Online Algorithms – II

Online Algorithms – II

Amrinder Arora

Permalink: http://standardwisdom.com/softwarejournal/presentations/

Sum

mar

y• Online Problem – Problem is revealed, one step at a

time.• Online Algorithm – Needs to respond to the request

sequence that has been shown, not knowing next items in the sequence

• Competitive Ratio – The worst ratio of the online algorithm to the optimal offline algorithm, considering all possible request sequences.

Basic Concepts

• Online algorithms show up in many practical problems.• Even if you are considering an offline problem, consider

what would be the online version of that problem.• Research areas including improving algorithms,

improving analysis of existing algorithms, proving tightness of analysis, considering problem variations, etc.

Importance and Research Topics

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Part II only makes sense, if it is better than part I..

Online Algorithms - II

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Two Options

Randomized version of online job scheduling

Online algorithms in machine learning

Online Graph Coloring


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Job Scheduling – Randomized

• No randomized algorithm for 2-machine scheduling can be better than 4/3-competitive

• Consider any randomized algorithm A• How can we prove a lower bound on

competitive ratio of algorithm A?


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Job Scheduling

• Consider the old job sequence of 1,1, 2• After first two jobs, E[L1] ≤ 4/3.• Therefore, E[L2] ≥ 2/3.• Therefore, after job of size 2 is scheduled,

then E[L2] ≥ 8/3.• OPT offline makespan = 2.• Therefore, competitive ratio of randomized

algorithm A ≥ 4/3.Online Algorithms - II

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Algorithm “Random Scheduler”

• 4/3 competitive randomized algorithm• Tries to keep machine loads in expected

ratio of 2:1.


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Online Algorithms in Machine Learning

• But first, let us understand classification techniques


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Classification• Given a collection of records (training set)

– 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.


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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

Learningalgorithm

Training Set


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Examples of Classification Task• Predict tax returns as “clean” or “auditable”

• 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


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Classification Techniques

• Decision Tree based Methods• Rule-based Methods• Memory based reasoning• Neural Networks• Naïve Bayes and Bayesian Belief

Networks• Support Vector Machines


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Example of a Decision Tree• 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


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Decision Tree Induction

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


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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.


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Measures of Node Impurity

• Gini Index

• Entropy

• Misclassification error

j

tjptGINI 2)]|([1)(

j

tjptjptEntropy )|(log)|()(

)|(max1)( tiPtErrori


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Different kinds of classifiers..• Different decision trees based on Hunt’s• C4.5• Naïve Bayes• Support Vector Machine


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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.


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“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)


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Per

form

ance

of W

M-A

1

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.


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Per

form

ance

of W

M-A

1 Proof (cont.)1/2m ≤ n(3/4)M

(4/3)M ≤ n 2m

M lg (4/3) ≤ lg n + mM ≤ [1 / lg (4/3)] [m + lg n]M ≤ 2.41 [m + lg n]

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, and n is number of experts.


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Are the experts independent?


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“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)


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Per

form

ance

of W

M-A

2

For β = ½, this is:1.39m + 2 ln n

For β = 3/4, this is:1.15m + 4 ln n

The number of mistakes made by Weighted 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.


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Per

form

ance

of W

M-A

2Proof 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 = F i]

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 )


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Per

form

ance

of W

M-A

2Proof (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}

The number of mistakes made by Weighted 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.

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Why does this all matter?

• http://www.fda.gov/predict


http://www.fda.gov/predict

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• Vertices arrive one by one• We need to assign a color immediately,

which cannot be changed.• What we are shown is an induced

subgraph – not merely a subgraph.– In other words, edges cannot arrive by

themselves.


Applications of Graph Coloring

• Tasks: { T1, ..., Tn}• Concurrency Constraint: unsharable resources• Conflict Matrix C:

– C(i, j) = 0: Ti and Tj need no common resources– C(i, j) = 1: otherwise

• Conflict Graph G: a graph with adjacency matrix C

• G is k-colorable iff tasks can be schedules in k time interval

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Graph Coloring – Offline Version

• Problem is NP-complete• Reduction from 3-SAT• Approximation algorithm?


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• What should we aim for?• 1-competitive seems unlike, n-competitive

is trivial and useless.


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Online Graph Coloring (cont.)

• For every positive integer k, there exists a tree Tk on 2k-1 vertices such that every on-line coloring algorithm A requires at least k colors.


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Party Time!


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Online Algorithms – II