PhD Viva Voce: SPOT heuristic method

PhD Viva Voce

SPOT: A suboptimum- and proportion-based
heuristic generation method for
combinatorial optimization problems
25th February 2013, Hong Kong

PhD Candidate: Fan XUE
Supervisor: Dr. C.Y. CHAN
Co-supervisors: Dr. W.H. IP and Dr. C.F. CHEUNG
Department of Industrial & Systems Engineering
Hong Kong Polytechnic University
Department of
Industrial and Systems Engineering
工業及系統工程學系

Outline

1 Introduction

2 The SPOT algorithm

3 Applications in two domains

4 Discussion and conclusions

Fan XUE: SPOT: A heuristic generation method (PhD viva) 2

Section 1
INTRODUCTION


Introduction: Optimization
Optimization is …
Minimizing inconvenience (e.g., cost)
Maximizing benefits (e.g., revenue)
A continuous example: Dido’s problem Ruins of Carthage (from Wikipedia)

The legend of foundation of Carthage (~800 y
B.C.) was recorded by A Roman poet (~29-
19B.C.)
The Queen Dido cut a bull hide into very
a b x
narrow strips and circumscribed a maximized Dido’s problem (variables: b, f(x))

size of land in a semicircle, with the
max
Mediterranean coast as a given edge.
The Dido’s problem (isoperimetric problem) s.t.

To enclose the maximum area with a given
perimeter.

Introduction: Optimization
A discrete example: Tian’s horse racing
(田忌賽馬)
(In 340s B.C.,) General Tian of Kingdom Qi
raced horses with other members of royal family
several times. His guest Sun told him a better Ruins of Ancestral Temple of Qi
(from qidu.net)
plan… (Sima, 91bc)
General Tian’s King’s
With Sun’s displacement strategy, Tian won 2/3 Faster
races.
The two successful solvers of optimization
problems became representatives of human
rationality and intelligence. Slower

Optimization: “soft” research, yet important Tian’s usual races
Sun’s strategy


Introduction: Combinatorial
optimization
To look for an object from a finite (or possibly countably
infinite) set (Papadimitriou & Steiglitz, 1982)
Typical object: an number, a permutation or a graph structure …
(C. Blum & Roli, 2003).
Term search space: the set of all objects
Term optimum: the best object
Examples
Traveling salesman (TSP) Flow shop scheduling (FSP)

An instance (from Wikipedia)


Introduction: Algorithms for difficult
combinatorial optimization
Many problems, including TSP and FSP, are very difficult
(NP-hard, Non-deterministic Polynomial time-hard) when in large-scale
Even NP-hard to approximate in polynomial time
TSP: when ratio < 220/219 (Papadimitriou and Vempala, 2006)
FSP: when ratio < 5/4 (Williamson et al., 1997)

Exact algorithms which guarantee to find the optimum
Including Brach-and-bound, integer (linear) programming-based,
relaxation-based, exponential neighborhood search, …
Typically face explosive running time (when prob. size is large)
Heuristics which concentrate on a (small) subset of the
solution space do not guarantee finding the optimum
Including greedy construction, hyper-heuristics, random, …
Typically return suboptimal solutions Thesis focuses on

Introduction: Dominance among
algorithms
Exact algorithms and
heuristics are indispensable
squeeze theorem
One algorithm is called
“dominated”
Iff. inferior both in time and
in objective value
Non-dominated = good
A few exact algorithms
A lot of heuristics

Examples of non-dominated and dominated heuristics for solving the 10,000-
Fan XUE: SPOT: A heuristic generation method (PhD viva) city Random Uniform Euclidean dataset (Johnson & McGeoch, 2002) 8

Introduction: Hyper-heuristics
A hyper-heuristic is a search method or learning
mechanism for selecting or generating heuristics to solve
computational search problems (Burke et al., 2010)
Formally, a hyper-heuristic is a 3-tuple (ag, A, l) where
S: problem search space
A: attribute space
H: low-level heuristic
(LLH) space
R+∪{0}: measurement
ag: attribute generation
l: learning
π: getting performance

Introduction: Hyper-heuristics (cont.)
Two subclasses in hyper-heuristics
Heuristic selection
Dated back to Crowston et al. (1963)’s early trial
Portfolios algorithm: SATzilla (Xu et al., 2008), Pearl Hunter (Chan
et al., 2012)
Heuristic generation
E.g., Genetic programming (GP) (Koza, 1992)
Many hyper-heuristics were easily dominated
But some (e.g., SATzilla, Pearl Hunter) are strong
Two main challenges for hyper-heuristics
Computation time (preparation+ learning + applying)
Generalization: “the biggest challenge” (Burke et al., 2010)


Introduction: No-Free-Lunch
Theorems
If an algorithm aims at solving general problems (latter
challenge), there is a pitfall called…
No-Free-Lunch Theorems (NFLs, Wolpert & Macready, 1997)
Over all possible problems, no search algorithm is better than any
other algorithm, on average.
Over all possible search algorithms, no search problem is harder
than any other problem, on average.
NFLs concern Black Box Optimization
No instance/domain-specific information is considered.
Question: How to keep a safe distance from
the pitfall of the NFLs?


Introduction: Motivations
Answer: For an optimization algorithm, there
are (at least) two options
To become a well-designed problem domain-specific
(ad-hoc) algorithm (Burke, et al., 2003)
To discover and to make use of domain-specific,
problem data set-specific and/or instance-specific
characteristics dynamically, from case to case Thesis focuses on

VLSI drilling data set Random data set
All in TSP domain


Introduction: Forerunners -- Fractals
using instance-specific information
An example is measuring the length of the coastline of
Great Britain by “self-similarity” (Mandelbrot, 1967)
L(G) = MG1-D
where M is a positive constant prefactor,
G is a variable of the measurement scale and
D is an instance-specific constant of the fractal dimension
D Geographic boundary

1.00 A straight line
1.25 The west coast of Great Britain
1.13 The Australian coast
1.02 The coast of South Africa, one of the smoothest in the atlas

Maps of Australia (left)
and South Africa (right)
Coastline of Britain (from Wikipedia) (from ducksters.com)

Introduction: Forerunners -- Sampling
using instance-specific information
Sampling/estimation is another example
to select some part of a population
to estimate something about the whole population
A sample is representative (correctly reflecting instance-
specific information) only when
accurate (with a mean of errors no more than a given threshold)
reproducible (with a variance of errors no more than another
given threshold)
Well-known applications
Census (with a sampling frame)
Production testing
(See also in this thesis)
Census (from ehow.com)

Introduction: Using instance-specific
information to adjust algorithms
Employing instance-specific information as “characteristic”
to adjust algorithms is common in literature, e.g., tour-merging
Typical ways of adjustments
algorithm selection
parameter optimization
(aka parameter tuning)
algorithm generation
In this thesis, define:
Heuristic selection: “always
returns n ≤ 0 possible heuristics”
Heuristic generation: “returns
n> 0 possible heuristics for at
least one (usually large) problem” (e.g., with 0 objects)
Thesis focuses on

Introduction: Supervised learning
Supervised learning (or classification)
The search for algorithms that reason from externally supplied
records (or instances) to produce general hypotheses, which then
make predictions about future instances (Kotsiantis, 2007).
Well known methods include
Naive Bayes, Logistic Regression, Support vector machines, C4.5,
Neural networks, Ripper, …
In thesis, three representative methods are employed (to
reduce errors of model)
Color Legs Result
black 8 Spider
gray 8 Spider
gray 4 Lizard
white/black 4 Other
… … …

Supervised learning comic
(Image from uq.edu.au)

Section 2
THE SPOT ALGORITHM


The SPOT algorithm: Objectives
SPOT
Suboptimum- and Proportion-based On-the-fly Training (SPOT)
Main objectives
On-the-fly learning based on sampled proportion & suboptimum
Using learned instance-specific information to generate (modify)
heuristics
Supportive objectives
To standardize the input model
A systematic way of compiling decision attributes
An indicator to guide development and to predict the
effectiveness approximately
Several effective ways of using instance-specific A proportion of an image of
a leopard with spots
information (from worldwildlife.org)

The SPOT algorithm: The U/EA model
The U/EA model
A combinatorial optimization problem is called Unconstrained
and with Equinumerous Assignments (U/EA), iff.
there are no hard constraints among different variables in the given
problem
all variables have equinumerous assignments in the given problem
Convenient for learning to predict the best assignment for each variable
A small proportion of variables (solved once) are sufficient

U/EA transformation
An injection (sometimes bijective) f is a U/EA transformation, iff.
The domain of f is a problem domain U/EA transformation

of combinatorial optimization A U/EA cable
(image from
cooldrives.com)
The image of f fulfills U/EA model
Problem in
U/EA model given domain

The SPOT algorithm: Formalization
 The SPOT algorithm is a 7-tuple
(sam, S’, T, S, ag, A, l), where
sam: sampling
S’: A proportion /subproblem’s
search space
T: Transformation to U/EA
∑: U/EA problem search space
rest symbols are similar to the
model of hyper-heuristics.
 The SPOT algorithm is a hyper-
heuristic
Proof: Define a mapping ag’(s) =
ag (T (sam(s))), it becomes a
hyper-heuristic (ag’, A, l).


The SPOT algorithm: Development
Three main phases to develop a SPOT application

An indicator resemblance r of two solutions is
r = ||same assignments|| / ||variables||
Compatibility: An evaluation of U/EA transformation
Given a set {r1, r2, …} of average resemblances of suboptima of
different instances (one average resemblance for each instance, at
least two suboptima), a U/EA problem is compatible with SPOT,
if the following can generally be held:
(Generality)ri ≈ rj ≈ … ≈ r (σ → 0), where ri, rj ∈{r1, r2, …}
(Approximability) r ≈1

The SPOT algorithm: Attribute
population
2p+3 attributes for i-th assignment from one raw attribute
p: ||variables in
the subproblem||
An example
One raw att.
raw a1 a2 a3 a4 …
2 2 1 T F …
3 3 2 F T …
3 3 2 F T …


The SPOT algorithm: Supervised
learning from library
Learning methods from Weka 3.6
Three supervised learning methods
J48, NaiveBayes and JRip implementing C4.5, Naïve Bayes and
Ripper methods, respectively
An attribute selection method
BestFirst search with the CfsSubsetEval evaluator
Parameters were their default values

Parameters of the methods were default values


The SPOT algorithm: Implementation
in Java
The main program SPOT calls
two components
SPOT_ML
3 + 1 learning methods
PD_UEA2
Extends problem domains from
HyFlex
HyFlex is a hyper-heuristic
development platform
 In CHeSC 2011 (1st cross-domain
heuristic search challenge)
 6 domains, tens of existing LLHs
 With some best-so-far hyper-
heuristic results

Section 3
APPLICATIONS IN TWO
DOMAINS


Application I: The FSP domain
FSP: An overview
To find a permutation of n jobs on m machines that
minimizes the makespan
Early research dated back to Johnson (1954)
Applications: shop floor, Internet multimedia transferring, …
Top heuristics: NEH (Nawaz et al., 1983), ...
FSP benchmark instances & heuristics in
HyFlex


Application I: Transformations
P1: Transformations
For each two jobs Ji, Jj in a permutation, define a function
“following” as

A permutation (J3, J2, J1) can be written as
Reverse transformation: summation of i-th row as Ji’s reverse order
Compatibility
r=79.58% high approx.
σ=5.80% high generality
Learning r in 4 test instances in FSP

Predicting the instance-specific 0-1 matrix

Application I: Attribute definitions
5m+3 attributes for each job pair (Ji ,Jj)


Application I: Parameters
P2: Parameter determination
n' : size of subproblem := 30
Reverse:= weighted permutation (50% NEH)
Sampling := Random selection  2

Average test results of resemblance (n’= 30, 100 runs, significant values of p in bold)

Fan XUE: SPOT: A heuristic generation method (PhD viva) Average test results (100 runs) 29

Application I: Modifying heuristics
Applying learning results back to LLHs

Become constructive heuristics
by set param 1 to 1.0

Random
Instance-specific
information guided


Application I: Individual results of new
LLHs
LLHs 0# and 1# were not significantly changed
LLHs 5#, 6#, 9# & 10# were significantly changed
3 new non-dominated LLHs 5’, 6’ & 9’


Application I: Comparing to world
leading hyper-heuristics
PHunter: Pearl Hunter (10 mins)
SPOT-PH: SPOT + PH (new
LLHs) (10 mins overall)
Observations
PH is generally improved
Score doubled and sur-
passed the best-known entries
Score against best results in CHeSC 2011
SPOT executed on-the-fly (5x31 runs, higher = better)

(Median)

Time spent by SPOT (31 runs) A comparison on median makespan (31 runs)

Application II: The TSP domain (fast-
forward)
P1: transform = Predict best two or more edge
candidates in edge sets
Compatibility
r=70.4% moderate approximability
σ=21.7% moderate to low generality

P2: Parameters determinations
n' := 400
Aspect ratio of subproblems := 1:1 Predicting promising candidates from a list of edge set
Sampling := rectangular subarea + random selection
P3: Applying back & runs
Let local search LLHs try the candidates suggested by SPOT first
Note: A light /general version of Xue et al. (2011)

Application II: Results in the TSP
domain
Observations
Improved in some instances
Worsen in some others
Overall improvement is
slight and insignificant
statistically
Comparing to Xue et al.
(2011) Score against best results in CHeSC 2011 (5x31 runs, higher = better)

LLHs: 2/3-Opt vs. 5-Opt
Candidate sets: 8-NN (99%
coverage of the optimum’s
edges) vs. 5-NN (97%)

A comparison on median tour length (31 runs)

Section 4
DISCUSSION AND CONCLUSIONS


Discussion: The SPOT
SPOT: a heuristic selection or heuristic generation?
A heuristic generation usually
A heuristic selection, in extreme cases, e.g., there is only one
Boolean attribute designed (insufficient resolution for learning)
and/or the domain is finite (e.g., 33 tic-tac-toe)
Assumptions for learning from parts?
ri ≈ rj ≈ … ≈ roverall ≈ 1, for instance i, j, …
Tic-tac-toe
Flexibly indicated in development (from cornell.edu)

SPOT can generate completely new LLHs like Genetic
Programming, instead of modifying existing LLHs
See the permutations in the “direct” mode in FSP. (Though it was
easily dominated)


Discussion: The SPOT (cont.)
Advantages of SPOT
SPOT is supposed to be on-the-fly
Solving asubproblem(s): on-the-fly
Learning on the results of the subproblem(s): on-the-fly
Applying back to LLHs: on-the-fly

SPOT is supposed to be cross-domain (with U/EA models)
But it depends on the U/EA transformation and attributes in each
application
Only a few parameters to determine in SPOT
Size of proportion (key parameter) , sampling & applying methods
Two of them can be quickly determined by resemblances instead of trial-and-
error tests


Discussion: The development
Difficulties/drawbacks
Designing a compatible U/EA transformation “cable” (Key issue)
Memory usage of stochastic learning (may exceed 32-bit limit)
Applying instance-specific information back to heuristics
SPOT’s instance-specific information can also be applied
into
Constructive heuristics
Some exact algorithms
E.g., branching (Xue et al., 2011)


Discussion: Errors in experiments
Possible sources of errors in tests
Number of test instances
Source of test instances
Unstable solutions
Instrument error in measuring time in Java
32/64-bit environment of Java
CPU and memory performance adjustment
…
So statistical techniques are employed generally to
identify
errors, deviations
Statistical significance

Discussion: The formal definitions
Definitions of heuristic selection and heuristic generation
Consistent with the existing taxonomy (Burke et al., 2010)
Extensions of both concepts are extended
Parameter tuning for heuristics now belongs to heuristic selection
Known issues:
Countability trap: Ifa problem domain contains a strictly finite
variables, each variable has a strictly finite domain, (e.g., all
atoms/quanta in our universe), then any heuristic generation method
exists there?
Not well-defined concept of learning: Define:
L0: If ||H||=1, skip learning
Meta-heuristics/Heuristics versus hyper-heuristics under the new
definition?


Conclusion
Thesis presents
The SPOT heuristic generation approach for combinatorial
optimization
And supportive models and indicators
Formal definitions of the hyper-heuristic and its subclasses
The results of tests were encouraging
Possible future works
Non-stochastic machine learning techniques
To re-examine the-state-of-the-art LLHs
A domain transformation map
Further investigation of formalization


Contributions of thesis
Contributions
A successful trial of learning instance-specific information from
proportions and suboptima
A successful trial of employing machine learning in optimizing
difficult combinatorial optimization problems
A trial of formalizing the hyper-heuristics
And the first clear separation between the heuristic selection and the
heuristic generation, as far as is concerned
Relations to previous publications
Extends Xue et al. (2011)’s algorithm to cross-domain
Embeds Chan et al. (2012)’s Pearl Hunter in learning preparation


References
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Appendix I: Coursework

Content Credits Score
ISE6830 On rostering 3 A
ISE6831 On demand modeling 3 A
ISE552 On logistics 3 A
ELC6001,6002 English - PASS
Credit transfer 6 -
Overall 15 A (4.0)


Appendix II: Demos
Mini demos
FSP benchmark instance TA051
n=50, m=20
n' :=15
Sampling := Random  1
30 seconds per run
Upper bound (BK) = 3850
Lower bound = 3771


Thank you!

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PhD Viva Voce: SPOT heuristic method

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