Transcript Intelligent Information Retrieval and Web Search

Machine Learning
Neural Networks
Tomorrow no lesson! (will recover on
Wednesday 11, WEKA !!)
1
Neural Networks
• Analogy to biological neural systems, the most
robust learning systems we know.
• Attempt to understand natural biological systems
through computational modeling.
• Massive parallelism allows for computational
efficiency.
• Help understand “distributed” nature of neural
representations (rather than “localist”
representation) that allow robustness and graceful
degradation.
• Intelligent behavior as an “emergent” property of
large number of simple units rather than from
explicitly encoded symbolic rules and algorithms.
2
Neural Speed Constraints
• Neurons have a “switching time” on the order of a
few milliseconds, compared to nanoseconds for
current computing hardware.
• However, neural systems can perform complex
cognitive tasks (vision, speech understanding) in
tenths of a second.
• Only time for performing 100 serial steps in this
time frame, compared to orders of magnitude
more for current computers.
• Must be exploiting “massive parallelism.”
• Human brain has about 1011 neurons with an
average of 104 connections each.
3
Neural Network Learning
• Learning approach based on modeling
adaptation in biological neural systems.
• Perceptron: Initial algorithm for learning
simple neural networks (single layer)
developed in the 1950’s.
• Backpropagation: More complex algorithm
for learning multi-layer neural networks
developed in the 1980’s.
4
Real Neurons
• Cell structures
–
–
–
–
Cell body
Dendrites
Axon
Synaptic terminals
5
Neural Communication
• Electrical potential across cell membrane exhibits spikes
called action potentials.
• Spike originates in cell body, travels down
axon, and causes synaptic terminals to
release neurotransmitters.
• Chemical diffuses across synapse to
dendrites of other neurons.
• Neurotransmitters can be excititory or
inhibitory.
• If net input of neurotransmitters to a neuron from other
neurons is excititory and exceeds some threshold, it fires
an action potential.
6
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.”
7
Artificial Neuron Model
• Model network as a graph with cells as nodes and synaptic
connections as weighted edges from node i to node j, wji
1
• Model net input to cell as
w12
net j   w jioi
w15
w16
w13 w14
2
i
3
4
5
6
• Cell output is:
0 if net j  T j
oj 
1 if neti  T j
(Tj is threshold for unit j)
oj
1
0
Tj
netj
8
Perceptron schema
Perceptron training phase= estimating edge weights and threshold
o1
o2
on
w1j
wnj
Threshold
Tj
∑
oj
netj
0 if net j  T j
oj 
1 if neti  T j
9
Perceptron Training
• Assume supervised training examples
giving the desired output for a unit given a
set of known input activations.
• Learn synaptic weights so that unit
produces the correct output for each
example.
• Perceptron uses iterative update algorithm
to learn a correct set of weights.
10
Perceptron Learning Rule
• Update weights by:
w = w + h(t j - o j )oi
(t )
ji
(t-1)
ji
0 if net j  T j
oj 
1 if neti  T j
where η is a constant named the “learning rate”
tj is the teacher specified output for unit j, (tj -oj) is
the error in iteration t
• Equivalent to rules:
– If output is correct do nothing.
– If output is “high”, lower weights on active inputs
– If output is “low”, increase weights on active inputs
• Also adjust threshold to compensate:
T = T - h(t j - o j )
t
j
t-1
j
11
Perceptron Learning Algorithm
• Iteratively update weights until convergence.
Initialize weights to random values
Until outputs of all training examples are correct
For each training pair, Ej <xj(o1,o2,on),tj>, do:
Compute current output oj for Ej given its inputs
Compare current output to target value, tj , for Ej
Update synaptic weights and threshold using learning rule
• Each execution of the outer loop is typically
called an epoch.
12
Perceptron as a Linear Separator
• Since perceptron uses linear threshold function, it is searching
for a linear separator that discriminates the classes.
w12
2
o3
1
w13
3
w12o2  w13o3  T1
??
w12
T1
o3  
o2 
w13
w13
o2
Or hyperplane in
n-dimensional space
13
Concept Perceptron Cannot Learn
• Cannot learn exclusive-or, or parity
function in general (not linearly separable!).
o3
1
+
??
–
–
+
0
1
o2
14
Perceptron Limits
• System obviously cannot learn concepts it
cannot represent.
• Minksy and Papert (1969) wrote a book
analyzing the perceptron and demonstrating
many functions it could not learn.
• These results discouraged further research
on neural nets; and symbolic AI became the
dominate paradigm.
15
Perceptron Convergence
and Cycling Theorems
• Perceptron convergence theorem: If the data is
linearly separable and therefore a set of weights
exist that are consistent with the data, then the
Perceptron algorithm will eventually converge to a
consistent set of weights.
• Perceptron cycling theorem: If the data is not
linearly separable, the Perceptron algorithm will
eventually repeat a set of weights and threshold at
the end of some epoch and therefore enter an
infinite loop.
– By checking for repeated weights+threshold, one can
guarantee termination with either a positive or negative
result.
16
Perceptron as Hill Climbing
• The hypothesis space being search is a set of weights and a
threshold.
• Objective is to minimize classification error on the training set.
• Perceptron effectively does hill-climbing (gradient descent) in
this space, changing the weights a small amount at each point
to decrease training set error.
• For a single model neuron, the space is well behaved with a
single minima.
training
error
0
weights
17
Example: Learn a NOR function
w1 x1 + w2 x2 > u
(u = Threshold)
w = w + 0.05(t j - o )o u = u - 0.05(t j - o )
(t )
i
(t-1)
i
(t-1)
i
j
t
j
t-1
j
(t-1)
j
18
19
20
21
22
23
After 20 iterations
24
Perceptron Performance
• In practice, converges fairly quickly for
linearly separable data.
• Can effectively use even incompletely
converged results when only a few outliers
are misclassified.
• Experimentally, Perceptron does quite well
on many benchmark data sets.
25
Multi-Layer Networks
• Multi-layer networks can represent arbitrary functions
• A typical multi-layer network consists of an input, hidden
and output layer, each fully connected to the next, with
activation feeding forward.
output
hidden
activation
input
• The weights determine the function computed.
• Given an arbitrary number of hidden units, any boolean
function can be computed with a single hidden layer.
26
Hill-Climbing in Multi-Layer Nets
• Since “greed is good” perhaps hill-climbing can be used to
learn multi-layer networks in practice although its
theoretical limits are clear.
• However, to do gradient descent with multiple neurons, we
need the output of a unit to be a differentiable function of
its input and weights.
• Standard linear threshold function is not differentiable
at the threshold.
training
error
oi
1
0
Tj
netj
weights 27
Differentiable Output Function
• Need non-linear output function to move beyond linear
functions.
– Standard solution is to use the non-linear, differentiable
sigmoidal “logistic” function:
oj 
1
1 e
net j   w jioi
1
 ( net j T j )
0
Tj
netj
i
Can also use tanh or Gaussian output function
28
• Structure of a node:
threshold
Thresholding function limits node output
(Tj=0 in the example):
29
Feeding data through the net
(1  0.25) + (0.5  (-1.5)) = 0.25 + (-0.75) = - 0.5
thresholding:
1
 0.3775
0 .5
1 e
30
Gradient Descent
• The objective is to minimize error:
E(W )   (tkd  okd )2
dD kK
where D is the set of training examples, K is the set of
output units, tkd and okd are, respectively, the teacher and
current output for unit k for example d.
• The derivative of a sigmoid unit with respect to net input is:
o j
 o j (1  o j )
1
o

net j
j
 ( net T )
1 e
• Learning rule to change weights to minimize error is:
E
w ji  
w ji
j
j
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Backpropagation Learning Rule
w ji   j oi
• Each weight changed by:
i
oi
j
oj
wij
d j = o j (1- o j )(t j - o j )
if j is an output unit
 j  o j (1  o j ) k wkj
if j is a hidden unit
k
where η is a constant called the learning rate
tj is the correct “teacher” output for unit j
δj is the error measure for unit j
32
Example for net1=w1x1+w2x2
net1 = w1 x1 + w2 x2
E =1/ 2(t - o)
2
w1
w2
x1
E
w ji  
w ji
x2
¶ E(w1 x1 + w2 x2 )
¶ E ¶ net1
¶E
Dw1 = -h
= -h
= -h
x1
¶ w1
¶ net1 ¶ w1
¶ net1
¶E
¶ E ¶o
=
¶ net1 ¶ o ¶ net1
¶E 1
¶ (t - o)
= 2(t - o)
= -(t - o)
¶o 2
¶o
¶o
¶s (net1 )
=
= o(1- o)
¶ net1
¶ net1
¶ (s (x)) = s (x)(1- s (x))
Dw1 = ho(1- o)(t - o)x1
Se x1 è l’output del nodo precedente, cioè oi, abbiamo:
w ji   j oi
33
Example
o j = s (x n j h1 w j1 + x n j h2 w j2 ) = s (o h1 w j1 + oh2 w j2 )
oh1 = s (x h1 n1 w n1 h1 + x h1 n 2 w n 2 h1 + ....+ x h1 n N w n N h1 )
oh2 = s (x h2 n1 w n1 h2 + x h2 n 2 w n 2 h2 + ....+ x h2 n N w n N h2 )
oj
nj
wj1
wj2
h1
h2
…….
n1
x h1 n1 = on1 = s (x11 )...on N = s (x1N )
…...
nj
x1: x11 x12…x1N
x ji = o ji
…..
nN
Step 1: feed with first example
and compute all oj
Example (cont’d)
d j = o j (1 - o j )(t 1 - o j )
oj
dh1 = x h1 (1 - x h1 )w j1d j =
n out
j
n1hidden
wj1
…….
n1in
n2hidden
…...
oh1 (1 - o h1 )w j 1d j
wj2
dh2 = o h2 (1 - o h2 )w j 2d j
…..
n in
N
Compute the error on the output node, and backpropagate
computation on hidden nodes
Example (cont’d)
oj
n out
j
n1hidden
wj1
wh1n1
…….
n1in
n2hidden
wh2n1
…...
wj2
…and on input nodes
…..
n in
N
d (n1in ) = o1in (1 - oin
1 )(wh1n1d h1 + wh2n1d h2 )
......
in
in
d (n in
)
=
o
(1
o
N
N
N )(wh1nN d h1 + wh2n N dh2 )
Example (cont’d)
w j1 ¬ w j1 + hd j x j 1
w j2 ¬ w j2 + hd j x j 2
wh1n1 ¬ wh1n1 + hd h1x h1 ......
n1
•Update all weights
•Consider the second example
•Compute input and output in all nodes
•Compute errors on outpus and on all nodes
•Re-update weights
•Untill all examples have been considered (Epoch)
•Repeat for n Epochs, until convergence
Another example (weight updates)
• Training set
–
–
–
–
((0.1, 0.1), 0.1)
((0.1, 0.9), 0.9)
((0.9, 0.1), 0.9)
((0.9, 0.9), 0.1)
0,1
-0,1
-0,1
3
-0,1
0,1
Initial weights
1
2
0,1
1. Compute output and error on output
node
First example:
((0.1, 0.1), 0.1)
0,1
0,1
s (1) = s (w1x1+ w2x2) =
-0,1
1
-0,1
0,1
-0,1
1
1+ e-(w1x1+w2x2)
3
0,5
0,1
0,1
2
0,5
0,1
0,5
1
Err = (0,1- 0,5) 2 = 0,08
2
d j = o j (1- o j )(t j - o j )
d (3) = 0,5(1- 0,5)(0.1- 0,5) = -0,5´ 0,5´ 0,40 = -0,1
2. Backpropagate error
d h = oh (1- oh ) å w khd k
n k ÎO
d (1) = 0,5(1- 0,5)[ 0,1(-0,1)] = -0,025
d (2) = 0,5(1- 0,5)[ 0,1(0,1)] = 0,025
1
-0,025
-0,1 0,1
0,1
0,5
2
0,1
0,025
0,5
3
0,5
3. Update weights
0,1
0,1
1
-0,025
-0,1
-0,1
0,5
-0,1
w 31 = -0,1+ 0,2 ´ 0,5 ´ (-0,1) = -0,11
w 32 = 0,1+ 0,2 ´ 0,5 ´ (-0,1) = 0,09
0,1
w1i1 = 0,1+ 0,2 ´ 0,1´ (0,025) = 0,1005
w1i2 = -0,1+ 0,2 ´ 0,1´ (0,025) = -0,0995
w 2i1 = -0,1+ 0,2 ´ 0,1´ (-0,025) = -0,1005
w 2i2 = 0,1+ 0,2 ´ 0,1´ (-0,025) = 0,0995
-0,1 0,1
3
0,1
0,1
2
0,025
0,5
0,5
Read second input from D
2. ((0,1,0,9), 0,9)
0,1
0,1005
1
-0,11
-0,0995
0,48
3
0,9
-0,1005
0,9
0,0995
0,09
2
0,501
0,52
..etc!!
Yet another example
• First calculate error of output units and use this to
change the top layer of weights.
Current output: oj=0.2
Correct output: tj=1.0
Error δj = oj(1–oj)(tj–oj)
0.2(1–0.2)(1–0.2)=0.128
output
Update weights into j
w ji   j oi
hidden
input
43
Error Backpropagation
• Next calculate error for hidden units based on
errors on the output units it feeds into.
output
 j  o j (1  o j ) k wkj
k
hidden
input
44
Error Backpropagation
• Finally update bottom layer of weights based on
errors calculated for hidden units.
output
 j  o j (1  o j ) k wkj
k
Update weights into j
hidden
w ji   j oi
input
45
Example: Voice Recognition
• Task: Learn to discriminate between two
different voices saying “Hello”
• Data
– Sources
• Steve Simpson
• David Raubenheimer
– Format
• Frequency distribution (60 bins)
• Analogy: cochlea
• Network architecture
– Feed forward network
• 60 input (one for each frequency bin)
• 6 hidden
• 2 output (0-1 for “Steve”, 1-0 for “David”)
• Presenting the data
Steve
David
• Presenting the data (untrained network)
Steve
0.43
0.26
David
0.73
0.55
• Calculate error
Steve
0.43 – 0
= 0.43
0.26 –1
= 0.74
0.73 – 1
= 0.27
0.55 – 0
= 0.55
David
• Backprop error and adjust weights
Steve
0.43 – 0
= 0.43
0.26 – 1
= 0.74
1.17
David
0.73 – 1
= 0.27
0.55 – 0
= 0.55
0.82
• Repeat process (sweep) for all training pairs
–
–
–
–
Present data
Calculate error
Backpropagate error
Adjust weights
• Repeat process multiple times
• Presenting the data (trained network)
Steve
0.01
0.99
David
0.99
0.01
Backpropagation Training Algorithm
Create the 3-layer network with H hidden units with full connectivity
between layers. Set weights to small random real values.
Until all training examples produce the correct value (within ε), or
mean squared error ceases to decrease, or other termination criteria:
Begin epoch
For each training example, d, do:
Calculate network output for d’s input values
Compute error between current output and correct output for d
Update weights by backpropagating error and using learning rule
End epoch
54
Comments on Training Algorithm
• Not guaranteed to converge to zero training error,
may converge to local optima or oscillate
indefinitely.
• However, in practice, does converge to low error
for many large networks on real data.
• Many epochs (thousands) may be required, hours
or days of training for large networks.
• To avoid local-minima problems, run several
trials starting with different random weights
(random restarts).
– Take results of trial with lowest training set error.
– Build a committee of results from multiple trials
(possibly weighting votes by training set accuracy).
55
Representational Power
• Boolean functions: Any boolean function can be
represented by a two-layer network with sufficient
hidden units.
• Continuous functions: Any bounded continuous
function can be approximated with arbitrarily
small error by a two-layer network.
– Sigmoid functions can act as a set of basis functions for
composing more complex functions, like sine waves in
Fourier analysis.
• Arbitrary function: Any function can be
approximated to arbitrary accuracy by a threelayer network.
56
Over-Training Prevention
error
• Running too many epochs can result in over-fitting.
on test data
on training data
0
# training epochs
• Keep a hold-out validation set and test accuracy on it after
every epoch. Stop training when additional epochs actually
increase validation error.
• To avoid losing training data for validation:
– Use internal 10-fold CV on the training set to compute the average
number of epochs that maximizes generalization accuracy.
– Train final network on complete training set for this many epochs.
57
Determining the Best
Number of Hidden Units
error
• Too few hidden units prevents the network from
adequately fitting the data.
• Too many hidden units can result in over-fitting.
on test data
on training data
0
# hidden units
• Use internal cross-validation to empirically determine an
optimal number of hidden units.
58
Successful Applications
• Text to Speech (NetTalk)
• Fraud detection
• Financial Applications
– HNC (eventually bought by Fair Isaac)
• Chemical Plant Control
– Pavillion Technologies
• Automated Vehicles
• Game Playing
– Neurogammon
• Handwriting recognition
59
Issues in Neural Nets
• More efficient training methods:
– Quickprop
– Conjugate gradient (exploits 2nd derivative)
• Learning the proper network architecture:
– Grow network until able to fit data
• Cascade Correlation
• Upstart
– Shrink large network until unable to fit data
• Optimal Brain Damage
• Recurrent networks that use feedback and can
learn finite state machines with “backpropagation
through time.”
60
Issues in Neural Nets (cont.)
• More biologically plausible learning
algorithms based on Hebbian learning.
• Unsupervised Learning
– Self-Organizing Feature Maps (SOMs)
• Reinforcement Learning
– Frequently used as function approximators for
learning value functions.
• Neuroevolution
61