Franciszek Seredynski, Damian Kurdej

1
APPLYING LEARNING CLASSIFIER SYSTEMS for
MULTIPROCESSOR SCHEDULING PROBLEM

• Franciszek Seredynski, Damian Kurdej
• Polish Academy of Sciences and
• Polish-Japanese Institute of Information
  Technology

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Motivations

• New scheduling algorithms are proposed near every
  day
• In the light of
• NP-hard compliteness of the scheduling problem,
  and
• No free lunch theorem concerning metaheuristics
• this situation may last forever, at least
  till the moment of appearing quantum computers
• Can we use the knowledge gained from the
  experience with already known scheduling
  algorithms (hypeheuristics approach) ?
• We will use GA-based Learning classifier systems
  (LCS) to extract some knowledge and use it in the
  scheduling process

3
Plan of the presentation

• Multiprocessor Scheduling Problem
• The idea of LCS
• The concept of LCS-based scheduling
• Experimental results
• Conclusions

4
Mutiprocessor scheduling problem
Examples of a precedence task graph (a) and a
system graph (b).
a) b)
Multiprocessor system undirected, unweighted
graph Gs(Vs,Es), called a system graph.
Parallel program weighted, directed, acyclic
graph GPltVP,EPgt, called a precedence task graph
or a program graph.
The purpose of the scheduling is to distribute
the tasks among the processors in such a way that
the precedence constraints are preserved and the
response time T (the total execution time) is
minimized.
T f (allocation, scheduling_policy const)
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Problem formulation

• Given a set of program graph instances
• Given a multiprocessor system
• Given a number of scheduling algorithms
  (heuristics) solving instances of the scheduling
  problem with some efficiency
• Is it possible to train LCS system to match a
  given instance of the scheduling problem with the
  best for it scheduling algorithm (to minimize the
  total exectution time) from a set of scheduling
  algorithms ?

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Idea of GA-based Learning Classifier System (LCS)
Learning Classifier System
System for discovery of new rules
Evaluation system
Decision system
Environment
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Idea of Learning Classifier System (LCS)
Learning Classifier System
System for discovery of new rules
Decision system
Evaluation system
Environment state or message e.g.10100
Environment
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Idea of Learning Classifier System (LCS)
Learning Classifier System
System for discovery of new rules
Decision system
Evaluation system
action e.g. Turn right
Environment state e.g.10100
Environment
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Idea of Learning Classifier System (LCS)
Learning Classifier System
System for discovery of new rules
Decision system
Evaluation system
reward e.g. 120
action e.g. Turn right
Environment state e.g.10100
Environment
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Classifier (rule) in classical LCS

• The structure of a classifier
• Condtition part C
• Action A
• Strength S
• Strength S
• Used when a classifier is selected from a set of
  classifiers to perform an action
• Used when GA creates new rules

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Classifier in XCS
a
e
exp
as
01000 1000 2,504 0,77 499 19924
146,76 109
C
p
f
ts
num

• C condition part
• a action
• p expected reward
• e prediction error
• f fitness
• exp experience of classifier
• ts remembers recent time when GA was applied to
  this classifier
• as expected population size A, in which
  appears classifier
• num numerosity of classifier

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XCS
Environment
5.
1.
a
s
2.
4.
3.
9.
8.
GA
Subsumption
7.
cover
Enforcement
6.
?
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Features of XCS

• Creates population of classifiers
• Processes messages received from environment
• Applies GA to evolve classifiers
• Sends action to environment
• Learns, generalizes and modifies the set of
  classifiers

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Our problem

• Given 200 program graph instances created on the
  base of the 15-tree graph training set
• Each instance is a tree with different random
  task and communication weights
• Two processor system is considered
• Given 5 scheduling heuristics
• We want to train LCS system to select in the best
  way the scheduling heuristic to solve given set
  of instance of the scheduling problem to provide
  the best possible solutions ?

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Set of list algorithms

• ISH (Insert Scheduling Heuristic)
• MCP (Modified Critical Path)
• STF (Shortest Time First)
• LTF (Longest Time First)
• own list algorithm priority of a task depends on
  a size of the subgraph
• We know how works each algorithm (response time)
  on the set of scheduling instances

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XCS-based scheduling system

1. XCS receives information about an instance of
   the scheduling problem

XCS
1.
Program graph System graph
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XCS-based scheduling system

1. XCS receives information about an instance of
   the scheduling problem
2. XCS selects the best available heuristic

XCS
1.
Program graph System graph
2.
scheduling algorithm
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XCS-based scheduling system

1. XCS receives information about an instance of
   the scheduling problem
2. XCS selects the best heuristic from the set of
   available heuristics
3. Program graph and a system graph become input
   data of scheduling algorithm

XCS
1.
Program graph System graph
2.
3.
scheduling algorithm
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XCS-based scheduling system

1. XCS receives information about an instance of
   the scheduling problem
2. XCS selects the best heuristic from the set of
   available heuristics
3. Program graph and a system graph become input
   data of scheduling algorithm
4. Scheduling algorithm delivers a solution

XCS
1.
Program graph System graph
2.
3.
scheduling algorithm
4.
Gantt diagram
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Program graph signature the basic information
concerning program graph

• LCS receives from environment a signature of
  program graph
• The signature codes some static properties of
  program graph
• comm/comp the averaged communication to
  computation time for a program graph (3 bits)
• information about distribution of tasks with a
  given computational requirements (12 bits)
• information about distribution of communication
  time requirements to communicate between tasks
  (12 bits)
• Information about critical path based on
  evaluation of comp/comm (16bits)
• The length of the signature 43 bits

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Distribution of tasks with a given computational
requirements/distribution of communication time
requirements
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Coding information concerning critical path based
on evaluation of comp/comm

• Computing ratios on critical path
• ratios0 1/4, ratios1 5/3, ratios2
  1/3
• Normalization
• ratios0 3/27, coding as 01,
• ratios1 20/27,coding as 11,
• ratios2 4/27, coding as 01.
• Coding signal concerning critical path
  0111010000000000

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Training LCS number of correct matching
scheduling algorithms to instances as function of
number of training cycles
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Training LCS population size of rules as
function of a number of cycles
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Training summary of experiments

• Nontrained system correctly matched heuristics
  with scheduling instances in 40-50 cases
• The system was able to learn to match correctly
  in 100 heuristics to instances
• It means that information about the matching
  process was extracted during the learning process
  
• Classifiers contain this information and during
  the learning process the process of
  generalization of rules was observed
• Learning process is a costly process, but the
  gained information can be used in the scheduling

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LCS-based scheduling system normal operation mode

• Modification of instances (program graphs) from
  training set testing set
• All computation and communication weights were
  scailed by 10
• Next, weights of k tasks or communications were
  changed by constant d

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Experiment k1, d1
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Experiment k2, d2
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Experiment k3, d3
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Normal operation mode summary of experiments
Number of correct matching heuristics to scheduling instances Number k of modified weights (tasks or communications) Difference d between initial weight value and the value after modification
90 1 1
80 2 2
75 3 3
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Conclusions

• LCS has been proposed to learn optimal matching
  scheduling algorithms to instances
• Instances were represented by specially
  signatures
• During the learning process the knowledge about
  matching was extracted in the shape of LCS rules,
  and next generalized
• Creating signatures is one of the most crucial
  issues in the proposed approach
• Performance of the system depends also on many
  parameters of LCS
• We believe that encouraging results of
  experiments open new possibilities in developing
  hyperheuristics