Feature Selection with Conjunctions of Decision Stumps and Learning from Microarray Data

Abstract

One of the objectives of designing feature selection learning algorithms is to obtain classifiers that depend on a small number of attributes and have verifiable future performance guarantees. There are few, if any, approaches that successfully address the two goals simultaneously. Performance guarantees become crucial for tasks such as microarray data analysis due to very small sample sizes resulting in limited empirical evaluation. To the best of our knowledge, such algorithms that give theoretical bounds on the future performance have not been proposed so far in the context of the classification of gene expression data. In this work, we investigate the premise of learning a conjunction (or disjunction) of decision stumps in Occam's Razor, Sample Compression, and PAC-Bayes learning settings for identifying a small subset of attributes that can be used to perform reliable classification tasks. We apply the proposed approaches for gene identification from DNA microarray data and compare our results to those of well known successful approaches proposed for the task. We show that our algorithm not only finds hypotheses with much smaller number of genes while giving competitive classification accuracy but also have tight risk guarantees on future performance unlike other approaches. The proposed approaches are general and extensible in terms of both designing novel algorithms and application to other domains.

Full text

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arXiv:1005.0530v1 [cs.LG] 4 May 2010
1
Feature Selection with Conjunctions of Decision
Stumps and Learning from Microarray Data
Mohak Shah, Mario Marchand, and Jacques Corbeil
Abstract
One of the objectives of designing feature selection learning algorithms is to obtain classifiers that depend on a small number
of attributes and have verifiable future performance guarantees. There are few, if any, approaches that successfully address the two
goals simultaneously. Performance guarantees become crucial for tasks such as microarray data analysis due to very small sample
sizes resulting in limited empirical evaluation. To the best of our knowledge, such algorithms that give theoretical bounds on the
future performance have not been proposed so far in the context of the classification of gene expression data. In this work, we
investigate the premise of learning a conjunction (or disjunction) of decision stumps in Occam’s Razor, Sample Compression, and
PAC-Bayes learning settings for identifying a small subset of attributes that can be used to perform reliable classification tasks.
We apply the proposed approaches for gene identification from DNA microarray data and compare our results to those of well
known successful approaches proposed for the task. We show that our algorithm not only finds hypotheses with much smaller
number of genes while giving competitive classification accuracy but also have tight risk guarantees on future performance unlike
other approaches. The proposed approaches are general and extensible in terms of both designing novel algorithms and application
to other domains.
Index Terms
Microarray data classification, Risk bounds, Feature selection, Gene identification.
I. INTRODUCTION
An important challenge in the problem of classification of high-dimensional data is to design a learning algorithm that can
construct an accurate classifier that depends on the smallest possible number of attributes. Further, it is often desired that
there be realizable guarantees associated with the future performance of such feature selection approaches. With the recent
explosion in various technologies generating huge amounts of measurements, the problem of obtaining learning algorithms
with performance guarantees has acquired a renewed interest.
Consider the case of biological domain where the advent of microarray technologies [Eisen and Brown, 1999, Lipshutz et al.,
1999] have revolutionized the outlook on the investigation and analysis of genetic diseases. In parallel, on the classification
front, many interesting results have appeared aiming to distinguish between two or more types of cells, (e.g. diseased vs.
normal, or cells with different types of cancers) based on gene expression data in the case of DNA microarrays (see, for
instance, [Alon et al., 1999] for results on Colon Cancer, [Golub et al., 1999] for Leukaemia). Focusing on very few genes to
give insight into the class association for a microarray sample is quite important owing to a variety of reasons. For instance, a
small subset of genes is easier to analyze as opposed to the set of genes output by the DNA microarray chips. It also makes
it relatively easier to deduce biological relationships among them as well as study their interactions. An approach able to
identify a very few number of genes can facilitate customization of chips and validation experiments– making the utilization
of microarray technology cheaper, affordable, and effective.
In the view of a diseased versus a normal sample, these genes can be considered as indicators of the disease’s cause.
Subsequent validation study focused on these genes, their behavior, and their interactions, can lead to better understanding of
the disease. Some attempts in this direction have yielded interesting results. See, for instance, a recent algorithm proposed
by Wang et al. [2007] involving the identification of a gene subset based on importance ranking and subsequently combinations
of genes for classification. Another example is the approach of Tibshirani et al. [2003] based on nearest shrunken centroids.
Some kernel based approaches such as the BAHSIC algorithm [Song et al., 2007] and their extensions (e.g., [Shah and Corbeil,
2010] for short time-series domains) have also appeared.
The traditional methods used for classifying high-dimensionaldata are often characterized as either “filters” (e.g. [Furey et al.,
2000, Wang et al., 2007] or “wrappers” (e.g. [Guyon et al., 2002]) depending on whether the attribute selection is performed
independent of, or in conjunction with, the base learning algorithm.
M. Shah is with the Centre for Intelligent Machines, McGill University, Montreal, Canada, H3A 2A7.
E-mail: mohak@cim.mcgill.ca
M. Marchand is with the Department of Computer Science and Software Engineering, Pav. Adrien Pouliot, Laval University, Quebec, Canada, G1V-0A6.
Email: Mario.Marchand@ift.ulaval.ca
J. Corbeil is with CHUL Research Center, Laval University, Quebec (QC) Canada, G1V-4G2.
Email: Jacques.Corbeil@crchul.ulaval.ca

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Despite the acceptable empirical results achieved by such approaches, there is no theoretical justification of their performance
nor do they come with a guarantee on how well will they perform in the future. What is really needed is a learning algorithm
that has provably good performance guarantees in the presence of many irrelevant attributes. This is the focus of the work
presented here.
A. Contributions
The main contributions of this work come in the form of formulation of feature selection strategies within well established
learning settings resulting in learning algorithms that combine the tasks of feature selection and discriminative learning.
Consequently, we obtain feature selection algorithms for classification with tight realizable guarantees on their generalization
error. The proposed approaches are a step towards more general learning strategies that combine feature selection with the
classification algorithm and have tight realizable guarantees. We apply the approaches to the task of classifying microarray
data where the attributes of the data sample correspond to the expression level measurements of various genes. In fact the
choice of decision stumps as learning bias has in part motivated by this application. The framework is general and extensible
in a variety of ways. For instance, the learning strategies proposed in this work can readily be extended to other similar tasks
that can benefit from this learning bias. An immediate example would be classifying data from other microarray technologies
such as in the case of Chromatin Immunoprecipitation experiments. Similarly, learning biases other than the conjunctions of
decision stumps, can also be explored in the same frameworks leading to novel learning algorithms.
B. Motivation
For learning the class of conjunctions of features, we draw motivation from the guarantee that exists for this class in the
following form: if there exists a conjunction, that depends on r out of the n input attributes and that correctly classifies a
training set of m examples, then the greedy covering algorithm of Haussler [1988] will find a conjunction of at most rlnm
attributes that makes no training errors. Note the absence of dependence on the number n of input attributes. The method is
guaranteed to find at most rlnm attributes and, hence, depends on the number of available samples m but not on the number
of attributes n to be analyzed.
We propose learning algorithms for building small conjunctions of decision stumps. We examine three approaches to obtain
an optimal classifier based on this premise that mainly vary in the coding strategies for the threshold of each decision stump.
The first two approaches attempt to do this by encoding the threshold either with message strings (Occam’s Razor) or by
using training examples (Sample Compression). The third strategy (PAC-Bayes) attempts to examine if an optimal classifier
can be obtained by trading off the sparsity1of the classifier with the magnitude of the separating margin of each decision
stump. In each case, we derive an upper bound on the generalization error of the classifier and subsequently use it to guide the
respective algorithm. Finally, we present empirical results on the microarray data classification tasks and compare our results
to the state-of-the-art approaches proposed for the task including the Support Vector Machine (SVM) coupled with feature
selectors, and Adaboost. The preliminary results of this work appeared in [Marchand and Shah, 2005].
C. Organization
Section II gives the basic definitions and notions of the learning setting that we utilize and also characterizes the hypothesis
class of conjunctions of decision stumps. All subsequent learning algorithms are proposed to learn this hypothesis class.
Section III proposes an Occam’s Razor approach to learn conjunctions of decision stumps leading to an upper bound on the
generalization error in this framework. Section IV then proposes an alternate encoding strategy for the message strings using
the Sample Compression framework and gives a corresponding risk bound. In Section V, we propose a PAC-Bayes approach
to learn conjunction of decision stumps that enables the learning algorithm to perform an explicit non-trivial margin-sparsity
trade-off to obtain more general classifiers. Section VI then proposes algorithms to learn in the three learning settings proposed
in Sections III, IV and V along with a time complexity analysis. Note that the learning (optimization) strategies proposed in
Section VI do not affect the respective theoretical guarantees of the learning settings. The algorithms are evaluated empirically
on real world microarray datasets in Section VII. Section VIII presents a discussion on the results and also provides an analysis
of the biological relevance of the selected genes in the case of each dataset, and their agreement with published findings. Finally,
we conclude in Section IX.
II. DEFINITIONS
The input space X consists of all n-dimensional vectors x = (x1,...,xn) where each real-valued component xi∈ [Ai,Bi]
for i = 1,...n. Each attribute xifor instance can refer to the expression level of gene i. Hence, Aiand Biare, respectively,
the a priori lower and upper bounds on values for xi. The output space Y is the set of classification labels that can be assigned
to any input vector x ∈ X. We focus here on binary classification problems. Thus Y = {0,1}. Each example z = (x,y) is an
1This refers to the number of decision stumps used.

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input vector x ∈ X with its classification label y ∈ Y chosen i.i.d. from an unknown distribution D on X × Y. The true risk
R(f) of any classifier f is defined as the probability that it misclassifies an example drawn according to D:
R(f)def
= Pr(x,y)∼D(f(x) ?= y) = E(x,y)∼DI(f(x) ?= y)
where I(a) = 1 if predicate a is true and 0 otherwise. Given a training set S = {z1,...,zm} of m examples, the empirical
risk RS(f) on S, of any classifier f, is defined according to:
RS(f)def
=
1
m
m
?
i=1
I(f(xi) ?= yi)def
= E(x,y)∼SI(f(x) ?= y)
The goal of any learning algorithm is to find the classifier with minimal true risk based on measuring empirical risk (and other
properties) on the training sample S.
We focus on learning algorithms that construct a conjunction of decision stumps from a training set. Each decision stump
is just a threshold classifier defined on a single attribute (component) xk. More formally, a decision stump is identified by an
attribute index k ∈ {1,...,n}, a threshold value t ∈ R, and a direction d ∈ {−1,+1} (that specifies whether class 1 is on
the largest or smallest values of xk). Given any input example x, the output rk
td(x) of a decision stump is defined as:
rk
td(x)
def
=
?
1
0
if
if
(xk− t)d > 0
(xk− t)d ≤ 0
We use a vector k
number of indices present in k (and thus the number of decision stumps in the conjunction)2. Furthermore, We use a vector
t = (tk1,tk2,...,tk|k|) of threshold values and a vector d = (dk1,dk2,...,dk|k|) of directions where kj ∈ {1,...,n} for
j ∈ {1,...,|k|}. On any input example x, the output Ck
def
= (k1,...,k|k|) of attribute indices kj∈ {1,...,n} such that k1< k2< ... < k|k|where |k| is the
td(x) of a conjunction of decision stumps is given by:
Ck
td(x)
def
=
?
1
0
if
if
rj
∃j ∈ k : rj
tjdj(x) = 1
∀j ∈ k
tjdj(x) = 0
Finally, any algorithm that builds a conjunction can be used to build a disjunction just by exchanging the role of the positive
and negative labeled examples. In order to keep our description simple, we describe here only the case of a conjunction.
However, the case of disjunction follows symmetrically.
III. AN OCCAM’S RAZOR APPROACH
Our first approach towards learning the conjunction (or disjunction) of decision stumps is the Occam’s Razor approach.
Basically, we wish to obtain a hypothesis that can be coded using the least number of bits. We first propose an Occam’s Razor
risk bound which will ultimately guide the learning algorithm.
In the case of zero-one loss, we can model the risk of the classifier as a binomial. Let Bin(κ,m,r) be the the binomial tail
associated with a classifier of (true) risk r. Then Bin(κ,m,r) is the probability that this classifier makes at most κ errors on
a set of m examples:
κ
?
The binomial tail inversion Bin(κ,m,δ) then gives the largest risk value that a classifier can have while still having a probability
of at least δ of observing at most κ errors out of m examples [Langford, 2005, Blum and Langford, 2003]:
Bin(κ,m,r)
def
=
i=0
?m
i
?
ri(1 − r)m−i
Bin(κ,m,δ)def
= sup{r : Bin(κ,m,r) ≥ δ}
From this definition, it follows that Bin(mRS(f),m,δ) is the smallest upper bound, which holds with probability at least
1 − δ, on the true risk of any classifier f with an observed empirical risk RS(f) on a test set of m examples:
∀f :PrS∼Dm?R(f) ≤ Bin?mRS(f),m,δ??≥ 1 − δ
Our starting point is the Occam’s razor bound of Langford [2005] which is a tighter version of the bound proposed
by Blumer et al. [1987]. It is also more general in the sense that it applies to any prior distribution P over any countable class
of classifiers.
2Although it is possible to use up to two decision stumps on any attribute, we limit ourselves here to the case where each attribute can be used for only
one decision stump.

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Theorem 1 (Langford [2005]). For any prior distribution P over any countable class F of classifiers, for any data-generating
distribution D, and for any δ ∈ (0,1], we have:
PrS∼Dm
?
∀f ∈ F : R(f) ≤ Bin?mRS(f),m,P(f)δ??
≥ 1 − δ
The proof (available in [Langford, 2005]) directly follows from a straightforward union bound argument and from the fact
that?
In our case, decision-stumps’ conjunctions are specified in terms of the discrete-valued vectors k and d and the continuous-
valued vector t. We will see below that we will use a finite-precision bit string σ to specify the set of threshold values t. Let
us denote by P(k,d,σ) the prior probability assigned to the conjunction Ck
the following form:
1
?n
where gk,d(σ) is the prior probability assigned to string σ given that we have chosen k and d. Let M(k,d) be the
set of all message strings that we can use given that we have chosen k and d. If I denotes the set of all 2npossi-
ble attribute index vectors and Dk denotes the set of all 2|k|binary direction vectors d of dimension |k|, we have that
?
classifier, constructed from the group of attributes specified by k, should depend only on the number |k| of attributes in this
group. If we have complete ignorance about the number of decision stumps the final classifier is likely to have, we should
choose p(d) = 1/(n + 1) for d ∈ {0,1,...,n}. However, we should choose a p that decreases as we increase d if we have
reasons to believe that the number of decision stumps of the final classifier will be much smaller than n. Since this is usually
our case, we propose to use:
6
π2(|k| + 1)−2
The third factor of P(k,d,σ) gives equal prior probabilities for each of the two possible values of direction dj.
To specify the distribution of strings gk,d(σ), consider the problem of coding a threshold value t ∈ [a,b] ⊂ [A,B] where
[A,B] is some predefined interval in which we are permitted to choose t and where [a,b] is an interval of “equally good”
threshold values.3We propose the following diadic coding scheme for the identification of a threshold value that belongs to
that interval. Let l be the number of bits that we use for the code. Then, a code of l bits specifies one value among the set Λl
of threshold values:
??
We denote by Aiand Bi, the respective a priori minimum and maximum values that the attribute i can take. These values are
obtained from the definition of data. Hence, for an attribute i ∈ k, given an interval [ai,bi] ⊂ [Ai,Bi] of threshold values, we
take the smallest number liof bits such that there exists a threshold value in Λlithat falls in the interval [ai,bi]. In that way,
we will need at most ⌊log2((Bi− Ai)/(bi− ai))⌋ bits to obtain a threshold value that falls in [ai,bi].
Hence, to specify the threshold for each decision stump i ∈ k, we need to specify the number liof bits and a li-bit string
sithat identifies one of the threshold values in Λli. The risk bound does not depend on how we actually code σ (for some
receiver). It only depends on the a priori probabilities we assign to each possible realization of σ. We choose the following
distribution:
f∈FP(f) = 1. To apply this bound for conjunctions of decision stumps we thus need to choose a suitable prior P for
this class. Moreover, Theorem 1 is valid when?
f∈FP(f) ≤ 1. Consequently, we will use a subprior P whose sum is ≤ 1.
σddescribed by (k,d,σ). We choose a prior of
P(k,d,σ) =
|k|
?p(|k|)
1
2|k|gk,d(σ)
k∈I
The reasons motivating this choice for the prior are the following. The first two factors come from the belief that the final
?
d∈Dk
?
σ∈M(k,d)P(k,d,σ) ≤ 1 whenever?n
d=0p(d) ≤ 1 and?
σ∈M(k,d)gk,d(σ) ≤ 1 ∀k,d.
p(|k|) =
Λl
def
=1 −2j − 1
2l+1
?
A +2j − 1
2l+1B
?2l
j=1
gk,d(σ)
def
=gk,d(l1,s1,...,l|k|,s|k|)
?
(1)
=
i∈k
ζ(li) · 2−li
(2)
where:
ζ(a)
def
=
6
π2(a + 1)−2
∀a ∈ N
(3)
The sum over all the possible realizations of σ gives 1 since?∞
The distribution ζ that we have chosen for each string length li has the advantage of decreasing slowly so that the risk
bound does not deteriorate too rapidly as li increases. Other choices are clearly possible. However, note that the dominant
contribution comes from the 2−literm yielding a risk bound that depends linearly in li.
i=1i−2= π2/6. Note that by giving equal a priori probability
to each of the 2listrings siof length li, we give no preference to any threshold value in Λli.
3By a “good” threshold value, we mean a threshold value for a decision stump that would cover many negative examples and very few positive examples
(see the learning algorithm).

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With this choice of prior, we have the following theorem:
Theorem 2. Given all our previous definitions and for any δ ∈ (0,1], we have:
Pr
S∼Dm
?
∀k,d,σ: R(Ck
σd) ≤ Bin
?
mRS(Ck
σd),m,p(|k|)gk,d(σ)δ
?n
|k|
?2|k|
??
≥ 1 − δ
Finally, we emphasize that the risk bound of Theorem 2, used in conjunction with the distribution of messages given by
gk,d(σ), provides a guide for choosing the optimal classifier. Note that the above risk bound suggests a non-trivial trade-off
between the number of attributes and the length of the message string used to encode the classifier. Indeed the risk bound
may be smaller for a conjunction having a large number of attributes with small message strings (i.e., small lis) than for a
conjunction having a small number of attributes but with large message strings.
IV. A SAMPLE COMPRESSION APPROACH
The basic idea of the Sample compression framework [Kuzmin and Warmuth, 2007] is to obtain learning algorithms with
the property that the generated classifier (with respect to some training data) can often be reconstructed with a very small
subset of training examples. More formally, a learning algorithm A is said to be a sample-compression algorithm iff there
exists a compression function C and a reconstruction function R such that for any training sample S = {z1,...,zm} (where
zi
= (xi,yi)), the classifier A(S) returned by A is given by:
def
A(S) = R(C(S))
∀S ∈ (X × Y)m
For a training set S, the compression function C of learning algorithm A outputs a subset ziof S, called the compression set,
and an information message σ, i.e., (zi,σ) = C(z1,...,zm). The information message σ contains the additional information
needed to reconstruct the classifier from the compression set zi. Given a training sample S, we define the compression set zi
by a vector of indices i such that idef
= (i1,i2,...,i|i|), with ij∈ {1,...,m}∀j and i1< i2< ... < i|i|and where |i| denotes
the number of indices present in i.
When given an arbitrary compression set zi and an arbitrary information message σ, the reconstruction function R of a
learning algorithm A must output a classifier. The information message σ is chosen from a set M(zi) that consists of all the
distinct messages that can be attached to the compression set zi. The existence of this reconstruction function R assures that
the classifier returned by A(S) is always identified by a compression set ziand an information message σ.
In sample compression settings for learning decision stumps’ conjunctions, the message string consists of the attributes and
directions defined above. However, the thresholds are now specified by training examples. Hence, if we have |k| attributes
where k is the set of thresholds, the compression set consists of |k| training examples (one per threshold).
Our starting point is the following generic Sample Compression bound [Marchand and Sokolova, 2005]:
Theorem 3. For any sample compression learning algorithm with a reconstruction function R that maps arbitrary subsets of
a training set and information messages to classifiers:
PS∼Dm {∀i ∈ I,σ ∈ M(Zi): R(R(σ,Zi)) ≤ ǫ(σ,Zi,|j|)} ≥ 1 − δ
where
ǫ(σ,zi,|j|)=1 − exp
?
−1
m − |i| − |j|
1
PM(Zi)(σ)
?
ln
?m
?
|i|
?
+ ln
?m − |i|
???
|j|
?
+ln
?
?
+ ln
1
ζ(|i|)ζ(|j|)δ
(4)
and ζ is defined by Equation 3.
Now, we need to specify the distribution of messages (PM(Zi)(σ)) for the conjunction of decision stumps. Note that in
order to specify a conjunction of decision stumps, the compression set consists of one example per decision stump. For each
decision stump we have one attribute and a corresponding threshold value determined by the numerical value that this attribute
takes on the training example.
The learner chooses an attribute whose threshold is identified by the associated training example. The set of these training
examples form the compression set. Finally, the learner chooses a direction for each attribute.
The subset of attributes that specifies the decision stumps in our compression set ziis given by the vector k defined in the
previous section. Moreover, since there is one decision stump corresponding to each example in the compression set, we have
|i| = |k|. Now, we assign equal probability to each possible set |k| of attributes (and hence thresholds) that can be selected
from n attributes. Moreover, we assign equal probability over the direction that each decision stump can have (+1,−1). Hence,
we get the following distribution of messages:

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PM(zi)(σ) =
?n
|k|
?−1
· 2−|k|∀σ
(5)
Equation 5 along with the Sample Compression Theorem completes the bound for the conjunction of decision stumps.
V. A PAC-BAYES APPROACH
The Occam’s Razor and Sample Compression, in a sense, aim at obtaining sparse classifiers with minimum number of stumps.
This sparsity is enforced by selecting the classifiers with minimal encoding of the message strings and the compression set in
respective cases.
We now examine if by sacrificing this sparsity in terms of a larger separating margin around the decision boundary (yielding
more confidence) can lead us to classifiers with smaller generalization error. The learning algorithm is based on the PAC-Bayes
approach [McAllester, 1999] that aims at providing Probably Approximately Correct (PAC) guarantees to “Bayesian” learning
algorithms specified in terms of a prior distribution P (before the observation of the data) and a data-dependent, posterior
distribution Q over a space of classifiers.
We formulate a learning algorithm that outputs a stochastic classifier, called the Gibbs Classifier GQ defined by a data-
dependent posterior Q. Our classifier will be partly stochastic in the sense that we will formulate a posterior over the threshold
values utilized by the decision stumps while still retaining the deterministic nature for the selected attributes and directions for
the decision stumps.
Given an input example x, the Gibbs classifier first selects a classifier h according to the posterior distribution Q and then
use h to assign the label h(x) to x. The risk of GQis defined as the expected risk of classifiers drawn according to Q:
R(GQ)
def
= Eh∼QR(h) = Eh∼QE(x,y)∼DI(h(x) ?= y)
Our starting point is the PAC-Bayes theorem [McAllester, 2003, Langford, 2005, Seeger, 2002] that provides a bound on the
risk of the Gibbs classifier:
Theorem 4. Given any space H of classifiers. For any data-independent prior distribution P over H, we have:
Pr
S∼Dm
?
∀Q : kl(RS(GQ)?R(GQ)) ≤KL(Q?P) + lnm+1
δ
m
?
≥ 1 − δ
where KL(Q?P) is the Kullback-Leibler divergence between distributions4Q and P:
KL(Q?P)def
= Eh∼QlnQ(h)
P(h)
and where kl(q?p) is the Kullback-Leibler divergence between the Bernoulli distributions with probabilities of success q and
p:
kl(q?p)def
= qlnq
p+ (1 − q)ln1 − q
1 − p
This bound for the risk of Gibbs classifiers can easily be turned into a bound for the risk of Bayes classifiers BQover the
posterior Q. BQbasically performs a majority vote (under measure Q) of binary classifiers in H. When BQmisclassifies an
example x, at least half of the binary classifiers (under measure Q) misclassifies x. It follows that the error rate of GQis at
least half of the error rate of BQ. Hence R(BQ) ≤ 2R(GQ).
In our case, we have seen that decision stump conjunctions are specified in terms of a mixture of discrete parameters k and
d and continuous parameters t. If we denote by Pk,d(t) the probability density function associated with a prior P over the
class of decision stump conjunctions, we consider here priors of the form:
Pk,d(t) =
1
?n
|k|
?p(|k|)
1
2|k|
?
j∈k
I(tj∈ [Aj,Bj])
Bj− Aj
As before, we have that:
?
k∈I
?
d∈Dk
?
j∈k
?Bj
Aj
dtjPk,d(t) = 1
whenever?n
The factors relating to the discrete components k and d have the same rationale as in the case of the Occam’s Razor
approach. However, in the case of the threshold for each decision stumps, we now consider an explicitly continuous uniform
e=0p(e) = 1.
4Here Q(h) denotes the probability density function associated with Q, evaluated at h.

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prior. As in the Occam’s Razor case, we assume each attribute value xkto be constrained, a priori, in [Ak,Bk] such that Ak
and Bkare obtained from the definition of the data. Hence, we have chosen a uniform prior probability density on [Ak,Bk]
for each tksuch that k ∈ k. This explains the last factors of Pk,d(t).
Given a training set S, the learner will choose an attribute group k and a direction vector d deterministically. We pose the
problem of choosing the threshold in a similar manner as in the case of Occam’s Razor approach of Section III with the only
difference that the learner identifies the interval and selects a threshold stochastically. For each attribute xk∈ [Ak,Bk] : k ∈ k,
a margin interval [ak,bk] ⊆ [Ak,Bk] is chosen by the learner. A deterministic decision stump conjunction classifier is then
specified by choosing the thresholds values tk∈ [ak,bk] uniformly. It is tempting at this point to choose tk= (ak+bk)/2 ∀k ∈ k
(i.e., in the middle of each interval). However, the PAC-Bayes theorem offers a better guarantee for another type of deterministic
classifier as we see below.
Hence, the Gibbs classifier is defined with a posterior distribution Q having all its weight on the same k and d as chosen
by the learner but where each tkis uniformly chosen in [ak,bk]. The KL divergence between this posterior Q and the prior
P is then given by:
??n
In this limit when [ak,bk] = [Ak,Bk] ∀k ∈ k, it can be seen that the KL divergence between the “continuous components”
of Q and P vanishes. Furthermore, the KL divergence between the “discrete components” of Q and P is small for small values
of |k| (whenever p(|k|) is not too small). Hence, this KL divergence between our choices for Q and P exhibits a tradeoff
between margins (bk− ak) and sparsity (small value of |k|) for Gibbs classifiers. Theorem 4 suggests that the GQwith the
smallest guarantee of risk R(GQ) should minimize a non trivial combination of KL(Q?P) and RS(GQ).
The posterior Q is identified by an attribute group vector k, a direction vector d, and intervals [ak,bk] ∀k ∈ k. We refine the
notation for our Gibbs classifier GQto reflect this. Hence, we use Gkd
of aks and bks respectively. We can obtain a closed-form expression for RS(Gkd
on a single example (x,y) since RS(Gkd
KL(Q?P) = ln
|k|
?
·
2|k|
p(|k|)
?
+
?
k∈k
ln
?Bk− Ak
bk− ak
?
abwhere a and b are the vectors formed by the unions
ab) by first considering the risk R(x,y)(Gkd
ab). From our definition for Q, we find that:
ab)
ab) = E(x,y)∼SR(x,y)(Gkd
R(x,y)(Gkd
ab) = (1 − 2y)
??
k∈k
σdk
ak,bk(xk) − y
?
(6)
where:
σd
a,b(x)
def
=







0
x−a
b−a
b−x
b−a
1
if (x < a and d = +1) or (b < x and d = −1)
if a ≤ x ≤ b and d = +1
if a ≤ x ≤ b and d = −1
if (b < x and d = +1) or (x < a and d = −1)
Note that the expression for R(x,y)(Ck
functions σdk
The PAC-Bayes theorem provides a risk bound for the Gibbs classifier Gkd
a majority vote under the same posterior distribution as the one used by Gkd
td) is identical to the expression for R(x,y)(Gkd
ab) except that the piece-wise linear
ak,bk(xk) are replaced by the indicator functions I((xk− tk)dk> 0).
ab. Since the Bayes classifier Bkd
ab, it follows that:
abjust performs
Bkd
ab(x) =
?
1
0
if
if
?
k∈kσdk
k∈kσdk
ak,bk(xk) > 1/2
ak,bk(xk) ≤ 1/2
?
(7)
Note that Bkd
stumps. There is, however, no computational difficulty at obtaining the output of Bkd
main theorem:
abhas an hyperbolic decision surface. Consequently, Bkd
abis not representable as a conjunction of decision
ab(x) for any x ∈ X. We now state our
Theorem 5. Given all our previous definitions, for any δ ∈ (0,1], and for any p satisfying?n
?
where
1
m
|k|
e=0p(e) = 1, we have, with
probability atleast 1 − δ over random draws of S ∼ Dm:
∀k,d,a,b: R(Gkd
ab) ≤ sup?ǫ: kl(RS(Gkd
2|k|
p(|k|)·m + 1
δ
ab)?ǫ) ≤ ψ??
?Bk− Ak
ψ =
?
ln
??n
?
·
?
+
?
k∈k
ln
bk− ak
??
Furthermore: R(Bkd
ab) ≤ 2R(Gkd
ab)
∀k,d,a,b.

Page 8

8
VI. THE LEARNING ALGORITHMS
Having proposed the theoretical frameworks attempting to obtain the optimal classifiers based on various optimization criteria,
we now detail the learning algorithms for these approaches. Ideally, we would like to find a conjunction of decision stumps
that minimizes the respective risk bounds for each approach. Unfortunately, this cannot be done efficiently in all cases since
this problem is at least as hard as the (NP-hard) minimum set cover problem as mentioned by Marchand and Shawe-Taylor
[2002]. Hence, we use a set covering greedy heuristic. It consists of choosing the decision stump i with the largest utility USC
where:
USC
i
= |Qi| − p|Ri|
i
(8)
where Qiis the set of negative examples covered (classified as 0) by feature i, Riis the set of positive examples misclassified
by this feature, and p is a learning parameter that gives a penalty p for each misclassified positive example. Once the feature
with the largest Uiis found, we remove Qiand Rifrom the training set S and then repeat (on the remaining examples) until
either no more negative examples are present or that a maximum number s of features has been reached. This heuristic was
also used by Marchand and Shawe-Taylor [2002] in the context of a sample compression classifier called the set covering
machine. For our sample compression approach (SC), we use the above utility function USC
However, for the Occam’s Razor and the PAC-Bayes approaches, we need utility functions that can incorporate the opti-
mization aspects suggested by these approaches.
i
.
A. The Occam’s Razor learning algorithm
We propose the following learning strategy for Occam’s Razor learning of conjunctions of decision stumps. For a fixed li
and η, let N be the set of negative examples and P be the set of positive examples. We start with N′= N and P′= P. Let
Qi be the subset of N′covered by decision stump i, let Ri be the subset of P′covered by decision stump i, and let li be
the number of bits used to code the threshold of decision stump i. We choose the decision stump i that maximizes the utility
UOccam
i
defined as:
UOccam
i
def
=
|Qi|
N′− p|Ri|
P
− η · li
where p is the penalty suffered by covering (and hence, misclassifying) a positive example and η is the cost of using libits
for decision stump i. Once we have found a decision stump maximizing Ui, we update N′= N′−Qiand P′= P′−Riand
repeat to find the next decision stump until either N′= ∅ or the maximum number v of decision stumps has been reached
(early stopping the greedy). The best values for the learning parameters p,η, and v are determined by cross-validation.
B. The PAC-Bayes Learning Algorithm
Theorem 5 suggests that the learner should try to find the Bayes classifier Bkd
a small |k|), each with a large separating margin (bk− ak), while keeping the empirical Gibbs risk RS(Gkd
As discussed earlier, we utilize the greedy set covering heuristic for learning.
In our case, however, we need to keep the Gibbs risk on S low instead of the risk of a deterministic classifier. Since the
Gibbs risk is a “soft measure” that uses the piece-wise linear functions σd
cannot make use of the hard utility function of Equation 8. Instead, we need a “softer” version of this utility function to take
into account covering (and erring on) an example partly. That is, a negative example that falls in the linear region of a σd
in fact partly covered and vice versa for the positive example.
Following this observation, let k′be the vector of indices of the attributes that we have used so far for the construction of
the classifier. Let us first define the covering value C(Gk′d
0 by Gk′d
abthat uses a small number of attributes (i.e.,
ab) at a low value.
a,binstead of the “hard” indicator functions, we
a,bis
ab) of Gk′d
abby the “amount” of negative examples assigned to class
ab:
C(Gk′d
ab)
def
=
?
(x,y)∈S
(1 − y)

1 −
?
j∈k′
σdj
aj,bj(xj)


We also define the positive-side error E(Gk′d
ab) of Gk′d
abas the “amount” of positive examples assigned to class 0 :
E(Gk′d
ab)
def
=
?
(x,y)∈S
y

1 −
?
j∈k′
σdj
aj,bj(xj)



Page 9

9
We now want to add another decision stump on another attribute, call it i, to obtain a new vector k′′containing this new
attribute in addition to those present in k′. Hence, we now introduce the covering contribution of decision stump i as:
Ck′d
ab(i)
def
=
C(Gk′′d′
?
a′b′ ) − C(Gk′d
ab)
=
(x,y)∈S
(1 − y)
?
1 − σdi
ai,bi(xi)
??
j∈k′
σdj
aj,bj(xj)
and the positive-side error contribution of decision stump i as:
Ek′d
ab(i)
def
=
E(Gk′′d′
?
a′b′ ) − E(Gk′d
?
ab)
=
(x,y)∈S
y1 − σdi
ai,bi(xi)
??
j∈k′
σdj
aj,bj(xj)
Typically, the covering contribution of decision stump i should increase its “utility” and its positive-side error should
decrease it. Moreover, we want to decrease the “utility” of decision stump i by an amount which would become large
whenever it has a small separating margin. Our expression for KL(Q?P) suggests that this amount should be proportional
to ln((Bi− Ai)/(bi− ai)). Furthermore we should compare this margin term with the fraction of the remaining negative
examples that decision stump i has covered (instead of the absolute amount of negative examples covered). Hence the covering
contribution Ck′d
covered before considering decision stump i:
ab(i) of decision stump i should be divided by the amount Nk′d
ab
of negative examples that remains to be
Nk′d
ab
def
=
?
(x,y)∈S
(1 − y)
?
j∈k′
σdj
aj,bj(xj)
which is simply the amount of negative examples that have been assigned to class 1 by Gk′d
examples, we define the utility Uk′d
ab. If P denotes the set of positive
ab(i) of adding decision stump i to Gk′d
abas:
Uk′d
ab(i)
def
=
Ck′d
ab(i)
Nk′d
ab
− pEk′d
ab(i)
|P|
− η lnBi− Ai
bi− ai
where parameter p represents the penalty of misclassifying a positive example and η is another parameter that controls the
importance of having a large margin. These learning parameters can be chosen by cross-validation. For fixed values of these
parameters, the “soft greedy” algorithm simply consists of adding, to the current Gibbs classifier, a decision stump with
maximum added utility until either the maximum number v of decision stumps has been reached or all the negative examples
have been (totally) covered. It is understood that, during this soft greedy algorithm, we can remove an example (x,y) from S
whenever it is totally covered. This occurs whenever?
we normalize the number of covered and erred examples so as to increase their sensitivity to the respective η terms.
1) Time Complexity Analysis: Let us analyze the time complexity of this algorithm for fixed p and η. For each attribute,
we first sort the m examples with respect to their values for the attribute under consideration. This takes O(mlogm) time.
Then, we examine each potential aivalue (defined by the values of that attribute on the examples). Corresponding to each ai,
we examine all the potential bivalues (all the values greater than ai). This gives us a time complexity of O(m2). Now if k is
the largest number of examples falling into [ai,bi], calculating the covering and error contributions and then finding the best
interval [ai,bi] takes O(km2) time. Moreover, we allow k ∈ O(m) giving us a time complexity of O(m3) for each attribute.
Finally, we do this over all the attributes. Hence, the overall time complexity of the algorithm is O(nm3). Note, however,
that for microarray data, we have n >> m (hence, we can consider m3to be a constant). Moreover once the best stump is
found, we remove the examples covered by this stump from the training set and repeat the algorithm. Now, we know that
greedy algorithms of this kind have the following guarantee: if there exist r decision stumps that covers all the m examples,
the greedy algorithm will find at most rln(m) decision stumps. Since we almost always have r ∈ O(1), the running time of
the whole algorithm will almost always be ∈ O(nm3log(m)). The good news is, since n >> m, the time complexity of our
algorithm is roughly linear in n.
2) Fixed-Margin Heuristic: In order to show why we prefer a uniformly distributed threshold as opposed to the one fixed
at the middle of the interval [ai,bi] for each stump i, we use an alternate algorithm that we call the fixed margin heuristic.
The algorithm is similar to the one described above but with an additional parameter γ. This parameter decides a fixed
margin boundary around the threshold, i.e. γ decides the length of the interval [ai,bi]. The algorithm still chooses the attribute
vector k, the direction vector d and the vectors a and b. However, the ai’s and bi’s for each stump i are chosen such that,
|bi− ai| = 2γ. The threshold tiis then fixed in the middle of this interval, that is ti=
interval [ai,bi] = [ti− γ,ti+ γ]. For fixed p and γ, a similar analysis as in the previous subsection yields a time complexity
of O(nm2log(m)) for this algorithm.
j∈k′σdj
aj,bj(xj) = 0.
Hence, we use the above utility function for the PAC-Bayes learning strategy. Note that, in the case of UPB
i
and UOccam
i
,
(ai+bi)
2
. Hence, for each stump i, the

Page 10

10
Data Set SVM
Errs
12.8±1.4
13.2±1
28.2±2.2
21.3±1.4
8.8±1.3
15.3±2.4
SVM+gs
Errs
14.4±3.5
7.2±2.6
23.1±2.8
14±2.8
6.8±1.9
10.3±2.7
SVM+rfe
Errs
15.4±4.8
10.4±2.4
28.2±2.2
21±3.2
7.2±1.8
11.2±2.8
Adaboost
Itrs
20
20
50
20
1
50
Name
Colon
B MD
C MD
Leuk
Lung
BreastER
ex
62
34
60
72
52
49
Genes
2000
7129
7129
7129
918
7129
SS Errs
15.2±2.1
9.8±1.1
21.2±2.4
17.8±1.8
2.4±1.4
9.8±1.7
256
32
1024
64
64
256
128
64
7129
256
32
256
TABLE I
RESULTS OF SVM, SVM COUPLED WITH GOLUB’S FEATURE SELECTION ALGORITHM (FILTER), SVM WITH RECURSIVE FEATURE ELIMINATION
(WRAPPER) AND ADABOOST ALGORITHMS ON GENE EXPRESSION DATASETS.
Data SetOccamSC
Name
Colon
B MD
C MD
Leuk
Lung
BreastER
ex
62
34
60
72
52
49
Genes
2000
7129
7129
7129
918
7129
Errs
23.6±1.2
17.2±1.8
28.6±1.8
27.8±1.7
21.7±1.1
25.4±1.2
Sbits
6
3
4
6
5
2
Errs
18.2±1.8
17.2±1.3
29.2±1.1
27.3±1.7
18±1.3
21.2±1.5
S
1.8±.6
1.2±.8
2.6±1.1
2.2±.8
1.8±1.2
3.2±.6
1.2±.6
1.4±.8
1.2±.6
1.4±.7
1.2±.5
1.4±.5
TABLE II
RESULTS OF THE PROPOSED OCCAM’S RAZOR AND SAMPLE COMPRESSION LEARNING ALGORITHMS ON GENE EXPRESSION DATASETS.
VII. EMPIRICAL RESULTS
The proposed approaches for learning conjunctions of decision stumps were tested on the six real-world binary microarray
datasets viz. the colon tumor [Alon et al., 1999], the Leukaemia [Golub et al., 1999], the B MD and C MD Medulloblastomas
data [Pomeroy et al., 2002], the Lung [Garber et al., 2001], and the BreastER data [West et al., 2001].
The colon tumor data set [Alon et al., 1999] provides the expression levels of 40 tumor and 22 normal colon tissues measured
for 6500 human genes. We use the set of 2000 genes identified to have the highest minimal intensity across the 62 tissues. The
Leuk data set [Golub et al., 1999] provides the expression levels of 7129 human genes for 47 samples of patients with Acute
Lymphoblastic Leukemia (ALL) and 25 samples of patients with Acute Myeloid Leukemia (AML). The B MD and C MD
data sets [Pomeroy et al., 2002] are microarray samples containing the expression levels of 7129 human genes. Data set B MD
contains 25 classic and 9 desmoplastic medulloblastomas whereas data set C MD contains 39 medulloblastomas survivors and
21 treatment failures (non-survivors). The Lung dataset consists of gene expression levels of 918 genes of 52 patients with 39
Adenocarcinoma and 13 Squamous Cell Cancer [Garber et al., 2001]. This data has some missing values which were replaced
by zeros. Finally, the BreastER dataset is the Breast Tumor data of West et al. [2001] used with Estrogen Receptor status
to label the various samples. The data consists of expression levels of 7129 genes of 49 patients with 25 positive Estrogen
Receptor samples and 24 negative Estrogen Receptor samples.
The number of examples and the number of genes in each data are given in the “ex” and “Genes” columns respectively under
the “Data Set” tab in each table. The algorithms are referred to as “Occam” (Occam’s Razor), “SC” (Sample Compression) and
“PAC-Bayes” (PAC-Bayes) in Tables II to V. They utilize the respective theoretical frameworks discussed in Sections III, IV
and V along with the respective learning strategies of Section VI.
We have compared our learning algorithm with a linear-kernel soft-margin SVM trained both on all the attributes (gene
expressions) and on a subset of attributes chosen by the filter method of Golub et al. [1999]. The filter method consists of
ranking the attributes as function of the difference between the positive-example mean and the negative-example mean and then
use only the first ℓ attributes. The resulting learning algorithm, named SVM+gs is the one used by Furey et al. [2000] for the
same task. Guyon et al. [2002] claimed obtaining better results with the recursive feature elimination method but, as pointed
out by Ambroise and McLachlan [2002], their work contained a methodological flaw. We use the SVM recursive feature
elimination algorithm with this bias removed and present these results as well for comparison (referred to as “SVM+rfe” in
Table I). Finally, we also compare our results with the state-of-the-art Adaboost algorithm. For this, we use the implementation
in the Weka data mining software [Witten and Frank, 2005].
Each algorithm was tested over 20 random permutations of the datasets, with the 5-fold cross validation (CV) method. Each
of the five training sets and testing sets was the same for all algorithms. The learning parameters of all algorithms and the
gene subsets (for “SVM+gs” and “SVM+rfe”) were chosen from the training sets only. This was done by performing a second
(nested) 5-fold CV on each training set.
For the gene subset selection procedure of SVM+gs, we have considered the first ℓ = 2igenes (for i = 0,1,...,12) ranked
according to the criterion of Golub et al. [1999] and have chosen the i value that gave the smallest 5-fold CV error on the
training set. The “Errs” column under each algorithm in Tables I to III refer to the average (nested) 5-fold cross-validation
error of the respective algorithm with one standard deviation two-sided confidence interval. The “bits” column in Table II

Page 11

11
Data SetPAC-Bayes
G-errs
14.68±1.8
8.89±1.65
23.8±1.7
24.4±1.5
4.4±.6
12.8±.8
Name
Colon
B MD
C MD
Leuk
Lung
BreastER
ex
62
34
60
72
52
49
Genes
2000
7129
7129
7129
918
7129
S
B-errs
14.65±1.8
8.6±1.4
22.9±1.65
23.6±1.6
4.2±.8
12.4±.78
1.53±.28
1.2±.25
3.4±1.8
3.2±1.4
1.2±.3
2.6±1.1
TABLE III
RESULTS OF THE PAC-BAYES LEARNING ALGORITHM ON GENE EXPRESSION DATASETS.
refer to the number of bits used for the Occam’ Razor approach. The “G-errs” and the “B-errs” columns in Table III refer to
the average nested 5-fold CV error of the optimal Gibbs classifier and the corresponding Bayes classifier with one standard
deviation two-sided interval respectively.
For Adaboost, 10, 20, 50, 100, 200, 500, 1000 and 2000 iterations for each datasets were tried and the reported results
correspond to the best obtained 5-fold CV error. The size values reported here (the “S” columns for “SVM+gs”and “SVM+rfe”,
and “Itr” column for “AdaBoost” in Table I) correspond to the number of attributes (genes) selected most frequently by the
respective algorithms over all the permutation runs.5Choosing, by cross-validation, the number of boosting iteration is somewhat
inconsistent with Adaboost’s goal of minimizing the empirical exponential risk. Indeed, to comply with Adaboost’s goal, we
should choose a large-enough number of boosting rounds that assures the convergence of the empirical exponential risk to its
minimum value. However, as shown by Zhang and Yu [2005], Boosting is known to overfit when the number of attributes
exceeds the number of examples. This happens in the case of microarray experiments frequently where the number of genes
far exceeds the number of samples, and is also the case in the datasets mentioned above. Early stopping is the recommended
approach in such cases and hence we have followed the method described above to obtained the best number of boosting
iterations.
Further, Table IV gives the result for a single run of the deterministic algorithm using the fixed-margin heuristic described
above. Table V gives the results for the PAC-Bayes bound values for the results obtained for a single run of the PAC-Bayes
algorithm on the respective microarray data sets. Recall that the PAC-Bayes bound provides a uniform upper bound on the
risk of the Gibbs classifier. The column labels refer to the same quantities as above although the errors reported are over a
single nested 5-fold CV run. The “Ratio” column of Table V refers to the average value of (bk− ak)/(Bk− Ak) obtained
over the decision stumps used by the classifiers over 5 testing folds and the “Bound” columns of Tables IV and V refer to the
average risk bound of Theorem 5 multiplied by the total number of examples in respective data sets. Note, again, that these
results are on a single permutation of the datasets and are presented just to illustrate the practicality of the risk bound and the
rationale of not choosing the fixed-margin heuristic over the current learning strategy.
A. A Note on the Risk Bound
Note that the risk bounds are quite effective and their relevance should not be misconstrued by observing the results in just
the current scenario. One of the most limiting factor in the current analysis is the unavailability of microarray data with larger
number of examples. As the number of examples increase, the risk bound of Theorem 5 gives tighter guarantees. Consider,
for instance, if the datasets for the Lung and Colon Cancer had 500 examples. A classifier with the same performance over
500 examples (i.e. with the same classification accuracy and number of features as currently) would have a bound of about 12
and 30 percent error instead of current 34.6 and 54.6 percent respectively. This only illustrates how the bound can be more
effective as a guarantee when used on datasets with more examples. Similarly, a dataset of 1000 examples for Breast Cancer
5There were no close ties with classifiers with fewer genes.
Data Set
Stumps:PAC-Bayes(fixed margin)
Size
Errors
1 14
17
328
221
29
3 11
Name
Colon
B MD
C MD
Leuk
Lung
BreastER
ex
62
34
60
72
52
49
Genes
2000
7129
7129
7129
918
7129
Bound
34
20
48
46
29
31
TABLE IV
RESULTS OF THE PAC-BAYES APPROACH WITH FIXED-MARGIN HEURISTIC ON GENE EXPRESSION DATASETS.

Page 12

12
Data SetStumps:PAC-Bayes
Size
G-errs
1 12
17
5 21
322
13
2 11
Name
Colon
B MD
C MD
Leuk
Lung
BreastER
ex
62
34
60
72
52
49
Genes
2000
7129
7129
7129
918
7129
Ratio
0.42
0.10
0.08
0.002
0.12
0.09
B-errs
11
7
20
21
3
11
Bound
33
20
45
48
18
29
TABLE V
AN ILLUSTRATION OF THE PAC-BAYES RISK BOUND ON A SAMPLE RUN OF THE PAC-BAYES ALGORITHM.
with a similar performance can have a bound of about 30 percent instead of current 63 percent. Hence, the current limitation
in the practical application of the bound comes from limited data availability. As the number of examples increase, the bounds
provides tighter guarantees and become more significant.
VIII. ANALYSIS
The results clearly show that even though “Occam” and “SC” are able to find sparse classifiers (with very few genes), they
are not able to obtain acceptable classification accuracies. One possible explanation is that these two approaches focus on
the most succinct classifier with their respective criterion. The Sample compression approach tries to minimize the number of
genes used but does not take into account the magnitude of the separating margin and hence compromises accuracy. On the
other hand, the Occam’s Razor approach tries to find a classifier that depends on margin only indirectly. Approaches based on
sample compression as well as minimum description length have shown encouraging results in various domains. An alternate
explanation for their suboptimal performance here can be seen in terms of extremely limited sample sizes. As a result, the
gain in accuracy does not offset the cost of adding additional features in the conjunction. The PAC-Bayes approach seems to
alleviate these problems by performing a significant margin-sparsity tradeoff. That is, the advantage of adding a new feature is
seen in terms of a combination of the gain in both margin and the empirical risk. This can be compared to the strategy used
by the regularization approaches. The classification accuracy of PAC-Bayes algorithm is competitive with the best performing
classifier but has an added advantage, quite importantly, of using very few genes.
For the PAC-Bayes approach, we expect the Bayes classifier to generally perform better than the Gibbs classifier. This is
reflected to some extent in the empirical results for Colon, C MD and Leukaemia datasets. However, there is no means to
prove that this will always be the case. It should be noted that there exist several different utility functions that we can use
for each of the proposed learning approaches. We have tried some of these and reported results only for the ones that were
found to be the best (and discussed in the description of the corresponding learning algorithms).
A noteworthy observation with regard to Adaboost is that the gene subset identified by this algorithm almost always include
the ones found by the proposed PAC-Bayes approach for decision stumps. Most notably, the only gene Cyclin D1, a well
known marker for Cancer, found for the lung cancer dataset is the most discriminating factor and is commonly found by both
approaches. In both cases, the size of the classifier is almost always restricted to 1. These observations not only give insights
into the absolute peaks worth investigating but also experimentally validates the proposed approaches.
Finally, many of the genes identified by the final6PAC-Bayes classifier include some prominent markers for the corresponding
diseases as detailed below.
A. Biological Relevance of the Selected Features
Table VI details the genes identified by the final PAC-Bayes classifier learned over each dataset after the parameter selection
phase. There are some prominent markers identified by the classifier. Some of the main genes identified by the PAC-Bayes
approach are the ones identified by previous studies for each disease— giving confidence in the proposed approach. Some
of the discovered genes in this case include Human monocyte-derived neutrophil-activating protein (MONAP) mRNA in the
case of Colon Cancer dataset and oestrogen receptor in the case of Breast Cancer data, D79205 at-Ribosomal protein L39,
D83542 at-Cadherin-15 and U29195 at-NPTX2 Neuronal pentraxin II in the case of Medulloblastomas datasets B MD and
C MD. Other genes identified have biological relevance, for instance, the identification of Adipsin, LAF-4 and HOX1C with
regard to ALL/AML by our algorithm is in agreement with that of the findings of Chow et al. [2001], Hiwatari et al. [2003]
and Lawrence and Largman [1992] respectively and the studies that followed.
Further, in the case of breast cancer, Estrogen receptors (ER) have shown to interact with BRCA1 to regulate VEGF
transcription and secretion in breast cancer cells [Kawai et al., 2002]. These interactions are further investigated by Ma et al.
[2005]. Further studies for ER have also been done. For instance, Moggs et al. [2005] discovered 3 putative estrogen-response
elements in Keratin6 (the second gene identified by the PAC-Bayes classifier in the case of BreastER data) in the context of
6This is the classifier learned after choosing the best parameters using nested 5-fold CV and trained on the full dataset.

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Dataset
Colon
B MD
C MD
Gene(s) identified by PAC-Bayes Classifier
1. Hsa˙627 M26383-Human monocyte-derived neutrophil-activating protein (MONAP) mRNA
1. D79205 at-Ribosomal protein L39
1. S71824 at-Neural Cell Adhesion Molecule, Phosphatidylinositol-Linked Isoform Precursor
2. D83542 at-Cadherin-15
3. U29195 at-NPTX2 Neuronal pentraxin II
4. X73358 s at-HAES-1 mRNA
5. L36069 at-High conductance inward rectifier potassium channel alpha subunit mRNA
1. M84526 at-DF D component of complement (adipsin)
2. U34360 at-Lymphoid nuclear protein (LAF-4) mRNA
3. M16937 at-Homeo box c1 protein, mRNA
1. GENE221X-IMAGE 841641-cyclin D1 (PRAD1-parathyroid adenomatosis 1) Hs˙82932 AA487486
1. X03635 at,X03635- class C, 20 probes, 20 in all X03635 5885 - 6402
Human mRNA for oestrogen receptor
2. L42611 f at, L42611- class A, 20 probes, 20 in L42611 1374-1954,
Homo sapiens keratin 6 isoform K6e KRT6E mRNA, complete cds
Leuk
Lung
BreastER
TABLE VI
GENES IDENTIFIED BY THE Final PAC-BAYES CLASSIFIER
E2-responsive genes identified by microarray analysis of MDA-MD-231 cells that re-express ERα. An important role played
by cytokeratins in cancer development is also widely known (see for instance Gusterson et al. [2005]).
Furthermore, the importance of MONAP in the case of colon cancer and Adipsin in the case of leukaemia data has further
been confirmed by various rank based algorithms as detailed by Su et al. [2003] in the implementation of “RankGene”, a
program that analyzes and ranks genes for the gene expression data using eight ranking criteria including Information Gain
(IG), Gini Index (GI), Max Minority (MM), Sum Minority (SM), Twoing Rule (TR), t-statistic (TT), Sum of variances (SV)
and one-dimensional Support Vector Machine (1S). In the case of Colon Cancer data, MONAP is identified as the top ranked
gene by four of the eight criteria (IG, SV, TR, GI), second by one (SM), eighth by one (MM) and in top 50 by 1S. Similarly,
in the case of Leukaemia data, Adipsin is top ranked by 1S, fifth by SM, seventh by IG, SV, TR, GI and MM and is in top
50 by TT. These observations provides a strong validation for our approaches.
Cyclin as identified in the case of Lung Cancer dataset is a well known marker for cell division whose perturbations are
considered to be one of the major factors causing cancer [Driscoll et al., 1999, Masaki et al., 2003].
Finally, the discovered genes in the case of Medulloblastomas are important with regard to the neuronal functioning (esp.
S71824, U29195 and L36039) and can have relevance for nervous system related tumors.
IX. CONCLUSION
Learning from high-dimensional data such as that from DNA microarrays can be quite challenging especially when the aim
is to identify only a few attributes that characterizes the differences between two classes of data. We investigated the premise of
learning conjunctions of decision stumps and proposed three formulations based on different learning principles. We observed
that the approaches that aim solely to optimize sparsity or the message code with regard to the classifier’s empirical risk limits
the algorithm in terms of its generalization performance, at least in the present case of small dataset sizes. By trading-off the
sparsity of the classifier with the separating margin in addition to the empirical risk, the PAC-Bayes approach seem to alleviate
this problem to a significant extent. This allows the PAC-Bayes algorithm to yield competitive classification performance while
at the same time utilizing significantly fewer attributes.
As opposed to the traditional feature selection methods, the proposedapproaches are accompaniedby a theoretical justification
of the performance. Moreover, the proposed algorithms embed the feature selection as a part of the learning process itself.7
Furthermore, the generalization error bounds are practical and can potentially guide the model (parameter) selection. When
applied to classify DNA microarray data, the genes identified by the proposed approaches are found to be biologically significant
as experimentallyvalidated by various studies, an empirical justification that the approaches can successfully perform meaningful
feature selection. Consequently, this represents a significant improvement in the direction of successful integration of machine
learning approaches for use in high-throughput data to provide meaningful, theoretically justifiable, and reliable results. Such
approaches that yield a compressed view in terms of a small number of biological markers can lead to a targeted and well
focussed study of the issue of interest. For instance, the approach can be utilized in identifying gene subsets from the microarray
experiments that should be further validated using focused RT-PCR techniques which are otherwise both costly and impractical
to perform on the full set of genes.
Finally, as mentioned previously, the approaches presented in this wor have a wider relevance, and can have significant
implications in the direction of designing theoretically justified feature selection algorithms. These are one of the few approaches
that combines the feature selection with the learning process and provide generalization guarantees over the resulting classifiers
simultaneously. This property assumes even more significance in the wake of limited size of microarray datasets since it limits
the amount of empirical evaluation that can be reliably performed otherwise. Most natural extensions of the approaches and
the learning bias proposed here would be in other similar domains including other forms of microarray experiments such as
7Note that Huang and Chang [2007] proposed one such approach. However, they need multiple SVM learning runs. Hence, their method basically works
as a wrapper.

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Chromatin Immunoprecipitation promoter arrays (chIP-Chip) and from Protein arrays. Within the same learning settings, other
learning biases can also be explored such as classifiers represented by features or sets of features built on subsets of attributes.
ACKNOWLEDGMENT
This work was supported by the National Science and Engineering Research Council (NSERC) of Canada [Discovery Grant
No. 122405 to MM], the Canadian Institutes of Health Research [operating grant to JC, training grant to MS while at CHUL]
and the Canada Research Chair in Medical Genomics to JC.
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