Transcript A Black-Box approach to machine learning

A Black-Box approach to
machine learning
Yoav Freund
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Why do we need learning?
• Computers need functions that map highly variable data:




Speech recognition:
Audio signal -> words
Image analysis:
Video signal -> objects
Bio-Informatics: Micro-array Images -> gene function
Data Mining: Transaction logs -> customer classification
• For accuracy, functions must be tuned to fit the data
source.
• For real-time processing, function computation has to be
very fast.
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The complexity/accuracy
tradeoff
Error
Trivial performance
Complexity
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Flexibility
The speed/flexibility tradeoff
Matlab Code
Java Code
Machine code
Digital Hardware
Analog Hardware
Speed
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Theory Vs. Practice
• Theoretician: I want a polynomial-time algorithm which is
guaranteed to perform arbitrarily well in “all” situations.
- I prove theorems.
• Practitioner: I want a real-time algorithm that performs
well on my problem.
- I experiment.
• My approach: I want combining algorithms whose
performance and speed is guaranteed relative to the
performance and speed of their components.
- I do both.
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Plan of talk
•
•
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•
•
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The black-box approach
Boosting
Alternating decision trees
A commercial application
Boosting the margin
Confidence rated predictions
Online learning
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The black-box approach
• Statistical models are not generators, they are
predictors.
• A predictor is a function from
observation X to action Z.
• After action is taken, outcome Y is observed
which implies loss L (a real valued number).
• Goal: find a predictor with small loss
(in expectation, with high probability, cumulative…)
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Main software components
A predictor
A learner
x
z
Training examples
x1,y1,x2 , y2 , ,xm ,ym 

We assume the predictor will be applied to
examples similar to those on which it was trained
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Learning in a system
Learning System
Training
Examples
predictor
Target System
Sensor Data
Action
feedback
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Special case: Classification
Observation X - arbitrary (measurable) space
Outcome Y - finite set {1,..,K}
yY
Prediction Z - {1,…,K}
yˆ  Z
Usually K=2 (binary classification)
1 if
Loss( yˆ,y)  
0 if

y  yˆ
y  yˆ

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batch learning for
binary classification
Data distribution:
x, y ~ D;
Generalization error:
y  1,1
h P
Ý x,y ~D h(x)  y
Training set: T  x1,y1,x2 ,y2 ,...,xm ,ym ; T ~ Dm
Training error: 
1
ˆ(h) 
Ý 1h(x)  y P
Ý x,y ~T h(x)  y

m x,y T
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Boosting
Combining weak learners
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A weighted training set
x1,y1,w1,x2 ,y2 ,w2 , ,xm ,ym ,wm 
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A weak learner
Weighted
training set
x1,y1,w1,x2 ,y2 ,w2 , ,xm ,ym ,wm 
A weak rule
Weak Leaner
instances
h
predictions
x1,x2 , ,xm
h
yˆ1, yˆ2 , , yˆm ; yˆi  {0,1}
 y yˆ w
 w
m
The weak requirement:


i1 i i
m
i1
i
i
 0
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The boosting process
(x1,y1,1/n), … (xn,yn,1/n)
(x1,y1,w1), … (xn,yn,wn)
weak learner
h1
weak learner
h2
h3
h4
h5
h6
h7
h8
h9
hT
(x1,y1,w1), … (xn,yn,wn)
(x1,y1,w1), … (xn,yn,wn)
(x1,y1,w1), … (xn,yn,wn)
(x1,y1,w1), … (xn,yn,wn)
(x1,y1,w1), … (xn,yn,wn)
(x1,y1,w1), … (xn,yn,wn)
(x1,y1,w1), … (xn,yn,wn)
(x1,y1,w1), … (xn,yn,wn)
FT x  1h1x  2h2 x 
Final rule:
 T hT x
fT (x)  sign FT x
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Adaboost
F0 x  0
for t 1..T
w  expyi Ft1(xi )
t
i
Get ht from weak  learner

t
t 
 t  ln i:h x 1,y 1 wi i:h x 1,y 1 wi 
 t  i  i

t i
i
Ft1  Ft  t ht
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Main property of Adaboost
If advantages of weak rules over random
guessing are: 1,2,..,T then
training error of final rule is at most
 T 2 
ˆ fT   exp   t 

 t1 

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Boosting block diagram
Strong Learner
Accurate
Rule
Weak
Learner
Weak
rule
Example
weights
Booster
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What is a good weak learner?
The set of weak rules (features) should be:
• flexible enough to be (weakly) correlated with most
conceivable relations between feature vector and label.
• Simple enough to allow efficient search for a rule with
non-trivial weighted training error.
• Small enough to avoid over-fitting.
Calculation of prediction from observations should
be very fast.
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Alternating decision trees
Freund, Mason 1997
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Decision Trees
Y
X>3
+1
5
-1
Y>5
-1
-1
-1
+1
3
X
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A decision tree as a sum of
weak rules.
Y
-0.2
X>3
sign
+1
+0.2
+0.1
-0.1
-0.1
-1
Y>5
-0.3
-0.2 +0.1
-1
-0.3
+0.2
X
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An alternating decision tree
Y
-0.2
Y<1
sign
0.0
+0.2
+1
X>3
+0.7
-1
-0.1
+0.1
-0.1
+0.1
-1
-0.3
Y>5
-0.3
+0.2
+0.7
+1
X
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Example: Medical Diagnostics
•
Cleve dataset from UC Irvine database.
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Heart disease diagnostics (+1=healthy,-1=sick)
•
13 features from tests (real valued and discrete).
•
303 instances.
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AD-tree for heart-disease diagnostics
>0 : Healthy
<0 : Sick
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Commercial Deployment.
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AT&T “buisosity” problem
Freund, Mason, Rogers, Pregibon, Cortes 2000
• Distinguish business/residence customers from call detail
information. (time of day, length of call …)
• 230M telephone numbers, label unknown for ~30%
• 260M calls / day
• Required computer resources:
• Huge: counting log entries to produce statistics -- use specialized
I/O efficient sorting algorithms (Hancock).
• Significant: Calculating the classification for ~70M customers.
• Negligible: Learning (2 Hours on 10K training examples on an offline computer).
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AD-tree for “buisosity”
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AD-tree (Detail)
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Precision/recall:
Accuracy
Quantifiable results
Score
• For accuracy 94%
increased coverage from 44% to 56%.
• Saved AT&T 15M$ in the year 2000 in operations costs
and missed opportunities.
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Adaboost’s resistance to
over fitting
Why statisticians find Adaboost
interesting.
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A very curious phenomenon
Boosting decision trees
Using <10,000 training examples we fit >2,000,000 parameters
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
Large margins
 h x

F x
(x,y) y
y
Ý

 
T
margin
FT
t1 t t
T
t1
margin
FT
(x,y)  0 
T
t
1
fT (x)  y
Thesis:
large margins => reliable predictions
Very similar to SVM.
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Experimental Evidence
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Theorem
Schapire, Freund, Bartlett & Lee / Annals of statistics 1998
H: set of binary functions with VC-dimension d
C   i hi | hi  H , i  0, i  1
T  x1,y1,x2 ,y2 ,...,xm ,ym ; T ~ Dm
c  C,   0,
with probability 1  w.r.t. T ~ D m
Px,y~D sign c(x)  y Px,y~T margin c x,y   
 d / m   1 
O˜ 
 Olog 
     
No dependence on no. of combined functions!!!
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Idea of Proof
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Confidence rated
predictions
Agreement gives confidence
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A motivating example
?
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-
-
- -
+ Unsure
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-
++
+
+
- +
+ ++ +
+
?+ +
+
++ ++ +
+ +
+ +
+
+
+ +
+
+
+
- -- + + + + +++ +
++
+
+
?
Unsure
+
+
-
-
+
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The algorithm
Freund, Mansour, Schapire 2001
Parameters   0,   0
w(h) e
Ý
Hypothesis weight:

Empirical
Log Ratio

Prediction rule:

:
ˆ h 
  w(h) 
h:h ( x)1 
1
lˆ (x) 
Ý ln 

   w(h) 
h:h ( x)1 
 1
if

pˆ , x  -1,+1 if

if
 1
lˆx  
lˆx  
lˆx  
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Suggested tuning
Suppose H is a finite set.
0    14
  ln 8 H m1 2
2
Yields:
m
8m1 2
*


 2   ln 8 H 
ln m 
 2h  O 1/2 
m

1) P
mistake   Px,y ~D y  pˆ (x)
2) for
ln

1/  

m =  ln 1  ln H  


 ln 1   ln H 


*


ˆ
P(abstain )  Px,y ~D p(x)  1,1  5h  O
1/2



m


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Confidence Rating block
diagram
Training examples
x1,y1,x2 , y2 , ,xm ,ym 
Confidence-rated
Rule

Candidate
Rules
RaterCombiner
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Face Detection
Viola & Jones 1999
• Paul Viola and Mike Jones developed a face detector that can work
in real time (15 frames per second).
QuickTime™ and a YUV420 codec dec ompres sor are needed to see this pic ture.
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Using confidence to save time
The detector combines 6000 simple features using Adaboost.
In most boxes, only 8-9 features are calculated.
All
boxes
Feature 1
Might be a face
Feature 2
Definitely
not a face
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Using confidence to train
car detectors
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Original Image Vs.
difference image
QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.
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Co-training
Blum and Mitchell 98
Partially trained
B/W based
Classifier
Confident
Predictions
Hwy
Images
Partially trained
Diff based
Classifier
Confident
Predictions
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Co-Training Results
Levin, Freund, Viola 2002
Raw Image detector
Before co-training
Difference Image detector
After co-training
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Selective sampling
Unlabeled data
Partially trained
classifier
Sample of unconfident
examples
Labeled
examples
Query-by-committee, Seung, Opper & Sompolinsky
Freund, Seung, Shamir & Tishby
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Online learning
Adapting to changes
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Online learning
So far, the only statistical assumption was
that data is generated IID.
Can we get rid of that assumption?
Yes, if we consider prediction as a repeating game
An expert is an algorithm that maps the
past (x1,y1 ),(x2 ,y2 ), ,(xt1,yt1 ),x
to ta prediction
zt
Suppose we have a set of experts,
we believe one is good, but we don’t know which one.


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Online prediction game
For t  1,
,T
1
t
2
t
z ,z , ,z
Experts generate predictions:
Algorithm makes its own prediction:
Total loss of algorithm:
t
yt
Nature generates outcome:
Total loss of expert i:
N
t
L   Lzti ,yt 

T
i
T
 L  ,y
t1
L 
A
T
T
t1
Goal: for any sequence
  of events

t
t

LTA  min LiT  oT 
i
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A very simple example
•
•
•
•
Binary classification
N experts
one expert is known to be perfect
Algorithm: predict like the majority of
experts that have made no mistake so far.
• Bound:
LA  log 2 N
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History of online learning
• Littlestone & Warmuth
• Vovk
• Vovk and Shafer’s recent book:
“Probability and Finance, its only a game!”
• Innumerable contributions from many fields:
Hannan, Blackwell, Davison, Gallager, Cover, Barron,
Foster & Vohra, Fudenberg & Levin, Feder & Merhav,
Starkov, Rissannen, Cesa-Bianchi, Lugosi, Blum,
Freund, Schapire, Valiant, Auer …
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Lossless compression
X - arbitrary input space.
Y - {0,1}
Z - [0,1]
Log Loss: LZ,Y   y log 2
1
1
 (1 y)log 2
z
1 z
Entropy, Lossless compression, MDL.
Statistical likelihood, standard probability theory.

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Bayesian averaging
Folk theorem in Information Theory
N
t 
w z
i
t
i
t
L
; w 2
i
t1
i
t
i1
N
i
w
 t
i1
N
N
T  0; LTA  log 2  w1i  log 2  wTi  min LiT  ln N

i1
i1
i
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Game theoretical loss
X - arbitrary space
Y - a loss for each of N actions
y  0,1
N
Z - a distribution over N actions
p  0,1 , p 1 1

Loss: L(p,y)  p y  Ep y
N

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Learning in games
Freund and Schapire 94
An algorithm which knows T in advance
guarantees:
L  min L  2T ln N  ln N
A
T
i
i
T

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Multi-arm bandits
Auer, Cesa-Bianchi, Freund, Schapire 95
Algorithm cannot observe full outcome
yt
Instead, a single it  1, , N  is chosen at
it
p t and yt is observed
random according to

We describe
 an algorithm that guarantees:

NT 
A
i
LT 
 min LT  O NT ln

i
 
 
With probability
1 
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Why isn’t online learning
practical?
• Prescriptions too similar to Bayesian
approach.
• Implementing low-level learning requires a
large number of experts.
• Computation increases linearly with the
number of experts.
• Potentially very powerful for combining a
few high-level experts.
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Online learning for detector
deployment
Face
code
Detector
Library
Merl frontal 1.0
B/W Frontal face detector
Indoor, neutral background
direct front-right lighting
Detector can be
Feedbackadaptive!!
Download
Images
OL
Adaptive
real-time
face detector
Face
Detections
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Summary
• By Combining predictors we can:
 Improve accuracy.
 Estimate prediction confidence.
 Adapt on-line.
• To make machine learning practical:




Speed-up the predictors.
Concentrate human feedback on hard cases.
Fuse data from several sources.
Share predictor libraries.
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