Online algorithms in Machine Learning

A M R I N D E R A R O R A
ONLINE ALGORITHMS IN
MACHINE LEARNING

BRIEF INTRODUCTION/CONTACT INFO
CTO at BizMerlin
aarora@bizmerlin.com
www.bizmerlin.com
Adjunct Faculty at GWU/CS
Algorithms
amrinder@gwu.edu
www.gwu.edu
+1 571 276 8807
Arora - Online Algorithms Machine Learning 2
Second Edition
ISBN: 978-1-63487-073-3

ONLINE ALGORITHMS IN
MACHINE LEARNING
• First, let us understand a basic machine learning
problem.
• For example, let us consider: classification

CLASSIFICATION
• Given: A collection of records (training set), where
each record contains a set of attributes, and a
class.
• Find: A model for class attribute as a function of the
values of other attributes.
• Goal: previously unseen records should be assigned
a class as accurately as possible.
• A test set is used to determine the accuracy of the
model. Usually, the given data set is divided into
training and test sets, with training set used to build
the model and test set used to validate it.

ILLUSTRATING CLASSIFICATION TASK
Apply
Model
Induction
Deduction
Learn
Model
Model
Tid Attrib1 Attrib2 Attrib3 Class
1 Yes Large 125K No
2 No Medium 100K No
3 No Small 70K No
4 Yes Medium 120K No
5 No Large 95K Yes
6 No Medium 60K No
7 Yes Large 220K No
8 No Small 85K Yes
9 No Medium 75K No
10 No Small 90K Yes
10
Tid Attrib1 Attrib2 Attrib3 Class
11 No Small 55K ?
12 Yes Medium 80K ?
13 Yes Large 110K ?
14 No Small 95K ?
15 No Large 67K ?
10
Test Set
Learning
algorithm
Training Set

EXAMPLES OF CLASSIFICATION TASK
• Predict tax returns as “clean” or “need an
audit”
• Predicting tumor cells as benign or
malignant
• Classifying credit card transactions
as legitimate or fraudulent
• Classifying secondary structures of protein
as alpha-helix, beta-sheet, or random
coil
• Categorizing news stories as finance,
weather, entertainment, sports, etc

CLASSIFICATION TECHNIQUES
• Decision Tree based Methods
• Rule-based Methods
• Memory based reasoning
• Neural Networks
• Naïve Bayes and Bayesian Belief Networks
• Support Vector Machines

EXAMPLE OF A DECISION TREE
• Decision Trees are an intuitive example of
classification techniques
• income < $40K
• job > 5 yrs then good risk
• job < 5 yrs then bad risk
• income > $40K
• high debt then bad risk
• low debt then good risk

SO, WE HAVE DIFFERENT KINDS OF
CLASSIFIERS..
• Different decision trees based on Hunt’s
• C4.5
• Naïve Bayes
• Support Vector Machine
• Each of these models can be considered as an “expert”.
• We do not know how good each “expert” will perform in
an actual setting
• This is where online algorithms in machine learning an
help us.

ONLINE ALGORITHMS IN MACHINE
LEARNING
• Given m experts, each given an output (0,1)
• We want to be able predict the output
• After each try, we are told the result.
• Goal: After some time, we want to be able to do
“not much worse” than the best expert (without
knowing beforehand who was a good expert)

“WEIGHTED MAJORITY” –
ALGORITHM 1
• Initialize the weights of all experts w1..wn to 1
• At each step, take the majority decision.
• That is, output 1 if weighted average of experts saying 1 is at
least 0.5
• After each step, halve the weight of each expert
who was wrong (leave the weight of correct
experts unchanged)

PERFORMANCEOFWM-A1
Proof
• Suppose WM-A1 makes M mistakes
• After each mistake, total weight goes down by ¼. So, it is no more
than n(3/4)M
• [All initial weights are 1, so initial total weight = n]
• After each mistake, best expert’s weight goes down by ½. So, it is
no more than 1/2m
• So, 1/2m ≤ n(3/4)M
• [Best expert’s weight is no more than the total weight.]
The number of mistakes made by Weighted
Majority- Algorithm 1 is never more than 2.41 (m
+ lg n), where m is the number of mistakes made
by best expert.

PERFORMANCE OF WM-A1
Proof (cont.)
1/2m ≤ n(3/4)M
 (4/3)M ≤ n 2m
 M lg (4/3) ≤ lg n + m
 M ≤ [1 / lg (4/3)] [m + lg n]
 M ≤ 2.41 [m + lg n]
Majority- Algorithm 1 is never more than 2.41 (m
+ lg n), where m is the number of mistakes made
by best expert, and n is number of experts.

“WEIGHTED MAJORITY” –
ALGORITHM 2
• Initialize the weights of all experts w1..wn to 1
• At each step, take the probability decision. That is,
output 1 with probability that is equal to sum of
weights of experts that say 1 (divided by total
weight).
• After each step, multiply the weight of each expert
who was wrong by β (leave the weight of correct
experts unchanged)

PERFORMANCEOFWM-A2
For β = ½, this is:
1.39m + 2 ln n
For β = 3/4, this is:
1.15m + 4 ln n
Majority- Algorithm 2 is never more than (m ln
(1/ β) + ln n)/(1- β), where m is the number of
mistakes made by best expert.

Proof
 Suppose we have seen t tries so far.
 Let Fi be the fraction of total weight on the wrong answers at the i-th
trial.
 Suppose WM-A2 makes M mistakes.
 Therefore M = {i=1 to t} { Fi }
 [Why? Because, in each try, probability of mistake = Fi]
 Suppose best expert makes m mistakes.
 After each mistake, best expert’s weight gets multiplied by β. So, it is
no more than βm
 During each round, the total weight changes as:
 W  W (1 – (1-β) Fi )

Proof (cont.)
 Therefore, at the end of t tries, total weight:
W = n  {i= 1 to t} {1 – (1 – β) Fi}
 Since total weight ≥ weight of best expert:
n  {i= 1 to t} {1 – (1 – β) Fi} ≥ βm
 Taking natural logs:
ln n + {i=1 to t} ln {1 – (1 – β) Fi} ≥ m ln β
 Reversing the inequality (multiply by -1):
– ln n – {i=1 to t} ln {1 – (1 – β) Fi} ≤ m ln (1/β)
 A bit of math: – ln (1 – x) > x
– ln n + (1 – β) {i=1 to t} {Fi} ≤ m ln (1/β)
 – ln n + (1 – β) M ≤ m ln (1/β)
 M ≤ {m ln (1/β) + ln n} / {1 – β}

SUMMARY
Majority- Algorithm 2 is never more than (m ln (1/
β) + ln n)/(1- β), where m is the number of
mistakes made by best expert.

WHY DOES THIS ALL MATTER?
• Many practical applications use techniques
such as ensemble models.
• Ensemble models are a generalization of the
simple majority algorithms we discussed in
this presentation
• There are many relevant practical
applications
• Pandora, Netflix and other Recommendation Engines
• Government and Commercial targeting systems
http://www.fda.gov/predict

Q&A
• Ask anything you want..

PIZZA TIME!
You better cut the pizza in four pieces
because I'm not hungry enough to eat
six.
--Yogi Berra

APPENDIX 1
MORE ON DECISION TREES

DECISION TREE INDUCTION
• Many Algorithms:
• Hunt’s Algorithm (one of the earliest)
• CART
• ID3, C4.5
• SLIQ,SPRINT

GENERAL STRUCTURE OF HUNT’S
ALGORITHM
• Let Dt be the set of training records that reach a
node t
• General Procedure:
• If Dt contains records that belong the same class
yt, then t is a leaf node labeled as yt
• If Dt is an empty set, then t is a leaf node labeled
by the default class, yd
• If Dt contains records that belong to more than
one class, use an attribute test to split the data
into smaller subsets. Recursively apply the
procedure to each subset.

MEASURES OF NODE IMPURITY
• Gini Index
• Entropy
• Misclassification error

j
tjptGINI 2
)]|([1)(
 j
tjptjptEntropy )|(log)|()(
)|(max1)( tiPtError i


Online algorithms in Machine Learning

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