Neural Networks

Regression and
Classification with
Neural Networks
Andrew W. Moore
Note to other teachers and users of
these slides. Andrew would be delighted

Professor
if you found this source material useful in
giving your own lectures. Feel free to use
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received.

412-268-7599

Copyright © 2001, 2003, Andrew W. Moore Sep 25th, 2001

Linear Regression
DATASET

inputs outputs
x1 = 1 y1 = 1
x2 = 3 y2 = 2.2
↑
x3 = 2 y3 = 2
w
↓
←1→
x4 = 1.5 y4 = 1.9
x5 = 4 y5 = 3.1

Linear regression assumes that the expected value of
the output given an input, E[y|x], is linear.
Simplest case: Out(x) = wx for some unknown w.
Given the data, we can estimate w.
Copyright © 2001, 2003, Andrew W. Moore Neural Networks: Slide 2

1-parameter linear regression
Assume that the data is formed by
yi = wxi + noisei

where…
• the noise signals are independent
• the noise has a normal distribution with mean 0
and unknown variance σ2

P(y|w,x) has a normal distribution with
• mean wx
• variance σ2

Bayesian Linear Regression
P(y|w,x) = Normal (mean wx, var σ2)

We have a set of datapoints (x1,y1) (x2,y2) … (xn,yn)
which are EVIDENCE about w.

We want to infer w from the data.
P(w|x1, x2, x3,…xn, y1, y2…yn)
•You can use BAYES rule to work out a posterior
distribution for w given the data.
•Or you could do Maximum Likelihood Estimation


Maximum likelihood estimation of w

Asks the question:
“For which value of w is this data most likely to have
happened?”
<=>
For what w is
P(y1, y2…yn |x1, x2, x3,…xn, w) maximized?
<=>
For what w is n

∏P( y w, x ) maximized?
i i
i =1

For what w is
n

∏ P( y w , x i ) maximized?
i
i =1

For what w is
n
1
∏ exp(− 2 (
yi − wxi
) 2 ) maximized?
σ
i =1
For what w is
2
1  y − wx i 
n

∑ −i  maximized?
σ
2 
i =1
For what w is 2
n

∑ (y )
− wx minimized?
i i
i =1


Linear Regression

The maximum
likelihood w is
the one that E(w)
w
minimizes sum-
Ε = ∑( yi − wxi )
2
of-squares of
residuals i

(∑ x )w
= ∑ yi − (2∑ xi yi )w +
2 2 2
i
i
We want to minimize a quadratic function of w.

Linear Regression
Easy to show the sum of
squares is minimized
when
∑x y i i
w=
∑x
2
i

The maximum likelihood
Out(x) = wx
model is

We can use it for
prediction

Linear Regression
Easy to show the sum of
squares is minimized
when
∑ xi yi
p(w)

w= w

∑x
2
Note:
i In Bayesian stats you’d have
ended up with a prob dist of w
The maximum likelihood
Out(x) = wx
model is And predictions would have given a prob
dist of expected output

Often useful to know your confidence.
We can use it for
Max likelihood can give some kinds of
prediction
confidence too.


Multivariate Regression
What if the inputs are vectors?
3.
.4
6.
2-d input
5
.
example
.8
x2 . 10

x1
Dataset has form
x1 y1
x2 y2
x3 y3
.: :
.
xR yR

Write matrix X and Y thus:

 .....x1 .....   x11 x1m   y1 
x12 ...
.....x .....  x  y 
x22 ... x2 m 
 =  21
x= y =  2
2

 
 M
M
M
 
 
.....x R .....  xR1 xR 2 ... xRm   yR 

(there are R datapoints. Each input has m components)
The linear regression model assumes a vector w such that
Out(x) = wTx = w1x[1] + w2x[2] + ….wmx[D]
The max. likelihood w is w = (XTX) -1(XTY)


Write matrix X and Y thus:

 .....x1 .....   x11 x1m   y1 
x12 ...
.....x .....  x ... x2 m  
x22  y =  y2 
 =  21
x= 2

 
 M
M
M
 
 
.....x R .....  xR1 xR 2 ... xRm   yR 

IMPORTANT EXERCISE:
(there are R datapoints. Each input hasPROVE IT !!!!!
m components)
The linear regression model assumes a vector w such that
Out(x) = wTx = w1x[1] + w2x[2] + ….wmx[D]
The max. likelihood w is w = (XTX) -1(XTY)


Multivariate Regression (con’t)

The max. likelihood w is w = (XTX)-1(XTY)
R

∑x x ki kj
XTX is an m x m matrix: i,j’th elt is
k =1
R

∑x y
XTY is an m-element vector: i’th elt ki k
k =1


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??

The constant term
• The trick is to create a fake input “X0” that
always takes the value 1

X1 X2 Y X0 X1 X2 Y
2 4 16 1 2 4 16
3 4 17 1 3 4 17
5 5 20 1 5 5 20
Before: After:
Y=w1X1+ w2X2 Y= w0X0+w1X1+ w2X2
…has to be a poor = w0+w1X1+ w2X2
In this example,
You should be able
model …has a fine constant
to see the MLE w0
, w1 and w2 by
term
inspection

Regression with varying noise
• Suppose you know the variance of the noise that
was added to each datapoint.
y=3
σ=2
σi2
xi yi
½ ½ 4 y=2
σ=1/2
1 1 1
σ=1
2 1 1/4 y=1
σ=1/2
σ=2
2 3 4 y=0

3 2 1/4 x=0 x=1 x=2 x=3

MLE
the w?
yi ~ N ( wxi , σ i2 ) at’s f
W h at e o
Assume
stim
e

MLE estimation with varying noise
argmax log p( y , y ,..., y | x1 , x2 ,..., xR , σ 12 , σ 2 ,..., σ R , w) =
2 2
1 2 R

w Assuming i.i.d. and
( yi − wxi ) 2
R then plugging in
argmin ∑ = equation for Gaussian
σ i2 and simplifying.
i =1
w
 
x ( y − wx ) Setting dLL/dw
R
 w such that ∑ i i 2 i = 0  = equal to zero
 
σi
 
i =1

 R xi yi  Trivial algebra
∑ 2 
 
 i =1 σ i 
 R xi2 
∑ 2 
 σ
 i =1 i 

This is Weighted Regression
• We are asking to minimize the weighted sum of
squares
y=3
σ=2
( yi − wxi ) 2
R

argmin ∑ σ i2 y=2
i =1 σ=1/2
w
σ=1
y=1
σ=1/2
σ=2
y=0
x=0 x=1 x=2 x=3

1
where weight for i’th datapoint is σ i2

Weighted Multivariate Regression

The max. likelihood w is w = (WXTWX)-1(WXTWY)

xki xkj
R

∑
(WXTWX) is an m x m matrix: i,j’th elt is
σ i2
k =1

R
xki yk
(WXTWY) is an m-element vector: i’th elt
∑ σ i2
k =1


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:
y=3
xi yi
½ ½ y=2

1 2.5
2 3 y=1

3 2 y=0

3 3 x=0 x=1 x=2 x=3

MLE
yi ~ N ( w + xi , σ ) the w?
2 at’s f
W h at e o
Assume
stim
e

Non-linear MLE estimation
argmax log p( y , y ,..., y | x1 , x2 ,..., xR , σ , w) =
1 2 R


argmin ∑ (y )
R then plugging in
2
− w + xi = equation for Gaussian
i
and simplifying.
i =1
w
 
y − w + xi Setting dLL/dw
R
 w such that ∑ i = 0 = equal to zero
 
w + xi
 
i =1


1 2 R


argmin ∑ (y )
R then plugging in
2
− w + xi = equation for Gaussian
i
and simplifying.
i =1
w
 
y − w + xi Setting dLL/dw
R
 w such that ∑ i = 0 = equal to zero
 
w + xi
 
i =1

We’re down the
algebraic toilet
t
wha
ess
u
So g e do?
w

1 2 R


argmin ∑ ( )
R
Common (but not only) approach: then plugging in
2
yi − w + xi = equation for Gaussian
Numerical Solutions: and simplifying.
i =1
w
• Line Search
• Simulated Annealing
 
yi − w + xi Setting dLL/dw
R

∑
 w such = 0 =
• Gradient Descent that equal to zero
 
w + xi
 
• Conjugate Gradient i =1

• Levenberg Marquart
We’re down the
• Newton’s Method
algebraic toilet
Also, special purpose statistical-
t
wha
optimization-specific tricks such as
uess ?
So g e do
E.M. (See Gaussian Mixtures lecture
for introduction) w

GRADIENT DESCENT
f(w): ℜ → ℜ
Suppose we have a scalar function

We want to find a local minimum.
Assume our current weight is w

∂
f (w)
GRADIENT DESCENT RULE: w ← w −η
∂w

η is called the LEARNING RATE. A small positive
number, e.g. η = 0.05


GRADIENT DESCENT
f(w): ℜ → ℜ
Suppose we have a scalar function

We want to find a local minimum.
Assume our current weight is w

∂
f (w)
GRADIENT DESCENT RULE: w ← w −η
∂w
Recall Andrew’s favorite
default value for anything
η is called the LEARNING RATE. A small positive
number, e.g. η = 0.05
QUESTION: Justify the Gradient Descent Rule

Gradient Descent in “m” Dimensions
f(w ) : ℜ m → ℜ
Given
∂ 
f (w ) 

 ∂w1 
 points in direction of steepest ascent.
∇f (w) =  M
∂ 
 ∂w f (w)
m 
∇f (w) is the gradient in that direction

w ← w -η∇f (w)
GRADIENT DESCENT RULE:
Equivalently ∂
f (w ) ….where wj is the jth weight
wj ← wj - η
∂w j
“just like a linear feedback system”

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.
our eyes.
ression in y
s exp
, a querulou
y?” you ask
“Wh
e later.”
ply: “We’ll se
“Aha!!” I re

Linear Perceptrons
They are multivariate linear models:

Out(x) = wTx

And “training” consists of minimizing sum-of-squared residuals
by gradient descent.

∑ (Out (x ) − )2
Ε= yk
k
k

∑ (w )
2
Τ
= x k − yk
k

QUESTION: Derive the perceptron training rule.

Linear Perceptron Training Rule
R
E = ∑ ( yk − w T x k ) 2
k =1

Gradient descent tells us
we should update w
thusly if we wish to
minimize E:
∂E
wj ← wj - η
∂w j

∂E
?
So what’s
∂w j

R
∂E ∂
R
E = ∑ ( yk − w T x k ) 2 =∑ ( yk − w T x k ) 2
∂w j k =1 ∂w j
k =1
∂
R
= ∑ 2( yk − w T x k )
Gradient descent tells us ( yk − w T x k )
∂w j
we should update w k =1

thusly if we wish to ∂
R
= −2∑ δk
minimize E: wT xk
∂w j
k =1
∂E …where…
wj ← wj - η δk = yk − w T x k
∂w j
∂
R m
= −2∑ δk ∑w x i ki
∂w j
k =1 i =1
∂E
?
So what’s R
∂w j = −2∑ δk xkj
k =1


R
E = ∑ ( yk − w T x k ) 2
k =1

Gradient descent tells us
we should update w
thusly if we wish to
minimize E:
∂E
wj ← wj - η R
w j ← w j + 2η∑ δk xkj
∂w j
…where…
k =1
∂E R
= −2∑ δk xkj
∂w j We frequently neglect the 2 (meaning
k =1
we halve the learning rate)

The “Batch” perceptron algorithm
1) Randomly initialize weights w1 w2 … wm

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
δ i := yi − wΤxi
4) for j = 1 to m R
w j ← w j + η ∑ δ i xij
i =1

5) if ∑ δ i 2 stops improving then stop. Else loop
back to 3.

δ i ← yi − w Τ x i A RULE KNOWN BY

w j ← w j + ηδ i xij
MANY NAMES

rule
off
e wH
Rul idro
LMS W
The
e
Th
The delta ru
le
Th
e ad
alin
er
ule
Classical
conditioning


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)

δ ← y − wΤx
∀j w j ← w j + η δ x j


Gradient Descent vs Matrix Inversion
for Linear Perceptrons
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.

GD Disadvantages (MI advantages):
• It’s moronic
be efficient.

But we’ll
GD Disadvantages (MIsoon see that
advantages):
GD
• It’s moronic
has an important extra
trick up its sleeve
be efficient.

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???



or

Blue = Out(x)
Our prediction is 0 if out(x)≤½
1 if out(x)>½
WHAT’S THE BIG PROBLEM WITH THIS???



or

Blue = Out(x)
Our prediction is 0 if out(x)≤½ Green = Classification
1 if out(x)>½


Classification with Perceptrons I
∑ (y )2
− w Τxi .
Don’t minimize i

Minimize number of misclassifications instead. [Assume outputs are

∑ (y ( ))
+1 & -1, not +1 & 0]
− Round w Τ x i
i

where Round(x) = -1 if x<0 NOTE: CUTE &
NON OBVIOUS WHY
1 if x≥0 THIS WORKS!!

The gradient descent rule can be changed to:
if (xi,yi) correctly classed, don’t change
w w - xi
if wrongly predicted as 1
w w + xi
if wrongly predicted as -1


Classification with Perceptrons II:
Sigmoid Functions

Least squares fit useless
This fit would classify much
better. But not a least
squares fit.


Classification with Perceptrons II:
Sigmoid Functions

Least squares fit useless
This fit would classify much
SOLUTION: better. But not a least
squares fit.
Instead of Out(x) = wTx
We’ll use Out(x) = g(wTx)
where g( x) : ℜ → (0,1) is a
squashing function

The Sigmoid
1
g ( h) =
1 + exp(− h)

Note that if you rotate this
curve through 180o
centered on (0,1/2) you get
the same curve.

i.e. g(h)=1-g(-h)

Can you prove this?

The Sigmoid
1
g ( h) =
1 + exp(− h)

Now we choose w to minimize

∑ [ yi − Out(x i )] = ∑ [yi − g (w Τ x i )]
R R
2
2

i =1 i =1


Linear Perceptron Classification
Regions
0 0
0
1
X2 1
1
X1

Out(x) = g(wT(x,1))
We’ll use the model
= g(w1x1 + w2x2 + w0)
Which region of above diagram classified with +1, and
which with 0 ??

Gradient descent with sigmoid on a perceptron
First, notice g ' ( x ) = g ( x )(1 − g ( x ))
− e− x
1
Because : g ( x ) = so g ' ( x ) =
1 + e− x 2
1 + e − x 
 
 
1 −1 − e− x −1  1
1 1
 = − g ( x )(1 − g ( x ))
1 −
= = − =
2 1 + e− x 1 + e− x  1 + e− x 
2  
1 + e − x  1 + e − x 
   
   
 
Out(x) = g  ∑ wk xk 
The sigmoid perceptron
k 
update rule:
2
 

Ε = ∑  yi − g  ∑ wk xik  
 
k 
i
R
w j ← w j + η ∑ δ i gi (1 − gi )xij
  ∂  

∂Ε 
= ∑ 2 yi − g  ∑ wk xik   − g  ∑ wk xik  
 
  ∂w j  k 
∂w j k
   i =1
i

m 
  
 ∂
= ∑ − 2 yi − g  ∑ wk xik  g '  ∑ wk xik  ∑w x gi = g  ∑ w j xij 
  k ik
where
 ∂w j
k   k
 

i k

 j =1 
= ∑ − 2δ i g (net i )(1 − g (net i ))xij

δ i = yi − gi
i

net i = ∑ wk xk
where δ i = yi − Out(x i )
k

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


Perceptrons and Boolean Functions
If inputs are all 0’s and 1’s and outputs are all 0’s and 1’s…

X2
• Can learn the function x1 ∧ x2
X1

X2
• Can learn the function x1 ∨ x2 .
X1
• Can learn any conjunction of literals, e.g.
x1 ∧ ~x2 ∧ ~x3 ∧ x4 ∧ x5

QUESTION: WHY?


Perceptrons and Boolean Functions
• Can learn any disjunction of literals
e.g. x1 ∧ ~x2 ∧ ~x3 ∧ x4 ∧ x5

• Can learn majority function
f(x1,x2 … xn) = 1 if n/2 xi’s or more are = 1
0 if less than n/2 xi’s are = 1

• What about the exclusive or function?
f(x1,x2) = x1 ∀ x2 =
(x1 ∧ ~x2) ∨ (~ x1 ∧ x2)


Multilayer Networks
The class of functions representable by perceptrons
is limited  
( )
Out(x) = g w Τ x = g  ∑ w j x j 
 
j 

Use a wider
representation !

 

Out(x) = g  ∑W j g  ∑ w jk x jk   This is a nonlinear function
 
k 
j Of a linear combination
Of non linear functions
Of linear combinations of inputs

A 1-HIDDEN LAYER NET
NINPUTS = 2 NHIDDEN = 3

 NINS 
v1 = g ∑ w1k xk 
 
w11
 k =1  w1
x1 w21

 NINS 
w31
v2 = g ∑ w2k xk   N HID 
w2
Out = g  ∑ Wk vk 
   
 k =1 
w12
 k =1 
w22
x2 w3

 NINS 
w32
v3 = g ∑ w3k xk 
 
 k =1 

OTHER NEURAL NETS
1

x1
x2
x3
2-Hidden layers + Constant Term

“JUMP” CONNECTIONS

x1

 N INS 
N HID
Out = g  ∑ w0 k xk + ∑Wk vk 
x2
 
 k =1 
k =1

Backpropagation
 

Out(x) = g  ∑W j g  ∑ w jk xk  
 
k 
j
Find a set of weights {W j },{w jk }
to minimize

∑ ( y − Out(x ))
2
i i
i

by gradient descent.
That’s it!
That’s it!
That’s the backpropagation
That’s the backpropagation
algorithm.
algorithm.


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 hj ok vj neti
etc…


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.


Learning-rate problems

From J. Hertz, A. Krogh, and R.
G. Palmer. Introduction to the
Theory of Neural Computation.
Addison-Wesley, 1994.


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
∂Ε
−η
∂w regular gradient descent.
Instead we use
∂Ε
∆w (t +1) = −η + α∆w (t )
∂w
w (t + 1) = w (t ) + ∆w (t )
Momentum damps oscillations. momentum parameter
A hack? Well, maybe.


Momentum illustration


Newton’s method
∂E 1 T ∂ 2 E
E ( w + h) = E ( w ) + hT +h h + O(| h |3 )
∂w 2 ∂w 2

If we neglect the O(h3) terms, this is a quadratic form

Quadratic form fun facts:
If y = c + bT x - 1/2 xT A x
And if A is SPD
Then
xopt = A-1b is the value of x that maximizes y

Newton’s method
∂E 1 T ∂ 2 E
E ( w + h) = E ( w ) + hT +h h + O(| h |3 )
∂w 2 ∂w 2

If we neglect the O(h3) terms, this is a quadratic form
−1
 ∂ 2 E  ∂E
w ←w− 2
 ∂w  ∂w
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


Newton’s method
∂E 1 T ∂ 2 E
E ( w + h) = E ( w ) + hT +h h + O(| h |3 )
∂w 2 ∂w 2

IfBUT neglect the O(h3) terms, this is a quadratic form
we
(and
it’s a
big b
That ut) −1
secon
d der w ←… −  ∂ E  ∂E
2
expen w  2
i va
s i ve a
nd fid tive matr  ∂w  ∂w
ix
If we dly to
comp can b
’r
we’ll e not alrea us directlyute.the e
Thisoshould sendy i to global minimum if the
g nu dn
ts the q
function is. truly quadratic. adr ua
tic bo
wl,
And it might get us close if it’s locally quadraticish


Conjugate Gradient
Another method which attempts to exploit the “local
quadratic bowl” assumption
∂E
But does so while only needing to use
∂w

and not ∂ E
2

∂w 2

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.

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


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


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.


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?


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)


Neural Networks

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