Machine Learning
Algorithms in
Computational Learning Theory
TIAN
HE
JI
GUAN
WANG
Shangxuan
Xiangnan
Kun
Peiyong
Hancheng
25th Jan 2013
Outlines
2
1.
2.
Introduction
Probably Approximately Correct Framework (PAC)



3.
PAC Framework
Weak PAC-Learnability
Error Reduction
Mistake Bound Model of Learning
 Mistake Bound Model
 Predicting from Expert Advice

The Weighted Majority Algorithm
 Online Learning from Examples

The Winnow Algorithm
4.
5.
6.
PAC versus Mistake Bound Model
Conclusion
Q&A
Machine Learning
3
Machine cannot learn but can be trained.
Machine Learning
4

Definition
"A computer program is said to learn from experience E with
respect to some class of tasks T and performance measure P,
if its performance at tasks in T, as measured by P, improves
with experience E".
---- Tom M.Mitchell

Algorithm types
Supervised learning : Regression, Label
 Unsupervised learning : Clustering, Data Mining
 Reinforcement learning : Act better with observations.

Machine Learning
5

Other Examples
 Medical diagnosis
 Handwritten character recognition
 Customer segmentation (marketing)
 Document segmentation (classifying news)
 Spam filtering
 Weather prediction and climate tracking
 Gene prediction
 Face recognition
Computational Learning Theory
6

Why learning works
 Under what conditions is successful learning
possible and impossible?
 Under what conditions is a particular learning
algorithm assured of learning successfully?

We need particular settings (models)
 Probably approximately correct (PAC)
 Mistake bound models
7
Probably Approximately Correct Framework (PAC)




PAC Learnability
Weak PAC-Learnability
Error Reduction
Occam’s Razor
PAC Learning
8

PAC Learning
 Any hypothesis that is consistent with a
sufficiently large set of training examples is
unlikely to be wrong.
 Stationarity : The future being like the past.
 Concept: An efficiently computable function of a
domain. Function : {0,1} n -> {0,1} .
 A concept class is a collection of concepts.
PAC Learnability
9

Learnability

Requirements for ALG
must, with arbitrarily high probability (1-d),
output a hypothesis having arbitrarily low error(e).
 ALG must do as efficiently as in time that grows at
most polynomially with 1/d and 1/e.
 ALG
PAC Learning for Decision Lists
10


A Decision List (DL) is a way of presenting certain
class of functions over n-tuples.
Example
x1
x2
x3
x4
x5
f(x)
0
1
1
0
0
1
0
0
0
1
0
0
1
1
0
0
1
1
1
0
1
0
1
0
0
1
1
1
1
0
if x4 = 1 then f(x) = 0
else if x2 = 1 then f(x) = 1
else f(x) = 0.
Upper bound on the number of all
possible boolean decision lists on n
variables is :
n!4 n = O(n n )
PAC Learning for Decision Lists
11

1.
2.
3.
4.

Algorithms : A greedy approach (Rivest, 1987)
If the example set S is empty, halt.
Examine each term of length k until a term t is
found s.t. all examples in S which make t true are
of the same type v.
Add (t, v) to decision list and remove those
examples from S.
Repeat 1-3.
Clearly, it runs in polynomial time.
What does PAC do?
12

A supervised learning framework to classify data
How can we use PAC?
13

Use PAC as a general framework to guide us on
efficient sampling for machine learning

Use PAC as a theoretical analyzer to distinguish
hard problems from easy problems

Use PAC to evaluate the performance of some
algorithms

Use PAC to solve some real problems
What we are going to cover?
14

Explore what PAC can learn

Apply PAC to real data with noise

Give a probabilistic analysis on the
performance of PAC
PAC Learning for Decision Lists
15

Algorithms : A greedy approach
x1
x2
x3
x4
x5
f(x)
0
1
1
0
0
1
0
0
0
1
0
0
1
1
0
0
1
1
1
0
1
0
1
0
0
1
1
1
1
0
Analysis of Greedy Algorithm
16

The output

Performance Guarantee
PAC Learning for Decision Lists
17
1. For a given S, by partitioning the set of all concepts that agree with f on S into a
“bad” set and a “good”, we want to achieve
2. Consider any h, the probability that we pick S such that h ends up in bad set is
3.
4. Putting together
The Limitation of PAC for DLs
18

What if the examples are like below?
x1
0
0
x2
0
1
f(x)
1
0
1
1
0
1
0
1
Other Concept Classes
19




Decision tree : Dts of restricted size are not PAClearnable, although those of arbitrary size are.
AND-formulas: PAC-learnable.
3-CNF formulas: PAC-learnable.
3-term DNF formulas: In fact, it turns out that it is
an NP-hard problem, given S, to come up with a 3term DNF formula that is consistent with S.
Therefore this concept class is not PAC-learnable—
but only for now, as we shall soon revisit this class
with a modified definition of PAC-learning.
Revised Definition for PAC Learning
20
Weak PAC-Learnability
21
Benefits:
To loose the requirements for a highly accurate
algorithm
To reduce the running time as |S| can be smaller
To find a “good” concept using the simple algorithm
A
Confidence Boosting Algorithm
22

Boosting the Confidence
23

Boosting the Confidence
24

Boosting the Confidence
25

Error Reduction by Boosting
26

The basic idea exploits the fact that you can learn a little on
every distribution and with more iterations we can get much
lower error rate.
Error Reduction by Boosting
27
Detailed Steps:
1. Some algorithm A produces a hypothesis that has
an error probability of no more than p = 1/2−γ (γ>0).
We would like to decrease this error probability to
1/2−γ′ with γ′> γ.
2. We invoke A three times, each time with a slightly
different distribution and get hypothesis h1, h2 and
h3, respectively.
3. Final hypothesis then becomes h=Maj(h1, h2,h3).

Error Reduction by Boosting
28



Learn h1 from D1 with error p
Modify D1 so that the total weight of incorrectly marked
examples are 1/2, thus we get D2. Pick sample S2 from this
distribution and use A to learn h2.
Modify D2 so that h1 and h2 always disagree, thus we get D3.
Pick sample S3 from this distribution and use A to learn h3.
Error Reduction by Boosting
29


The total error probability h is at most 3p^2−2p^3, which is less
than p when p∈(0,1/2). The proof of how to get this probability
is shown in [1].
Thus there exists γ′> γ such that the error probability of our new
hypothesis is at most 1/2−γ′.
[1] http://courses.csail.mit.edu/6.858/lecture-12.ps
Error Reduction by Boosting
30
Adaboost
31

Defines a classifier using an additive model:
Adaboost
32
Adaboost Example
33
Adaboost Example
34
Adaboost Example
35
Adaboost Example
36
Adaboost Example
37
Adaboost Example
38
Error Reduction by Boosting
39
Fig. Error curves for boosting C4.5 on the letter dataset as reported by
Schapire et al.[]. Training and test error curves are lower and upper curves
respectively.
PAC learning conclusion
40



Strong PAC learning
Weak PAC learning
Error reduction and boosting
41
Mistake Bound Model of Learning


Mistake Bound Model
Predicting from Expert Advice


The Weighted Majority Algorithm
Online Learning from Examples

The Winnow Algorithm
Mistake Bound Model of Learning | Basic Settings
42




x – examples
c – the target function, ct ∈ C
x1, x2… xt an input series
at the tth stage
1.
2.
3.

The algorithm receives xt
The algorithm predicts a classification for xt, bt
The algorithm receives the true classification, ct(x).
a mistake occurs if ct(xt) ≠ bt
Mistake Bound Model of Learning | Basic Settings
43

A hypotheses class C has an algorithm A with
mistake M:
if for any concept c ∈ C, and
 for any ordering of examples,
 the total number of mistakes ever made by A is bounded
by M.

Mistake Bound Model of Learning | Basic Settings
44

Predicting from Expert Advice


The Weighted Majority Algorithm
Online Learning from Examples

The Winnow Algorithm


45
Predicting from Expert Advice
The Weighted Majority Algorithm
 Deterministic
 Randomized
Predicting from Expert Advice
Predicting from Expert Advice | Basic Flow
46
Combining Expert Advice
Prediction
Assumption: prediction ∈ {0, 1}.
Truth
Predicting from Expert Advice | Trial
47
(1)
Receiving
prediction from
experts
(2)
Making its own
prediction
(3)
Being told the
correct answer
Predicting from Expert Advice | An Example
48




Task
Input
Output
Goal
: predicting whether it will rain today.
: advices of n experts ∈ {1 (yes), 0 (no)}.
: 1 or 0.
: make the least number of mistakes.
Expert 1
Expert 2
Expert 3
Truth
21 Jan 2013
1
0
1
1
22 Jan 2013
0
1
0
1
23 Jan 2013
1
0
1
1
24 Jan 2013
0
1
1
1
25 Jan 2013
1
0
1
1
The Weighted Majority Algorithm | Deterministic
49
The Weighted Majority Algorithm | Deterministic
50
Date
Expert Advice
Weight
∑wi
Prediction
Correct
Answer
21 Jan 2013
x1
1
x2
0
x3
1
w1
1
w2
1
w3
1
(xi=0)
1
(xi=1)
2
1
1
22 Jan 2013
0
1
0
1
0.50
1
2
0.50
0
1
23 Jan 2013
1
0
1
0.50 0.50
0.50
0.50
1
1
1
24 Jan 2013
0
1
1
0.50 0.25
0.50
0.50
0.75
1
1
25 Jan 2013
1
0
1
0.25 0.25
0.50
0.25
0.75
1
1
The Weighted Majority Algorithm | Deterministic
51

Proof:

Let



A mistaken prediction:


W ≤ n(¾)M
Assuming the best expert made m mistakes.


At least ½ W predicted incorrectly.
In step 3 total weight reduced by a factor of ¼ (= ½ W x ½).


M := # of mistakes made by Weight Majority algorithm.
W := total weight of all experts (initially = n).
W ≥ ½m
So, ½m ≤ n(¾)M  M ≤ 2.41(m + lgn).
The Weighted Majority Algorithm | Randomized
52
View as
Probability
Multiply by
β
The Weighted Majority Algorithm | Randomized
53

Advantages

Dilutes the worst case:
Worst Case: slightly > ½ of the weights predicted incorrectly.
 Simple Version: make a mistake and reduce total weight by ¼.
 Randomized Version: has a 50/50 chance to predicate correctly.


Selecting an expert with probability proportional to its
weight:
Feasibility: (e.g., when weights cannot be easily combined).
 Efficiency: (e.g., when the experts are programs to be run or
evaluated).

The Weighted Majority Algorithm | Randomized
54
The Weighted Majority Algorithm | Randomized
55



β = ½ then M < 1.39m + 2ln n
β = ¾ then M < 1.15m + 4ln n
Adjusting β to make the “competitive ratio” as close to 1 as desired.
56
Mistake Bound Model of Learning


Mistake Bound Model
Predicting from Expert Advice


The Weighted Majority Algorithm
Online Learning from Examples

The Winnow Algorithm
Online Learning from Examples
57



Weighted Majority Algorithm is “learning from expert
advice”
Recall Offline learning V.S. Online Learning

Offline learning: training dataset => learning the model parameters

Online learning: each “training” instance comes like a stream =>
update the model parameters after receiving each instance
Recall the Mistake Bound Model (Online learning):

At each iteration:
Receive a
feature vector
x
Predict x’s
label: b
Receive x’s
true label: c
 Goal: Minimize the number of mistakes made.
Update the
model params
Winnow Algorithm
58

Simple Winnow Algorithm:
input vector x = {x1, x2, … xn}, xi ∈ {0, 1}
 Assume the target function is the disjunction of r
relevant variables. I.e. f(x) = xt1 V xt2 V … V xtr
 Each

Winnow algorithm provides
a linear separator
Winnow Algorithm
59


Initialize: weights w1 = w2 = … = wn=1
Iterate:
Receive an example vector x = {x1, x2, … xn}
 Predict:

Output 1 if
 Output 0 otherwise

Get the true label
 Update if making a mistake:

False positive error: for each xi = 1:
 False negative error: for each xi = 1:

wi = 2*wi
wi = wi/2
Difference with Weighted Majority Algorithm?
Analysis of Simple Winnow
60



Assumption: the target function is the disjunction of
r relevant variables and remains unchanged
Bound: #total mistakes ≤ 2 + 3r(1+log n)
Analysis & Proof

False Positive errors:
The true label is +1 because at least ONE relevant variable(xt) is 1
 Error is caused by x ∙ w < n => wt < n
 Update rule: wt = wt * 2 => #error when xt=1 is at most (1+log n)
 The false positive error bound is r(1+log n)


False Negative error:

The similar analysis procedure can get the bound 2+2r(1+log n)
Extensions of Winnow…
61

Examples are not exactly match the target function:
Error bound is O(r∙mc + r∙lg n)
mc is the errors made by the target function
 It means Winnow has a O(r)-competitive error bound.


The target function may change with time:
Imagine an adversary(enemy) changes the target function by
adding or removing variables.
 The error bound is O(cA ∙ log n)

Extensions of Winnow…
62

The feature variable is continuous rather than boolean


Theorem: if the target function is embedding-closed, then
Winnow can also be learned in the Infinite-Attribute model
By adding some randomness in the algorithm, the
bound can be further improved.
PAC versus MBM (Mistake Bound Model)
63

Intuitively, MBM is stronger than PAC
 MBM
gives the determinant error upper bound
 PAC guarantees the mistake with constant probability

A natural question: if we know A learns some
concept class C in the MBM, can A learn the same
C in the PAC model?
 Answer:
Of course! We can construct APAC in a
principled way [1]
Conclusion
64

PAC (Probably Approximately Correct)
Easier than MBM model since examples are restricted to
be coming from a distribution.
 Strong PAC and weak PAC
 Error reduction and boosting


MBM(Mistake Bound Model)
Stronger bound than PAC: exactly upper bound of errors
 2 representative algorithms

Weighted Majority: for online expert learning
 Winnow: for online linear classifier learning


Relationship between PAC and MBM
Q&A
65
References
66
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
















[1] Tel Aviv University’s Machine Lecture: http://www.cs.tau.ac.il/~mansour/ml-course-10/scribe4.pdf
Machine Learning Theory
http://www.cs.ucla.edu/~jenn/courses/F11.html
http://www.staff.science.uu.nl/~leeuw112/soiaML.pdf
PAC
www.cs.cmu.edu/~avrim/Talks/FOCS03/tutorial.ppt
http://www.autonlab.org/tutorials/pac05.pdf
http://www.cis.temple.edu/~giorgio/cis587/readings/pac.html
Occam’s Razor
http://www.cs.iastate.edu/~honavar/occam.pdf
The Weighted Majority Algorithm
http://www.mit.edu/~9.520/spring08/Classes/online_learning_2008.pdf
http://users.soe.ucsc.edu/~manfred/pubs/C50.pdf
http://users.soe.ucsc.edu/~manfred/pubs/J24.pdf
The Winnow Algorithm
http://www.cc.gatech.edu/~ninamf/ML11/lect0906.pdf
http://stat.wharton.upenn.edu/~skakade/courses/stat928/lectures/lecture19.pdf
http://www.cc.gatech.edu/~ninamf/ML10/lect0121.pdf
67
Supplementary Slides
Chernoff’s Bound
68

Chernoff bounds are another kind of tail bound. Like Markoff and
Chebyshev, they bound the total amount of probability of some random
variable Y that is in the “tail”, i.e. far from the mean.
For detailed derivation for Chernoff bounds, you may refer to
http://www.cs.cmu.edu/afs/cs/academic/class/15859-f04/www/scribes/lec9.pdf
http://www.cs.berkeley.edu/~jfc/cs174/lecs/lec10/lec10.pdf
Proof of Theorem 2 (1/2)
69
Proof of Theorem 2 (2/2)
70
Corollary 3:
71