Thạc Sĩ Nghiên cứu khả năng khái quát hóa của lập trình di truyền

Contents
Abstract ii
List of Figures v
List of Tables vii
Abbreviations ix
0.1 Motivations . 2
0.2 Research Perspectives 3
0.3 Contributions of the Thesis 3
0.4 Organization of the Thesis . 4
1 Backgrounds 6
1.1 Genetic Programming . 6
1.1.1 Representation, Initialization and Operators in GP . 7
1.1.2 Some Variants of GP 17
1.2 An Example of Problem Domain . 19
1.3 Machine Learning 19
1.4 GP and Machine Learning Issues . 21
1.4.1 Search Space 21
1.4.2 Bloat . 22
ii
1.4.3 Generalization and Complexity 22
1.5 Generalization in GP 23
1.5.1 Overview of Studies on GP Generalization Ability 24
1.5.2 Methods for Improving GP Generalization Ability 25
1.5.3 Problem of Improving Training Efficiency . 33
1.6 Summary 35
2 Early Stopping, Progressive Sampling, and Layered Learning 36
2.1 Over-training, Early Stopping and Regularization 36
2.1.1 Over-training 37
2.1.2 Early Stopping . 37
2.1.3 Regularization Learning 39
2.2 Progressive Sampling 41
2.2.1 Types of Progressive Sampling Schedule . 44
2.2.2 Detection of Convergence . 47
2.3 Layered Learning 49
2.4 Conclusion 50
3 An Empirical Study of Early Stopping for GP 51
3.1 Method . 53
3.1.1 Proposed Classes of Stopping Criteria . 53
3.2 Experimental Settings . 57
3.3 Results and Discussions 59
3.3.1 Comparison with GP 59
3.3.2 Comparison with Tarpeian . 76
3.4 Conclusion 77
iii
4 A Progressive Sampling Framework for GP - An Empirical Approach 79
4.1 The Proposed Method . 82
4.1.1 The Sampling Schedule . 84
4.1.2 The Initial Sample Set Size 84
4.1.3 The Number of Learning Layers 86
4.1.4 The Stopping Criterion for Each Layer 86
4.2 Experimental Settings . 87
4.2.1 Data Preparation 87
4.2.2 GP Systems Configurations 87
4.3 Results and Discussions 89
4.3.1 Learning Effectiveness Comparison 89
4.3.2 Learning Efficiency . 91
4.3.3 Solution Complexity 92
4.3.4 Distribution of Best Solutions by Layers . 93
4.3.5 Synergies between System Components 94
4.4 Conclusion 98
5 Conclusions and Future Work 99
5.1 Contributions 100
5.2 Future Directions 101
Bibliography 104
iv
List of Figures
1.1 GP’s main loop [57] . 7
1.2 GP expression tree representing max(x*x,x+3y). [57] . 8
1.3 Creation of a full tree having maximum depth 2 (and therefore a total of
seven nodes) using the Full initialisation method (t=time). [57] . 10
1.4 Creation of a five node tree using the Grow initialisation method with a
maximum depth of 2 (t=time). A terminal is chosen at t = 2, causing
the left branch of the root to be terminated at that point even though the
maximum depth has not been reached. [57] 12
1.5 Example of subtree crossover 14
1.6 Example of subtree mutation. [57] . 16
2.1 Idealized training and validation error curves. Vertical axis: errors; horizontal axis:time [88] 38
2.2 A real validation error curve. Vertical: validation set error; horizontal: time
(in training epochs) [88] . 38
2.3 Learning curves and progressive sample 44
2.4 Regions of the Efficiency of Arithmetic and Geometric Schedule Sampling [89]. 46
3.1 Distribution of stop time (last generation of the run) of the OF criterion
and GPM 74
v
3.2 Distribution of stop time (last generation of the run) of the VF criterion
and GPM 74
3.3 Distribution of stop time (last generation of the run) of the GL criterion
and GPM 75
3.4 Distribution of stop time (last generation of the run) of the PQ criterion
and GPM 75
3.5 Distribution of stop time (last generation of the run) of the UP criterion
and GPM 75
4.1 PS framework for GP 84
4.2 GP Evolution of each layer 87
vi
List of Tables
1.1 Test Functions 20
1.2 The real-world data sets 20
3.1 Parameter settings for the GP systems 58
3.2 Data Sets for Problems. For the synthetic problems, the notation [min, max
] defines the range from which the data points are sampled 59
3.3 Generalization error and run time of VF, OF, True Adap stopping criterion
and GP, Tar . 61
3.4 Size of best individual of VF, OF and True Adap stopping criterion and GP,
Tar 62
3.5 p-value of run time and GE of VF, OF, True Adap stopping criterion and
Tar vs GP 63
3.6 p-value of complexity of VF, OF, True Adap stopping criterion and Tar vs
GP 64
3.7 Relative Generalization Errors and Run Time of OF, VF and True Adap
stopping criterion (vs GP) . 65
3.8 Generalization error and run time of GL, PQ and UP stopping criterion 66
3.9 Size of the best individual of GL, PQ and UP stopping criterion 67
3.10 p-value of GL, PQ and UP stopping criterion vs GP . 68
vii
3.11 p-value of size of best individual of comparison GL, PQ and UP stopping
criterion with GP 69
3.12 Relative Generalization Errors and Run Time of GL, PQ and UP stopping
criterion (vs GP) 70
3.13 p-value of GE, run time and size of best solution of True Adap, PQ vs Tarpeian 77
4.1 Data Sets for the Test Functions 88
4.2 Evolutionary Parameter Settings for the Genetic Programming Systems 89
4.3 Mean and SD of Generalization Error, GPLL and GP (14 Problems) 90
4.4 Relative Average Generalization Errors and p-values (GPLL vs GP) (14
Problems) 90
4.5 Mean Run Time, GPLL and GP (14 Problems) . 91
4.6 Relative Run Times and p-values (GPLLs vs GP) (14 Problems) 92
4.7 Mean Solution Size and p-values, GPLL vs GP (14 Problems) . 93
4.8 Number of End-of-run Best Solutions First Found by GPLL in Each Layer 94
4.9 Average Generalization Error on 14 Problems 95
4.10 Average Run Time of All Systems . 96
4.11 Average Size of Solutions Found by all Systems . 97
viii
Abbreviations
Abbreviation Meaning
ARMA AutoregressiveMoving-Average
EA Evolutionary Algorithm
DSS Dynamic Subset Selection
GA Genetic Algorithms
GGGP Grammar Guided Genetic Programming
GP Genetic Programming
GPLL Layered Learning Genetic Programming
ILP Inductive Logic Programming
KDD Knowledge Discovery and Data Mining
LL Layered Learning
ML Machine Learning
MSE Mean Squared Error
PDGP Parallel Distributed GP
PS Progressive Sampling
RST Random Sampling Technique
ix
Introduction
Genetic programming (GP) paradigm was first proposed by Koza [53] can be viewed as the
discovery of computer programs which produce desired outputs for particular inputs. A
computer program could be an expression, formula, plan, control strategy, decision tree or
a model depending on the type of problem. GP is a simple and powerful technique which
has been applied to a wide range of problems in combinatorial optimization, automatic
programming and model induction. The direct encoding of solutions as variable-length
computer programs allows GP to provide solutions that can be evaluated and also examined
to understand their internal workings. In the field of evolutionary and natural computation,
GP plays a special role. In the Koza’s seminal book, the ambition of GP was to evolve in a
population of programs that can learn automatically from the training data. Since its birth,
GP has been developed fast with various fruitful applications. Some of the developments
are in terms of the novel mechanisms regarding to GP evolutionary process, others were on
finding potential areas on which GP could be successfully applied. Despite these advances
in GP research, when it is applied to learning tasks, the issue of generalization of GP, as a
learning machine, has not been taken the attention that it deserves. This thesis focuses on
the generalization aspect of GP and proposes mechanisms to improve GP generalization
ability.
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0.1. MOTIVATIONS
0.1 Motivations
GP is one of the evolutionary algorithms (EAs). The common underlying idea behind all
EA techniques is the same: given a population of individuals the environmental pressure
causes natural selection (survival of the fittest) and this causes a rise in the fitness of the
population. Given a quality function to be maximised we can randomly create a set of
candidate solutions, i.e., elements of the function’s domain, and apply the quality function
as an abstract fitness measure - the higher the better. Based on this fitness, some of the
better candidates are chosen to seed the next generation by applying recombination and/or
mutation to them. Recombination is an operator applied to two or more selected candidates
(the so-called parents) and results one or more new candidates (the children). Mutation
is applied to one candidate and results in one new candidate. Executing recombination
and mutation leads to a set of new candidates (the offspring) that compete - based on
their fitness (and possibly age) - with the old ones for a place in the next generation. This
process can be iterated until a candidate with sufficient quality (a solution) is found or a
previously set computational limit is reached. GP has been proposed as a machine learning
(ML) method [53], and solutions to various learning problems are sought by means of an
evolutionary process of program discovery. Since GP often tackles ML issues, generalization
that is synonymous with robustness has been the desirable property at the end of the GP
evolutionary process. The main goal when using GP or other ML techniques is not only to
create a program that exactly cover the training examples (as in many GP applications so
far), but also to have a good generalization ability: for unseen and future samples of data,
the program should output values that closely resemble the underlying function. Although,
recently, the issue of generalization in GP has received more attention than it deserves,
the use of more traditional machine learning techniques has been rather limited. It is
strange that generalization has been an old time studied topic in machine learning with
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0.2. RESEARCH PERSPECTIVES
many proposed practical techniques for improving learning machines (such as for decision
trees, neural networks, etc). It is hoped that adapting these techniques to GP would help
to improve GP generalization ability.
0.2 Research Perspectives
The focus of this thesis is on understanding the generalization of GP, the ways it can be
improved and the role it plays in learning. Specifically, the generalization of GP is analysed
to uncover key features. Based on that the thesis proposes two new learning mechanisms
for GP, early stopping and progressive sampling. In the process of developing the better
learning frameworks for GP, a clearer description of the scalability of the framework emerges
to facilitate and motivate future enhancements. The current theoretical models of GP
are limited in use and applicability due to their complexity. Therefore, the majority of
theoretical work has been derived from experimentation. The approach taken in this thesis
is also based on the careful designed experiments and the analysis of experimental results.
0.3 Contributions of the Thesis
This thesis makes the following contributions:
1. An early stopping approach for GP that is based on the estimate of the generalization
error as well as propose some new criteria apply to early stopping for GP. The GP
training process will be stopped early at the time which promises to bring the solution
with the smallest generalization error.
2. A progressive sampling framework for GP, which divides GP learning process into
several layers with the training set size, starting from being small, gradually get
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0.4. ORGANIZATION OF THE THESIS
bigger at each layer. The termination of training on each layer will be based on
certain early stopping criteria.
0.4 Organization of the Thesis
The organization of the rest of the thesis is as follows:
Chapter 1 first introduces the basic components of GP - the algorithm, representation,
and operators as well as some benchmark problem domains (some are used for the experiments in this thesis). Two important research issues and a metaphor of GP search
are then discussed. Next, the major issues of GP are discussed especially when the GP
is considered as a machine learning system. It first overviews the approaches proposed
in the literature that help to improve the generalization ability of GP. Then, the solution
complexity (code bloat), in the particular context of GP, and its relations to GP learning
performance (generalization, training time, solution comprehensibility) are discussed.
Chapter 2 provides a backgrounds on a number of concepts and techniques subsequently
used in the thesis for improving GP learning performance. They include progressive sampling, layer learning, and early stopping in machine learning.
Chapter 3 presents one of the main contributions of the thesis. It proposes some criteria used to determine when to stop the training of GP with an aim to avoid over-fitting.
Experiments are conducted to assess the effectiveness of the proposed early stopping criteria. The results emphasise that GP with early stopping could find solutions with at least
similar generalization errors with standard GP but in much shorter training time and lower
complexity of solutions.
Chapter 4 develops a learning framework for GP that is based layer learning and progressive sampling. The results show that the solutions of GP with progressive sampling
significantly reduce in size and the training time is much faster when compared to standard
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0.4. ORGANIZATION OF THE THESIS
GP while the generalization error of the obtained solutions is often smaller.
Chapter 5 concludes the thesis summarizing the main results and proposing suggestions
for future works that extend the research in this thesis.