Figure 1.doc

Heuristic Search Based Stacking of Classifiers
By Ricardo Aler, Daniel Borrajo, and Agapito Ledezma.
Universidad Carlos III
Avda. Universidad, 30
28911 Leganés (Madrid)
Spain

e-mails: aler@inf.uc3m.es, dborrajo@ia.uc3m.es, ledezma@elrond.uc3m.es
fax: +34-91-624 9430

INTRODUCTION

One of the most active and promising fields in inductive Machine Learning is the
ensemble of classifiers approach. An ensemble of classifiers is a set of classifiers whose
individual decisions are combined in some way to classify new examples (Dietterich,
1997). The purpose of combining classifiers consists on improving the accuracy of a
single classifier. Experimental results show that this is usually achieved.

There are several ways to construct such ensembles, but currently the most used ones
are bagging (Breiman, 1996), boosting (Freund & Schapire, 1995) and less widely used,
stacking (Wolpert, 1992). Bagging constructs a set of classifiers by subsampling the
training examples to generate different hypotheses. After the different hypotheses are
generated, they are combined by a voting mechanism. Boosting also uses the voting
system to combine the classifiers. But, instead of subsampling the training examples, it
generates the hypotheses sequentially. In each repetition, a new classifier is generated
which focus in those instances that were handled incorrectly by the previous classifier.
This is achieved by giving a weight to each instance in the training examples and
adjusting these weights according to its importance after every iteration. Both, bagging
and boosting use classifiers generated by the same base-learning algorithm and obtained
from the same data. Finally, stacking can combine classifiers obtained from different
learning algorithms using a high level classifier –the metaclassifier- to combine the
lower level models. This is based on the fact that different classifiers are obtained from
the same data and different learning algorithms use different biases to search the
hypothesis space. This approach expects that the metaclassifier will be able to learn how
to decide between the predictions provided by the base classifiers, in order to get
accuracies better than any of them, much in the same way as a committee of experts.

One problem of stacked generalization consists in identifying which learning algorithm
should be used to obtain the metaclassifier, and which ones should be the base
classifiers. The approach we present in this chapter consists to pose this problem as an
optimization task, and then use optimization techniques based on heuristic search to
solve it. In particular, we apply genetic algorithms (Holland, 1975) to automatically
obtain the ideal combination of learning methods for the stacking system.

BACKGROUND

The purpose of this section is to give enough background to understand the rest of the
paper. Here, we will explain concepts related to ensembles of classifiers, bagging,
boosting, stacking, and genetic algorithms.

Ensemble of Classifiers

The combination of multiple classifiers to improve the accuracy of a single classifier
has had good results over several datasets that appear in recent papers about ensembles
of classifiers (Bauer & Kohavi, 1999; Breiman, 1996; Freund & Schapire, 1996;
Quinlan, 1996). According to Dietterich (1997), an ensemble of classifiers is a set of
classifiers whose individual decisions are combined in some way to classify new
examples. There are many ways to construct an ensemble of classifiers. Bauer and
Kohavi (1999) have made a comparison of algorithms based in voting systems.
Dietterich (2000) carried out a survey of the main methods to construct an ensemble of
classifiers. One way to construct an ensemble of classifiers is based on subsampling the
training set to generate a different set of hypotheses and then combine them. This is
called bagging (Breiman, 1996). The second way is to create classifiers sequentially,
giving more importance to examples that missclassified the previous classifier. The
latter is called boosting (Schapire, 1990). Both bagging and boosting use a single
learning algorithm to generate all the classifiers, whereas stacking combines classifiers
from different learning algorithm.

There are other methods to combine a set of classifiers (Dietterich, 2000), but for
experimental purposes in this study we consider here the most well known methods,
bagging, boosting and stacking.

Bagging

Breiman (1996) introduced the Bootstrap aggregating Algorithm (Bagging) that
generates different classifiers from different bootstrap samples and combines decisions
from the different classifiers into a single prediction by voting (the class that gets more
votes from the classifiers wins). Figure 1 shows the bagging algorithm.

Each bootstrap sample is generated by sampling uniformly (with replacement) the m-
sized training set, until a new set with m instances is obtained. Obviously, some of the
instances of the training set will be cloned in the bootstrap sample, and some will be
missing. Then, a classifier Ci is built from each of the bootstrap samples B1, B2, … BT.
Each classifier corresponding to each bootstrap sample will focus on some instances of
the training set, and therefore the resulting classifiers will usually differ from each
other. The final classifier C* is the ensemble of the Ci classifiers combined by means of
the uniform voting system (all classifiers have equal weight).

The Bagging Algorithm
Input:
Training set S
Base Learning Algorithm B
Number of bootstrap samples T

Procedure:
For i = 1 to T {
S' = bootstrap sample from S (S' is a sample with replacement from S)
Ci = B (S') (create a new classifier from S')
}
C*(x) = arg max
y∈Y
∑(the most often predicted label y)
1
i:Ci( x) =y
Output
Classifier C*

Figure 1. The Bagging Algorithm

Since an instance has probability 1-(1-1/m) m of being in the bootstrap sample, each
bootstrap replicate has, in average, 63.2% of unique training instances (1 - 1/e = 63.2%).
For this reason the generated classifiers will be different if the base learning algorithm
is unstable (e.g. decision tree or neural networks). If the generated classifiers are good
and do not correlate, the performance will be improved. This will occur if they make
few mistakes and different classifiers do not missclassify the same instances frequently.

Boosting

Another method to construct an ensemble of classifiers is know as boosting (Schapire,
1990), which is used to boost the performance of a weak learner. A weak learner is a
simple classifier whose error is less than 50% on training instances. There are many
variants of the boosting idea, but in this study we will describe the widely used
AdaBoost Algorithm (Adaptive Boosting) that was introduced by Freund & Schapire
(1995). Sometimes the AdaBoost algorithm is also known as AdaBoostM1.

Similarly to bagging, boosting manipulates the training set to generate a set of
classifiers of the same type (e.g. decision trees). But while in bagging classifiers are
generated independently from each other, in boosting they are generated sequentially, in
such a way that each one complements the previous one. First, a classifier is generated
using the original training set. Then, those instances misclassified by this classifier are
given a larger weight (therefore, missclassifying this instance makes the error larger).
Next, a new classifier is obtained by using the previously weighted training set. The
training error is calculated and a weight is assigned to the classifier in accordance with
its performance on the training set. Therefore, the new classifier should do better on the
misclassified instances than the previous one, hence complementing it. This process is
repeated several times. Finally, to combine the set of classifiers in the final classifier,
AdaBoost (like bagging) uses the voting method. But unlike bagging that use equal
weights for all classifiers, in boosting the classifiers with lower training error have a
higher weight in the final decision.

More formally, let S be the training set and T the number of trials, the classifiers C 1, C2,
…. CT are built in sequence from the weighted training samples (S1, S2,…,ST). The final
classifier is the combination of the set of classifiers in which the weight of each
classifier is calculated by its accuracy on its training sample. The AdaBoost algorithm is
displayed in Figure 2

The AdaBoostM1 Algorithm
Input:
Training set S, of labelled instances: S = {(xi , yi), i = 1,2, …, m},
classes yi ∈ Y = {1, …, K}
Base Learning Algorithm (weak learner) B
Number of iteration T

Procedure:
Initialise for all i: w1 (i) = 1/m Assign equal weights to
all training instances
For t = 1 to T {
for i: pt (i) = w t(i)/(Σ i wt (i)) Normalise the weight
of the instances

C t = B(p t ) Apply the base learning
algorithm with the normalised
weights

εi =
m
1
∑ weight(x) weighted error on training set
xj ∈ S ':Ci ( xj ) ≠ yj
If ε i > 1/2 then
T= t -1
exit
βt = εi/(1- ε i)
1− [Ci ( xi ) ≠ yi ]
for all i: wt+1 (i) = wt(i) β t Compute the new weights

}
T
 1
Classifier C* = argmax
y∈Y
∑  log βt [C ( x) = y]
t

t =1

Output
Classifier C*
.
Figure 2 AdaBoostM1 Algorithm.

An important characteristic of the AdaBoost algorithm is that it can combine some
weak learners to produce a strong learner, so long as the error on the weighted training
set is smaller than 0.5. If the error is greater than 0.5, the algorithm finishes the
execution. Were it not so, the final prediction of the ensemble would be random.

Stacking

Stacking is the abbreviation to refer to Stacked Generalization (Wolpert, 1992). Unlike
bagging and boosting, it uses different learning algorithms to generate the ensemble of
classifiers. The main idea of stacking is to combine classifiers from different learners
such as decision trees, instance-based learners, etc. Since each one uses different
knowledge representation and different learning biases, the hypothesis space will be
explored differently, and different classifiers will be obtained. Thus, it is expected that
they will not be correlated.

Once the classifiers have been generated, they must be combined. Unlike bagging and
boosting, stacking does not use a voting system because, for example, if the majority of
the classifiers make bad predictions this will lead to a final bad classification. To solve
this problem, stacking uses the concept of meta learner. The meta learner (or level-1
model), tries to learn, using a learning algorithm, how the decisions of the base
classifiers (or level-0 models) should be combined.

Given a data set S, stacking first generates a subset of training sets S1,…,ST and then
follows something similar to a cross-validation process: it leaves one of the subsets out
(e.g. Sj ) to use later. The remaining instances S(-j)= S - Sj are used to generate the
level-0 classifiers by applying K different learning algorithms, k = 1, …, K, to obtain K
classifiers. After the level-0 models have been generated, the Sj set is used to train the
meta learner (level-1 classifier). Level-1 training data is built from the predictions of the
level-0 models over the instances in Sj , that were left out for this purpose. Level-1 data
has K attributes, whose values are the predictions of each one of the K level-0
classifiers for every instance in Sj . Therefore, a level-1 training example is made of K
attributes (the K predictions) and the target class, which is the right class for every
particular instance in Sj. Once the level-1 data has been built from all instances in Sj,
any learning algorithm can be used to generate the level-1 model. To complete the
process, the level-0 models are re-generated from the whole data set S (this way, it is
expected that classifiers will be slightly more accurate). To classify a new instance, the
level-0 models produce a vector of predictions that is the input to the level-1 model,
which in turn predicts the class.

There are many ways to apply the general idea of stacked generalization. Ting and
Witten (1997) use probability outputs from level-0 models instead a of simple class
prediction as inputs to the level-1 model. LeBlanc and Tibshirani (1993) analyze the
stacked generalization with some regularization (non-negative constraint) to improve
the prediction performance on one artificial dataset. Other works on stacked
generalization have developed a different focus (Chan & Stolfo, 1995; Fan, Chan &
Stolfo, 1996).

Genetic Algorithms for Optimization Problems

Genetic algorithms (GA's) are search procedures loosely connected to the theory of
evolution by means of artificial selection (Holland, 1975). In classical search terms,
GA's can be viewed as a kind of beam search procedure. Its main three components are:

• The beam (or population). It contains the set of points (candidate solutions or
individuals) in the search space that the algorithm is currently exploring. All points
are usually represented by means of bit strings. This domain independent
representation of candidate solutions makes GA's very flexible.
• The search operators. They transform current candidate solutions into new candidate
solutions. Their main characteristic is that, as they operate on bit strings, they are
also domain independent. That is, they can be used to search in any domain. GA's
operators are also based in biological analogies. The three most frequently used
operators are:
• Reproduction: it just copies the candidate solution without modification.
• Crossover: it takes two candidate solutions, mixes them and generates two new
candidate solutions. There are many variations of this operator (mainly, single
point, two point and uniform). See Figure 3 .

00110000100 00110000100

11100100010 11100100010

00100100010 00100100100

11110000100 11110000010

One point crossover Two point crossover

Figure 3 One and Two Point Crossover Operations

• Mutation: it flips a single bit of a candidate solution (it mutates from 0 to 1, or
from 1 to 0). The bit is selected randomly from the bits in the individual.

• The heuristic function (or fitness function). This function measures the worth of a
candidate solution. The goal of a GA is to find candidate solutions that maximize
this function.

GA's start from a population made of randomly generated bit strings. Then, genetic
operators are applied to worthy candidate solutions (according to the heuristic function)
until a new population is built (the new generation). A GA continues producing new
generations until a candidate solution is found that is considered to be good enough, or
when the algorithm seems to be unable to improve candidate solutions (or until the
number of generations reaches a predefined limit). More exactly, a GA follows the
algorithm below:

1. Randomly generate initial population G(0)
2. Evaluate candidate solutions in G(0) with the heuristic function
3. Repeat until a solution is found or population converges
3.1. Apply selection-reproduction: G(i) -> Ga(0)
3.2. Apply crossover: Ga(i) -> Gb(i)
3.3. Apply mutation: Gb(i) -> Gc(i)
3.4. Obtain new generation G(i+1) = Gc(i)
3.5. Evaluate new generation G(i+1)
3.6. Let i=i+1

The production of a new generation G(i+1) from G(i) (steps 3.1, 3.2, and 3.3) is as
follows. First, a new population (Ga(i)) is generated by means of selection. In order to
fill up a population of n individuals for Ga(0), candidate solutions are stochastically
selected with replacement from G(i) n times. The probability for selecting a candidate
solution is the ratio between its fitness and the total fitness of the population. This
means that there will be several copies of very good candidate solutions in Ga(0),
whereas there might be none of those whose heuristic evaluation is poor. However, due
to stochasticity, even bad candidate solutions have a chance of being present in Ga(0).
This method is called ‘selection proportional to fitness’, but there are several more, like
tournament and ranking (Goldberg,1989).

Gb(i) is produced by applying crossover to a fixed percentage pc of randomly selected
candidate solutions in Ga(i). As each crossover event takes two parents and produces
two offspring, crossover happens pc/2 times. In a similar manner, Gc(i) is obtained from
Gb(0) by applying mutation to a percentage pm of candidate solutions.

GA-STACKING APPROACH

In this section, the approach we have taken to build stacking systems will be explained.
Given that stacked generalization is composed by a set of classifiers from a different set
of learning algorithms, the question is: what algorithm should be used to generate the
level-0 and the level-1 models? In principle, any algorithm can be used to generate
them. For instance, Ting and Witten (1997) showed that a linear model is useful to
generate the level-1 model using probabilistic outputs from level-0 models. In our work
we start from the original stacked generalization idea proposed by Wolpert (1992), in
which the outputs from level-0 models are nominal (predictions). Also, our classifiers at
level-0 are hetereogeneous, and any algorithm can be used to build the metaclassifier. In
the present study, to determine the optimum distribution of learning algorithms for the
level-0 and level-1 models, we have used a GA.

GA-Stacking

Two sets of experiment have been carried out to determine whether stacked
generalization combinations of classifiers can be successfully found by genetic
algorithms. We also wanted to know whether GA stacking can obtain improvements
over the most popular ensemble techniques (bagging and boosting) as well as any of the
standalone learning algorithms.

As indicated previously, the application of genetic algorithms to optimization problems
requires to define:

• the representation of the individuals (candidate solutions)
• the fitness function that is used to evaluate the individuals
• the selection-scheme (e.g., selection proportional to fitness)
• the genetic operators that will be used to evolve the individuals
• the parameters (e.g., size of population, generations, probability of crossover, etc.)

In the two sets of performed experiments, the representation of the possible solutions
(individuals) are chromosomes with five genes (see Figure 4). They contain the codes of
the four algorithms to build the level-0 classifiers (firsts 4 genes) and the metaclassifier
(last gene). A gene has 3 bits, because we want to use 7 possible learning algorithms.

To evaluate every individual, we used the classification accuracy of the stacking system
as the fitness function.

level-1 model
010111000110001
level-0 models

Figure 4. Individual description.

EXPERIMENTAL RESULTS

Introduction

In this section, we present the empirical results for our GA-Stacking system. GA-
Stacking has been evaluated in several domains of the well-known UCI database (Blake
& Merz, 1998). There are two sets of experiments. In the first one, we show some
preliminary results, which are quite good but suffer some overfitting. In the second set
of experiments, steps are taken to avoid said overfitting. Both sets of experiments differ
only in the way GA individuals are tested (i.e. the fitness function). In this work, we
have used the algorithms implemented in Weka (Witten & Frank, 2000). This includes
all the classifiers above, bagging, boosting and stacking.

Basically, the implementation of GA-Stacking combines two parts: a part coming from
Weka that includes all the base learning algorithms and another part, which was
integrated into Weka, that implements a GA. The parameters used for the GA in the
experiments are shown in Table 1. The elite rate is the proportion of the population
carried forward unchanged from each generation. Cull rate refers to the proportion of
the population that is deemed unfit for reproduction. Finally, the mutation rate is the
proportion of the population that can suffer change in configuration at random.

Parameters Values
Population size 10
Generations 10
Elite rate 0.1
Cull rate 0.4
Mutation rate 0.067

Table 1. Genetic algorithms parameters.
For the experimental test of our approach we have used seven data sets from the
repository of machine learning databases at UCI. Table 2 shows the data sets
characteristics.

Dataset Attributes Attributes Type Instances Classes
Heart disease 13 Numeric-nominal 303 2
Sonar classification 60 Numeric 208 2
Musk 166 Numeric 476 2
Ionosphere 34 Numeric 351 2
Dermatology 34 Numeric-nominal 366 6
DNA splice 60 Nominal 3190 3
Satellite images 36 Numeric 6435 6
Table 2. Datasets description.
The candidate learning algorithms for GA stacking we used were:

• C4.5 (Quinlan, 1993). It generates decision trees - (C4.5)
• PART (Frank & Witten, 1998). It forms a decision list from pruned partial decision
trees generated using the C4.5 heuristic - (PART)
• A probabilistic Naive Bayesian classifier (John & Langley, 1995) - (NB)
• IB1. This is Aha’s instance based learning algorithm (Aha & Kibler, 1991) - (IB1)
• Decision Table. It is a simple decision table majority classifier (Kohavi, 1995) -
(DT)

• Decision Stump. It generates single-level decision trees - (DS)

Preliminary Experiments

Experiments in this subsection are a first evaluation of GA-Stacking. They use the GA-
Stacking scheme described in previous sections. Here, we will describe in more detail
how individuals are tested (the fitness function), and how the whole system is tested.
Each data set was split randomly in two sets. One of them was used as the training set
for the GA fitness function. It contained about 85% of all the instances. Individual
fitness was measured as the accuracy of the stacking system codified in it. The rest was
used as testing set to validate the stacked hypothesis obtained. In these experiments we
used only two of the data sets shown in Table 2 (dermatology and ionosphere).

Table 3 shows the results for this preliminary approach. Training and test accuracy
columns display how many instances in the training and testing sets were correctly
predicted for each of the systems. The first part of the table contains the standalone
algorithms. The second part shows results corresponding to traditional bagging and
boosting approaches (with C4.5 as the base classifier). Finally, results for GA-Stacking
are shown. The best GA-Stacking individual in the last generation has a 100% in the
training set in both domains, but drops to 85.71% and 95.45% in testing. On the other
hand, some individuals from previous generations manage to get a 94.29% and 98.48%
testing accuracies. This means that, GA-Stacking overfits the data as generations pass
on. Despite this overfitting, GA-Stacking found individuals which are better than most
of the base classifiers. Only in the Ionosphere domain, the Decision Table is better in
testing than the overfitted individual (88.57 vs. 85.71). Also, performance of GA-
Stacking is very close to bagging and boosting (both using C4.5 to generated base
classifiers). Non-overfitted individuals (those obtained in previous generations) show an
accuracy better than any of the other ensemble systems. In addition, stacking systems
only use four level-0 classifiers whereas bagging and boosting use at least 10 base
classifiers. This shows that the approach is solid. Next section will show how we
overcame the problem of overfitting.

Ionosphere Dermatology
Algorithm Training Test Accuracy Training Test
Accuracy Accuracy Accuracy
C4.5 98.42 82.86 96.67 92.42
Naive Bayes 85.13 82.86 98.67 96.97
PART 98.73 88.57 96.67 93.94
IBk (one neighbor) 100.00 82.86 100.00 92.42
IB1 100.00 82.86 100.00 92.42
Decision Stump 84.18 80.00 51.33 45.45
Decision Table 94.94 88.57 96.67 87.88
Ensembles
Bagging with C4.5 97.78 85.71 97.00 93.94
Boosting with C4.5 100.00 91.43 100.00 96.97
GA-Stacking (last
100.00 85.71 100.00 95.45
generation solutions)
GA-Stacking (solutions of
97.78 94.29 98.67 98.48
previous generations)
Table 3. Preliminary Results.

Main Experiments

In the preliminary experiments, individuals overfit because they are evaluated with the
same training instances that were used to build the stacking associated with the
individual. Obviously, the individual will do very well with the instances it has been
trained with. Therefore, in order to avoid the overfitting, the fitness value was
calculated with a separate validation set (different from the testing set used to evaluate
GA-Stacking itself). To do so, the set of training instances was partitioned randomly
into two parts. 80% of training instances were used to build the stacking system
associated with each individual, and the remaining 20% –the validation set- was used to
give a non-biased estimation of the individual accuracy. The latter was used as the
fitness of the individual. It is important to highlight that now fewer instances are used to
build the stacking systems, as some of them are used to ward off overfitting.

For testing all the systems we used the testing data set, just as we did in the preliminary
experiments. However, now a five-fold cross-validation was used. Therefore, results
shown are the average of the five cross-validation iterations. In the satellite images
domain no cross-validation was carried out because the sets for training and testing are
available separately and the data set donor indicated that it is better not to use cross-
validation.

Results are divided in two tables. Table 4 shows the testing accuracy of all the base
algorithms over the data sets. Table 5 displays testing results for the three ensemble
methods, including GA-Stacking. The best results for each domain are highlighted in
bold. The configuration of the stacked generalization found by the genetic algorithms
obtained more accuracy in four of the seven domains used in the experiment with
comparison with others ensemble methods. Boosting only won two times, and Bagging
once.

Dataset C4.5 PART NB IB1 DT DS

Heart 74.00 82.00 83.00 78.00 76.33 70.67
Sonar 72.38 73.81 67.62 79.52 71.90 73.93
Musk 83.33 85.21 74.38 86.46 82.08 71.46
Ionosphere 90.14 89.30 82.82 87.61 89.30 82.82
Dermatology 94.33 94.59 97.03 94.59 87.83 50.40
DNA splice 94.12 92.11 95.63 75.62 92.49 62.42
Satellite images* 85.20 84.10 79.60 89.00 80.70 41.75
Table 4. Average accuracy rate of independent classifiers. *not cross-validation.

Dataset Bagging Boosting GA-Stacking
Heart 76.33 79.67 80.67
Sonar 80.00 79.05 80.47
Musk 87.29 88.96 83.96
Ionosphere 92.11 91.83 90.42
Dermatology 94.59 97.03 98.11
DNA splice 94.56 94.43 95.72
Satellite images 88.80 89.10 88.60
Table 5. Average accuracy rate of ensemble methods
.

CONCLUSIONS

In this chapter, we have presented a new way of building hetereogeneous stacking
systems. To do so, a search on the space of stacking systems is carried out by means of
a genetic algorithm. Empirical results show that GA-Stacking is competitive with more
sophisticated ensemble systems such as Bagging and Boosting. Also, our simple
approach is able to obtain high accuracy classifiers which are very small (they contain
only 5 classifiers) in comparison with other ensemble approaches.

Even though the GA-Stacking approach has already produced very good results, we
believe that future research will show its worth even more clearly:

• Currently, GA-Stacking searches the space of stacked configurations. However, the
behavior of some learning algorithms is controlled by parameters. For instance, the
rules produced by C4.5 can be very different depending on the degree of pruning
carried out. Our GA-Stacking approach would only have to include a binary coding
of the desired parameters in the chromosome.
• Search depends on the representation of candidate hypothesis. Therefore, it would
be interesting to use other representations for our chromosomes. For instance, the
chromosome could be divided into two parts, one for the level 0 learners and
another for the level 1 metaclassifier. A bit in the first part would indicate whether a
particular classifier is included in level 0 or not. The metaclassifier would be coded
in the same way we have done in this paper (i.e. if there are 8 possible
metaclassifiers, 3 bits would be used).
• Different classifiers could be better suited for certain regions of the instance space.
Therefore, instead of having a metaclassifer that determines what output is
appropriate according to the level 0 outputs (as currently), it would be interesting to
have a metaclassifier that knows what level 0 classifier to trust according to some
features of the instance to be classified. In that case, the inputs to the metaclassifier
would be the attributes of the instance and the output the right level 0 classifier to
use.
• We would like to test our approach in domains where the cooperation between
hetereogeneous learning systems is required.

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