½
Blue = Out(x)
Green = Classification
21
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 41
Classification with Perceptrons I
( )
.xw
2

Τ

ii
y
Don’t minimize
Minimize number of misclassifications instead. [Assume outputs are
+1 & -1, not +1 & 0]
where Round(x) = -1 if x<0
1 if x≥0
The gradient descent rule can be changed to:
if (x
i
,y
i
) correctly classed, don’t change
if wrongly predicted as 1 w w -x
i
if wrongly predicted as -1 w w +x
i
(
)
(
)

Τ

ii
y xw Round
NOTE: CUTE &
NON OBVIOUS WHY
THIS WORKS!!
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 42
Classification with Perceptrons II:
Sigmoid Functions
Least squares fit useless
This fit would classify much
better. But not a least
squares fit.
22
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 43
Classification with Perceptrons II:
Sigmoid Functions
Least squares fit useless
This fit would classify much
better. But not a least
squares fit.
SOLUTION:
Instead of Out(x) = w
T
x
We’ll use Out(x) = g(w
T
x)
where is a
squashing function
(
) ( )
1,0:→ℜxg
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 44
The Sigmoid
)exp(1
1
)(
h
hg
−+
=
Note that if you rotate this
curve through 180
o
centered on (0,1/2) you get
the same curve.
i.e. g(h)=1-g(-h)
Can you prove this?
23
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 45
The Sigmoid
Now we choose w to minimize
[ ]
[
]
∑∑
=
Τ
=
−=−
R
i
ii
R
i
ii
gyy
1
2
1
2
)xw()x(Out
)exp(1
1
)(
h
hg
−+
=
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 46
Linear Perceptron Classification
Regions
0 0
0
1
1
1
X
2
X
1
We’ll use the model Out(x) = g(w
T
(x,1))
= g(w
1
x
1
+w
2
x
2
+ w
0
)
Which region of above diagram classified with +1, and
which with 0 ??
24
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 47
Gradient descent with sigmoid on a perceptron
(
)
(
)
(
)
(
)
( )
( )
( ) ( )( )
( ) ( )( )


∑∑∑ ∑
∑∑ ∑
∑ ∑

=−=
−−=






















−−=































−=

Ε∂














−=Ε






=
−−=









+


+

=

+








+
=







+

−−
=







+


=

+
=
−=
k
kkiiii
iji
i
ii
k
ikk
j
k
ikk
i k
ikki
k
ikk
j
i k
ikki
j
i k
ikki
k
kk
xwy
xgg
xw
w
xwgxwgy
xwg
w
xwgy
w
xwgy
xwg
xgxg
x
e
x
e
x
e
x
e
x
e
x
e
x
e
x
e
xg
x
e
xg
xgxgxg
net )Out(x where
net1net2
'2
2
Out(x)
1
1
1
1
1
1
1
1
2
1
1
2
1
11

2
1
' so
1
1
:Because
1' notice First,
2
δ
δ
( )

=
−+←
R
i
ijiiijj
xggww
1
1δη








=

=
m
j
ijji
xwgg
1
iii
gy

=
δ
The sigmoid perceptron
update rule:
where
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 48
Other Things about Perceptrons
• Invented and popularized by Rosenblatt (1962)
• Even with sigmoid nonlinearity, correct
convergence is guaranteed
• Stable behavior for overconstrained and
underconstrained problems
25
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 49
Perceptrons and Boolean Functions
If inputs are all 0’s and 1’s and outputs are all 0’s and 1’s…
• Can learn the function x
1
∧ x
2
• Can learn the function x
1
∨ x
2
.
• Can learn any
conjunction of literals, e.g.
x
1
∧ ~x
2
∧ ~x
3
∧ x
4
∧ x
5
QUESTION: WHY?
X
1
X
2
X
1
X
2
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 50
Perceptrons and Boolean Functions
• Can learn any disjunction of literals
e.g. x
1
∧ ~x
2
∧ ~x
3
∧ x
4
∧ x
5
• Can learn majority function
f(x
1
,x
2
…x
n
) = 1 if n/2 x
i
’s or more are = 1
0 if less than n/2 x
i
’s are = 1
• What about the exclusive or function?
f(x
1
,x
2
) = x
1
∀ x
2
=
(x
1
∧ ~x
2
) ∨ (~ x
1
∧ x
2
)
26
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 51
Multilayer Networks
The class of functions representable by perceptrons
is limited
( )








==

Τ
j
jj
xwgg Out(x) xw
Use a wider
representation !














=
∑∑
k
jkjk
j
j
xwgWg Out(x)
This is a nonlinear function
Of a linear combination
Of non linear functions
Of linear combinations of inputs
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 52
A 1-HIDDEN LAYER NET
N
INPUTS
= 2 N
HIDDEN
= 3








=

=
HID
N
k
kk
vWg
1
Out








=








=








=



=
=
=
INS
INS
INS
N
k
kk
N
k
kk
N
k
kk
xwgv
xwgv
xwgv
1
33
1
22
1
11
x
1
x
2
w
11
w
21
w
31
w
1
w
2
w
3
w
32
w
22
w
12
27
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 53
OTHER NEURAL NETS
2-Hidden layers + Constant Term
1
x
1
x
2
x
3
x
2
x
1
“JUMP” CONNECTIONS








+=
∑∑
==
HID
INS
N
k
kk
N
k
kk
vWxwg
11
0
Out
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 54
Backpropagation
( )( )
descent.gradient by
xOut
minimize to
}{,}{ weightsofset a Find
Out(x)
2

∑ ∑















=
i
ii
jkj
j k
kjkj
y
wW
xwgWg
That’s it!
That’s the backpropagation
algorithm.
That’s it!
That’s the backpropagation
algorithm.
28
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 55
Backpropagation Convergence
Convergence to a global minimum is not
guaranteed.
•In practice, this is not a problem, apparently.
Tweaking to find the right number of hidden
units, or a useful learning rate η, is more
hassle, apparently.
IMPLEMENTING BACKPROP: Differentiate Monster sum-square residual 
Write down the Gradient Descent Rule  It turns out to be easier &
computationally efficient to use lots of local variables with names like h
j
o
k
v
j
net
i
etc…
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 56
Choosing the learning rate
• This is a subtle art.
• Too small: can take days instead of minutes
to converge
• Too large: diverges (MSE gets larger and
larger while the weights increase and
usually oscillate)
• Sometimes the “just right” value is hard to
find.
29
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 57
Learning-rate problems
From J. Hertz, A. Krogh, and R.
G. Palmer. Introduction to the
Theory of Neural Computation.
Addison-Wesley, 1994.
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 58
Improving Simple Gradient Descent
Momentum
Don’t just change weights according to the current datapoint.
Re-use changes from earlier iterations.
Let ∆w(t) = weight changes at time t.
Let be the change we would make with
regular gradient descent.
Instead we use
Momentum damps oscillations.
A hack? Well, maybe.
w∂
Ε∂
−η
( )
( )
tt ∆w
w
∆w αη +

Ε

−=+1
momentum parameter
( )
(
)
(
)
ttt ∆www
+
=
+
1
30
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 59
Momentum illustration
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 60
Improving Simple Gradient Descent
Newton’s method
)|(|
2
1
)()(
3
2
2
hh
w
h
w
hwhw O
EE
EE
TT
+


+


+=+
If we neglect the O(h
3
) terms, this is a quadratic form
Quadratic form fun facts:
If y = c + b
T
x-1/2 x
T
A x
And if Ais SPD
Then
x
opt
= A
-1
bis the value of xthat maximizes y
31
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 61
Improving Simple Gradient Descent
Newton’s method
)|(|
2
1
)()(
3
2
2
hh
w
h
w
hwhw O
EE
EE
TT
+


+


+=+
If we neglect the O(h
3
) terms, this is a quadratic form
ww
ww










−←

EE
1
2
2
This should send us directly to the global minimum if the
function is truly quadratic.
And it might get us close if it’s locally quadraticish
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 62
Improving Simple Gradient Descent
Newton’s method
)|(|
2
1
)()(
3
2
2
hh
w
h
w
hwhw O
EE
EE
TT
+


+


+=+
If we neglect the O(h
3
) terms, this is a quadratic form
ww
ww










−←

EE
1
2
2
This should send us directly to the global minimum if the
function is truly quadratic.
And it might get us close if it’s locally quadraticish
B
U
T

(a
n
d

i
t

s

a

b
i
g

b
u
t
)…
T
h
a
t

s
e
c
o
n
d

d
e
r
i
v
a
t
i
v
e

m
a
t
r
i
x

c
a
n

b
e

e
x
p
e
n
s
i
v
e

a
n
d

f
i
d
d
l
y

t
o

c
o
m
p
u
t
e
.
I
f

w
e

r
e

n
o
t

a
l
r
e
a
d
y

i
n

t
h
e

q
u
a
d
r
a
t
i
c

b
o
w
l
,

w
e

l
l

g
o

n
u
t
s
.
32
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 63
Improving Simple Gradient Descent
Conjugate Gradient
Another method which attempts to exploit the “local
quadratic bowl” assumption
But does so while only needing to use
and not
2
2
w∂
∂ E
It is also more stable than Newton’s method if the local
quadratic bowl assumption is violated.
It’s complicated, outside our scope, but it often works well.
More details in Numerical Recipes in C.
w∂

E
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 64
BEST GENERALIZATION
Intuitively, you want to use the smallest,
simplest net that seems to fit the data.
HOW TO FORMALIZE THIS INTUITION?
1.Don’t. Just use intuition
2.Bayesian Methods Get it Right
3.Statistical Analysis explains what’s going on
4.Cross-validation
Discussed in the next
lecture
33
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 65
What You Should Know
• How to implement multivariate Least-
squares linear regression.
• Derivation of least squares as max.
likelihood estimator of linear coefficients
• The general gradient descent rule
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 66
What You Should Know
• Perceptrons
Linear output, least squares
Sigmoid output, least squares
• Multilayer nets
The idea behind back prop
 Awareness of better minimization methods
• Generalization. What it means.
34
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 67
APPLICATIONS
To Discuss:
• What can non-linear regression be useful for?
• What can neural nets (used as non-linear
regressors) be useful for?
• What are the advantages of N. Nets for
nonlinear regression?
• What are the disadvantages?
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 68
Other Uses of Neural Nets…
• Time series with recurrent nets
• Unsupervised learning (clustering principal
components and non-linear versions
thereof)
• Combinatorial optimization with Hopfield
nets, Boltzmann Machines
• Evaluation function learning (in
reinforcement learning)