Journal of Machine Learning Research 2 (2001) 265-292

Submitted 03/01; Published 12/01

On the Algorithmic Implementation ofMulticlass Kernel-based Vector Machines
Koby Crammer
Yoram Singer

kobics@cs.huji.ac.il
singer@cs.huji.ac.il

School of Computer Science & Engineering
Hebrew University, Jerusalem 91904, Israel

Editors: Nello Cristianini, John Shawe-Taylor and Bob Williamson

Abstract
In this paper we describe the algorithmic implementation of multiclass kernel-based vector
machines. Our starting point is a generalized notion of the margin to multiclass problems.
Using this notion we cast multiclass categorization problems as a constrained optimization
problem with a quadratic objective function. Unlike most of previous approaches which
typically decompose a multiclass problem into multiple independent binary classiﬁcation
tasks, our notion of margin yields a direct method for training multiclass predictors. By
using the dual of the optimization problem we are able to incorporate kernels with acompact set of constraints and decompose the dual problem into multiple optimization
problems of reduced size. We describe an eﬃcient ﬁxed-point algorithm for solving the
reduced optimization problems and prove its convergence. We then discuss technical details
that yield signiﬁcant running time improvements for large datasets. Finally, we describe
various experiments with our approach comparing it to previously studied kernel-based
methods. Our experiments indicate that for multiclass problems we attain state-of-the-art
accuracy.
Keywords: Multiclass problems, SVM, Kernel Machines

1. Introduction
Supervised machine learning tasks often boil down to the problem of assigning labels to
instances where the labels are drawn from a ﬁnite set of elements. This task is referred
to as multiclass learning. Numerous specialized algorithms have been devised for multiclass problems by building upon classiﬁcation learning algorithms for binary problems, i.e.,
problems in which the set of possible labels is of size two. Notable examples for multiclasslearning algorithms are the multiclass extensions for decision tree learning (Breiman et al.,1984, Quinlan, 1993) and various specialized versions of boosting such as AdaBoost.M2 and
AdaBoost.MH (Freund and Schapire, 1997, Schapire and Singer, 1999). However, the dominating approach for solving multiclass problems using support vector machines has been
based on reducing a single multiclass problems into multiple binary problems. For instance,
a common method is to build a set of binary classiﬁers where each classiﬁer distinguishes
between one of the labels to the rest. This approach is a special case of using output codesfor solving multiclass problems (Dietterich and Bakiri, 1995). However, while multiclasslearning using output codes provides a simple and powerful framework it cannot capturec
2001
Koby Crammer and Yoram Singer.

Crammer and Singer

correlations between the diﬀerent classes since it breaks a multiclass problem into multiple
independent binary problems.
In this paper we develop and discuss in detail a direct approach for learning multiclass
support vector machines (SVM). SVMs have gained an enormous popularity in statistics,
learning theory, and engineering (see for instance Vapnik, 1998, Schölkopf et al., 1998, Cristianini and Shawe-Taylor, 2000, and the many references therein). With a few exceptions
most support vector learning algorithms have been designed for binary (two class) problems. A few attempts have been made to generalize SVM to multiclass problems (Weston
and Watkins, 1999, Vapnik, 1998). These attempts to extend the binary case are acheived
by adding constraints for every class and thus the size of the quadratic optimization is
proportional to the number categories in the classiﬁcation problems. The result is often ahomogeneous quadratic problem which is hard to solve and diﬃcult to store.
The starting point of our approach is a simple generalization of separating hyperplanes
and, analogously, a generalized notion of margins for multiclass problems. This notion of amargin has been employed in previous research (Allwein et al., 2000) but not in the context
of SVM. Using the deﬁnition of a margin for multiclass problems we describe in Section 3 acompact quadratic optimization problem. We then discuss its dual problem and the form of
the resulting multiclass predictor. In Section 4 we give a decomposition of the dual problem
into multiple small optimization problems. This decomposition yields a memory and time
eﬃcient representation of multiclass problems. We proceed and describe an iterative solution
for the set of the reduced optimization problems. We ﬁrst discuss in Section 5 the means ofchoosing which reduced problem to solve on each round of the algorithm. We then discussin Section 6 an eﬃcient ﬁxed-point algorithm for ﬁnding an approximate solution for the
reduced problem that was chosen. We analyze the algorithm and derive a bound on its rateof convergence to the optimal solution. The baseline algorithm is based on a main loop whichis composed of an example selection for optimization followed by an invocation of the ﬁxedpoint algorithm with the example that was chosen. This baseline algorithm can be usedwith small datasets but to make it practical for large ones, several technical improvements
had to be sought. We therefore devote Section 7 to a description of the diﬀerent technicalimprovements we have taken in order to make our approach applicable to large datasets. We
also discuss the running time and accuracy results achieved in experiments that underscore
the technical improvements. In addition, we report in Section 8 the results achieved in
evaluation experiments, comparing them to previous work. Finally, we give conclusions in
Section 9.
Related work Naturally, our work builds on previous research and advances in learning
using support vector machines. The space is clearly too limited to mention all the relevant
work, and thus we refer the reader to the books and collections mentioned above. As wehave already mentioned, the idea of casting multiclass problems as a single constrained optimization with a quadratic objective function was proposed by Vapnik (1998), Weston andWatkins (1999), Bredensteiner and Bennet (1999), and Guermeur et. al (2000). However,the size of the resulting optimization problems devised in the above papers is typically large
and complex. The idea of breaking a large constrained optimization problem into small
problems, where each of which employs a subset of the constraints was ﬁrst explored in
the context of support vector machines by Boser et al. (1992). These ideas were further
266

Multiclass Kernel-based Vector Machinesdeveloped by several researchers (see Joachims, 1998 for an overview). However, the rootsof this line of research go back to the seminal work of Lev Bregman (1967) which was further developed by Yair Censor and colleagues (see Censor and Zenios, 1997 for an excellent
overview). These ideas distilled in Platt’s method, called SMO, for sequential minimal optimization. SMO works with reduced problems that are derived from a pair of examples
while our approach employs a single example for each reduced optimization problem. The
result is a simple optimization problem which can be solved analytically in binary classiﬁcation problems (see Platt, 1998) and leads to an eﬃcient numerical algorithm (that is
guaranteed to converge) in multiclass settings. Furthermore, although not explored in thispaper, it deems possible that the single-example reduction can be used in parallel applications. Many of the technical improvements we discuss in this paper have been proposed
in previous work. In particular ideas such as using a working set and caching have been
described by Burges (1998), Platt (1998), Joachims (1998), and others. Finally, we would
like to note that this work is part of a general line of research on multiclass learning wehave been involved with. Allwein et al. (2000) described and analyzed a general approachfor multiclass problems using error correcting output codes (Dietterich and Bakiri, 1995).
Building on that work, we investigated the problem of designing good output codes for multiclass problems (Crammer and Singer, 2000). Although the model of learning using output
codes diﬀers from the framework studied in this paper, some of the techniques presented inthis paper build upon results from an earlier paper (Crammer and Singer, 2000). Finally,some of the ideas presented in this paper can also be used to build multiclass predictors
in online settings using the mistake bound model as the means of analysis. Our current
research on multiclass problems concentrates on analogous online approaches (Crammer
and Singer, 2001).

2. Preliminaries
Let S = {(x̄1 , y1 ), . . . , (x̄m , ym )} be a set of m training examples. We assume that eachexample x̄i is drawn from a domain X ⊆ n and that each label yi is an integer from theset Y = {1, . . . , k}. A (multiclass) classiﬁer is a function H : X → Y that maps an instancex̄ to an element y of Y. In this paper we focus on a framework that uses classiﬁers of theformkHM (x̄) = arg max{M̄r · x̄} ,r=1

where M is a matrix of size k × n over  and M̄r is the rth row of M. We interchangeably
call the value of the inner-product of the rth row of M with the instance x̄ the conﬁdence
and the similarity score for the r class. Therefore, according to our deﬁnition above, thepredicted label is the index of the row attaining the highest similarity score with x̄. Thissetting is a generalization of linear binary classiﬁers. Using the notation introduced above,
linear binary classiﬁers predict that the label of an instance x̄ is 1 if w̄· x̄ > 0 and 2 otherwise(w̄ · x̄ ≤ 0). Such a classiﬁer can be implemented using a matrix of size 2 × n where M̄1 = w̄and M̄2 = −w̄. Note, however, that this representation is less eﬃcient as it occupies twice
the memory needed. Our model becomes parsimonious when k ≥ 3 in which we maintain k
prototypes M̄1 , M̄2 , . . . , M̄k and set the label of a new input instance by choosing the index
of the most similar row of M.
267

Crammer and Singer

Figure 1: Illustration of the margin bound employed by the optimization problem.
Given a classiﬁer HM (x̄) (parametrized by a matrix M) and an example (x̄, y), we saythat HM (x̄) misclassiﬁes an example x̄ if HM (x̄) = y. Let [[π]] be 1 if the predicate π holds
and 0 otherwise. Thus, the empirical error for a multiclass problem is given bym

1 [[HM (xi ) = yi ]] .
S (M ) =
m

(1)

i=1

Our goal is to ﬁnd a matrix M that attains a small empirical error on the sample S andalso generalizes well. Direct approaches that attempt to minimize the empirical error arecomputationally expensive (see for instance Höﬀgen and Simon, 1992, Crammer and Singer,
2000). Building on Vapnik’s work on support vector machines (Vapnik, 1998), we describe
in the next section our paradigm for ﬁnding a good matrix M by replacing the discrete
empirical error minimization problem with a quadratic optimization problem. As we see
later, recasting the problem as a minimization problem also enables us to replace innerproducts of the form ā · b̄ with kernel-based inner-products of the form K(ā, b̄) = φ̄(ā) · φ̄(b̄).

3. Constructing multiclass kernel-based predictors
To construct multiclass predictors we replace the misclassiﬁcation error of an example,([[HM (x) = y]]), with the following piecewise linear bound,max{M̄r · x̄ + 1 − δy,r } − M̄y · x̄ ,rwhere δp,q is equal 1 if p = q and 0 otherwise. The above bound is zero if the conﬁdence
value for the correct label is larger by at least one than the conﬁdences assigned to the restof the labels. Otherwise, we suﬀer a loss which is linearly proportional to the diﬀerence
between the conﬁdence of the correct label and the maximum among the conﬁdences of theother labels. A graphical illustration of the above is given in Figure 1. The circles in theﬁgure denote diﬀerent labels and the correct label is plotted in dark grey while the restof the labels are plotted in light gray. The height of each label designates its conﬁdence.
Three settings are plotted in the ﬁgure. The left plot corresponds to the case when themargin is larger than one, and therefore the bound maxr {M̄r · x̄ + 1 − δy,r } − M̄y · x̄ equals
zero, and hence the example is correctly classiﬁed. The middle ﬁgure shows a case where
the example is correctly classiﬁed but with a small margin and we suﬀer some loss. The
right plot depicts the loss of a misclassiﬁed example.
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Multiclass Kernel-based Vector Machines

Summing over all the examples in S we get an upper bound on the empirical loss,
1 
max{M̄r · x̄i + 1 − δyi ,r } − M̄yi · x̄i .
rmmS (M ) ≤

(2)

i=1

We say that a sample S is linearly separable by a multiclass machine if there exists a
matrix M such that the above loss is equal to zero for all the examples in S, that is,∀i max{M̄r · x̄i + 1 − δyi ,r } − M̄yi · x̄i = 0 .
r(3)Therefore, a matrix M that satisﬁes Eq. (3) would also satisfy the constraints,∀i, r M̄yi · x̄i + δyi ,r − M̄r · x̄i ≥ 1 .

(4)of the vector represented by the conDeﬁne the l2 -norm of a matrix M to be the l2 -norm
2 . Note that if the constraints
catenation of M’s rows, M 22 = (M̄1 , . . . , M̄k )22 = i,j Mi,jgiven by Eq. (4) are satisﬁed, we can make the diﬀerences between M̄yi · x̄i and M̄r · x̄i arbitrarily large. Furthermore, previous work on the generalization properties of large margin
DAGs (Platt et al., 2000) for multiclass problems showed that the generalization properties
depend on the l2 -norm of M (see also Crammer and Singer, 2000). We therefore would liketo seek a matrix M of a small norm that satisﬁes Eq. (4). When the sample S is linearly
separable by a multiclass machine, we seek a matrix M of the smallest norm that satisﬁes
Eq. (4). The result is the following optimization problem,1
M 222
subject to : ∀i, r M̄yi · x̄i + δyi ,r − M̄r · x̄i ≥ 1 .
min
M

(5)

Note that m of the constraints for r = yi are automatically satisﬁed since,M̄yi · x̄i + δyi ,yi − M̄yi · x̄i = 1 .
This property is an artifact of the separable case. In the general case the sample S might
not be linearly separable by a multiclass machine. We therefore add slack variables ξi ≥ 0
and modify Eq. (3) to be,∀i max{M̄r · x̄i + 1 − δyi ,r } − M̄yi · x̄i = ξi .
r

(6)

We now replace the optimization problem deﬁned by Eq. (5) with the following primal
optimization problem,
m

min
M,ξ


1
βM 22 +
ξi
2

(7)

i=1

subject to : ∀i, r M̄yi · x̄i + δyi ,r − M̄r · x̄i ≥ 1 − ξi .
where β > 0 is a regularization constant and for r = yi the inequality constraints becomeξi ≥ 0. This is an optimization problem with “soft” constraints. We would like to note in269

Crammer and Singer

passing that it is possible to cast an analogous optimization problem with “hard” constraints
as in (Vapnik, 1998).
To solve the optimization problem we use the Karush-Kuhn-Tucker theorem (see for
instance Vapnik, 1998, Cristianini and Shawe-Taylor, 2000). We add a dual set of variables,
one for each constraint and get the Lagrangian of the optimization problem,
m

1 
β
M̄r 22 +
ξi
2 r
i=1



ηi,r M̄r · x̄i − M̄yi · x̄i − δyi ,r + 1 − ξi+L(M, ξ, η) =

(8)

i,r

∀i, r ηi,r ≥ 0 .

subject to :We now seek a saddle point of the Lagrangian, which would be the minimum for the primal
variables {M, ξ} and the maximum for the dual variables η. To ﬁnd the minimum over the
primal variables we require,∂
L=1−
ηi,r = 0
∂ξi
r



⇒

ηi,r = 1 .

(9)rSimilarly, for M̄r we require,




 
∂
L =
ηi,r x̄i −
ηi,q x̄i + β M̄r
∂ M̄r
q
i
i,y =ri=ηi,r x̄i −

i



=1

δyi r x̄i + β M̄r = 0 ,iwhich results in the following formM̄r = β −1(δyi ,r − ηi,r )x̄i

.

(10)

i

Eq. (10) implies that the solution of the optimization problem given by Eq. (5) is a matrix
M whose rows are linear combinations of the instances x̄1 . . . x̄m . Note that from Eq. (10)we get that the contribution of an instance x̄i to M̄r is δyi ,r − ηi,r . We say that an examplex̄i is a support pattern if there is a row r for which this coeﬃcient is not zero. For eachrow M̄r of the matrix M we can partition the patterns with nonzero coeﬃcients into two
subsets by rewriting Eq. (10) as follows,(1 − ηi,r )x̄i +(−ηi,r )x̄i  .
M̄r = β −1 i:yi =ri:yi =r

The ﬁrst sum is over all patterns that belong to the rth class. Hence, an example x̄i labeled
yi = r is a support pattern only if ηi,r = ηi,yi < 1. The second sum is over the rest of thepatterns whose labels are diﬀerent from r. In this case, an example x̄i is a support pattern
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Multiclass Kernel-based Vector Machines

only if ηi,r > 0. Put another way, since
for each pattern x̄i the set {ηi,1 , ηi,2 , . . . , ηi,k } satisﬁes
the constraints ηi,1 , . . . , ηi,k ≥ 0 and r ηi,r = 1, each set can be viewed as a probability
distribution over the labels {1 . . . k}. Under this probabilistic interpretation an examplex̄i is a support pattern if and only if its corresponding distribution is not concentrated on
the correct label yi . Therefore, the classiﬁer is constructed using patterns whose labels are
uncertain; the rest of the input patterns are ignored.
Next, we develop the Lagrangian using only the dual variables by substituting Eqs. (9)and (10) into Eq. (8). Since the derivation is rather technical we defer the complete derivation to App. A. We obtain the following objective function of the dual program,
1 −1 
(x̄i · x̄j )(δyi ,r − ηi,r )(δyj ,r − ηj,r ) −ηi,r δyi ,r .
Q(η) = − β
2
r
i,j

i,r

Let 1̄i be the vector whose components are all zero except for the ith component which isequal to one, and let 1̄ be the vector whose components are all one. Using this notation we
can rewrite the dual program in the following vector form,
max
η



 
1
Q(η) = − β −1(x̄i · x̄j ) (1̄yi − η̄i ) · (1̄yj − η̄j ) −η̄i · 1̄yi
2i,jsubject to : ∀i : η̄i ≥ 0

(11)

i

η̄i · 1̄ = 1 .

and

It is easy to verify that Q(η) is concave in η. Since the set of constraints is convex, there
is a unique maximum value of Q(η). To simplify the problem we now perform the followingchange of variables. Let τ̄i = 1̄yi − η̄i be the diﬀerence between the point distribution 1̄yiconcentrating on the correct label and the distribution η̄i obtained by the optimization
problem. Then Eq. (10) that describes the form of M becomes,τi,r x̄i .
(12)M̄r = β −1iSince we search for the value of the variables which maximize the objective function Q (andnot the optimum value of Q itself), we can omit any additive and positive multiplicative
constants and write the dual problem given by Eq. (11) as,maxτQ(τ ) = −1(x̄i · x̄j )(τ̄i · τ̄j ) + β
τ̄i · 1̄yi
2i,jsubject to : ∀i τ̄i ≤ 1̄yiand

τ̄i · 1̄ = 0 .

Finally, we rewrite the classiﬁer H(x̄) in terms of the variable τ ,




k 
k
τi,r (x̄i · x̄) .
H(x̄) = arg max M̄r · x̄ = arg max
r=1

(13)

i

r=1

(14)

i

As in Support Vector Machines (Cortes and Vapnik, 1995), the dual program and theresulting classiﬁer depend only on inner products of the form (x̄i · x̄). Therefore, we canperform inner-product calculations in some high dimensional inner-product space Z by271

Crammer and Singer

replacing the inner-products in Eq. (13) and in Eq. (14) with a kernel function K(·, ·)that satisﬁes Mercer’s conditions (Vapnik, 1998). The general dual program using kernel
functions is therefore,maxτQ(τ ) = −1
K (x̄i , x̄j ) (τ̄i · τ̄j ) + β
τ̄i · 1̄yi
2i,jsubject to : ∀i τ̄i ≤ 1̄yi

(15)

i

and

τ̄i · 1̄ = 0 ,and the classiﬁcation rule H(x̄) becomes,kH(x̄) = arg max
r=1




τi,r K (x̄, x̄i )

.

(16)

i

Therefore, constructing a multiclass predictor using kernel-based inner-products is as simple
as using standard inner-products.
Note the classiﬁer of Eq. (16) does not contain a bias parameter br for r = 1 . . . k.
Augmenting these terms will add m more equality constraints to the dual optimization
problem, increasing the complexity of the optimization problem. However, one can always
use inner-productsof the form K(ā, b̄) + 1 which is equivalent of using bias parameters, and1  2adding 2 β r br to the objective function.
Also note that in the special case of k = 2 Eq. (7) reduces to the primal program of SVM
by setting w̄ = M̄1 − M̄2 and C = β −1 . As mentioned above, Weston and Watkins (1999)
also developed a multiclass version for SVM. Their approach compared the conﬁdence M̄y · x̄of the correct label to the conﬁdences of all the other labels M̄r ·x̄ and therefore used m(k−1)
slack variables in the primal problem. In contrast, in our framework the conﬁdence of thecorrect label is compared to the highest similarity-score among the rest of the labels and uses
only m slack variables in the primal program. As we describe in the sequel our compact
formalization leads to a memory and time eﬃcient algorithm for the above optimization
problem.

4. Decomposing the optimization problem
The dual quadratic program given by Eq. (15) can be solved using standard quadratic
programming (QP) techniques. However, since it employs mk variables, converting thedual program given by Eq. (15) into a standard QP form yields a representation that
employs a matrix of size mk × mk, which leads to a very large scale problem in general.
Clearly, storing a matrix of that size is intractable for large problems. We now introduce
a simple, memory eﬃcient algorithm for solving the quadratic optimization problem given
by Eq. (15) by decomposing it into small problems.
The core idea of our algorithm is based on separating the constraints of Eq. (15) intom disjoint sets, {τ̄i |τ̄i ≤ 1̄yi , τ̄i · 1̄ = 0}mi=1 . The algorithm we propose works in rounds.
On each round the algorithm chooses a pattern p and improves the value of the objectivefunction by updating the variables τ̄p under the set of constraints, τ̄p ≤ 1̄yp and τ̄p · 1̄ = 0.
Let us ﬁx an example index p and write the objective function only in terms of thevariables τ̄p . For brevity we use Ki,j to denote K (x̄i , x̄j ). We now isolate the contribution
272

Multiclass Kernel-based Vector Machines

Input {(x̄1 , y1 ), . . . , (x̄m , ym )}.
Initialize τ̄1 = 0̄, . . . , τ̄m = 0̄.
Loop:
1. Choose an example p.
2. Calculate the constants for the reduced problem:• Ap = K(x̄p , x̄p )• B̄p = i=p K(x̄i , x̄p )τ̄i − β 1̄yp
3. Set τ̄p to be the solution of the reduced problem :1
Q(τ̄p ) = Ap (τ̄p · τ̄p ) + B̄p · τ̄p2
subject to : τ̄p ≤ 1̄yp and τ̄p · 1̄ = 0kτi,r K (x̄, x̄i ) .
Output : H(x̄) = arg max
min
τp

r=1

i

Figure 2: Skeleton of the algorithm for learning multiclass support vector machine.
of τ̄p in Q.
Qp (τ̄p )

def

=
=


1
Ki,j (τ̄i · τ̄j ) + β
τ̄i · 1̄yi
2
i,j
i

1
− Kp,p (τ̄p · τ̄p ) −Ki,p (τ̄p · τ̄i )
2
−

i=p

=


1 
−
Ki,j (τ̄i · τ̄j ) + β τ̄p · 1̄yp + βτ̄i · 1̄yi
2i=p,j=p
i=p



1
− Kp,p (τ̄p · τ̄p ) − τ̄p · −β 1̄yp +Ki,p τ̄i 
2
i=p




1
Ki,j (τ̄i · τ̄j ) + β
τ̄i · 1̄yi  .
+ −2
i=p,j=p

(17)

i=p

Let us now deﬁne the following variables,
Ap = Kp,p > 0
B̄p = −β 1̄yp +(18)
Ki,p τ̄i

(19)

i=p

Cp = −

1 
Ki,j (τ̄i · τ̄j ) + β
τ̄i · 1̄yi .
2i,j=pi=pUsing the variables deﬁned above the objective function becomes,
1
Qp (τ̄p ) = − Ap (τ̄p · τ̄p ) − B̄p · τ̄p + Cp .
2273

Crammer and Singer

For brevity, let us now omit all constants that do not aﬀect the solution. Each reduced
optimization problem has k variables and k + 1 constraints,
1
Q(τ̄ ) = Ap (τ̄p · τ̄p ) + B̄p · τ̄p
(20)
min
τ
2
subject to : τ̄p ≤ 1̄yp and τ̄p · 1̄ = 0 .
The skeleton of the algorithm is given in Figure 2. The algorithm is initialized with
τ̄i = 0̄ for i = 1 . . . m which, as we discuss later, leads to a simple initialization of internalvariables the algorithm employs for eﬃcient implementation. To complete the details of thealgorithm we need to discuss the following issues. First, we need a stopping criterion forthe loop. A simple method is to run the algorithm for a ﬁxed number of rounds. A better
approach which we discuss in the sequel is to continue iterating as long as the algorithm
does not meet a predeﬁned accuracy condition. Second, we need a scheme for choosing
the pattern p on each round which then induces the reduced optimization problem given in
Eq. (20). Two commonly used methods are to scan the patterns sequentially or to choose apattern uniformly at random. In this paper we describe a scheme for choosing an example
p in a greedy manner. This scheme appears to perform better empirically than other naive
schemes. We address these two issues in Section 5.
The third issue we need to address is how to solve eﬃciently the reduced problem given
by Eq. (20). Since this problem constitutes the core and the inner-loop of the algorithm wedevelop an eﬃcient method for solving the reduced quadratic optimization problem. Thismethod is more eﬃcient than using the standard QP techniques, especially when it suﬃces
to ﬁnd an approximation to the optimal solution. Our specialized solution enables us tosolve problems with a large number of classes k when a straightforward approach could not
be applicable. This method is described in Section 6.

5. Example selection for optimization
To remind the reader, we need to solve Eq. (15),1min
Q(τ ) =
Ki,j (τ̄i · τ̄j ) − β
τ̄i · 1̄yiτ2i,jsubject to : ∀i τ̄i ≤ 1̄yiiandτ̄i · 1̄ = 0 ,where as before Ki,j = K (x̄i , x̄j ). We use the Karush-Kuhn-Tucker theorem (see Cristianiniand Shawe-Taylor, 2000) to ﬁnd the necessary conditions for a point τ to be an optimumof Eq. (15). The Lagrangian of the problem is,


1
Ki,j
τi,r τj,r − βτi,r δyi ,r(21)
L(τ, u, v) =
2
r
i,j
i,r
 

ui,r (τi,r − δyi ,r ) −
vi
τi,r
+
i,r

i

r

∀i, r ui,r ≥ 0 .

subject to :The ﬁrst condition is,∂L =Ki,j τj,r − βδyi ,r + ui,r − vi = 0 .
∂τi,r
j

274

(22)

Multiclass Kernel-based Vector Machines

Let us now deﬁne the following auxiliary set of variables,Ki,j τj,r − βδyi ,r .
Fi,r =(23)j

For each instance x̄i , the value of Fi,r designates the conﬁdence in assigning the label r tox̄i . A value of β is subtracted from the correct label conﬁdence in order to obtain a marginof at least β. Note that from Eq. (19) we get,Fp,r = Bp,r + kp,p τp,r .

(24)

We will make use of this relation between the variables F and B in the next section inwhich we discuss an eﬃcient solution to the quadratic problem.
Taking the derivative with respect to the dual variables of the Lagrangian given by
Eq. (21) and using the deﬁnition of Fi,r from Eq. (23) and KKT conditions we get thefollowing set of equality constraints on a feasible solution for the quadratic optimization
problem,∀i, rFi,r + ui,r = vi ,

(25)

∀i, r

ui,r (τi,r − δyi ,r ) = 0 ,

(26)

∀i, r

ui,r ≥ 0 .

(27)

We now further simplify the equations above. We do so by considering two cases. The ﬁrst
case is when τi,r = δyi ,r . In this case Eq. (26) holds automatically. By combining Eq. (27)and Eq. (25) we get that,
Fi,r ≤ vi .
(28)
In the second case τi,r < δyi ,r . In order for Eq. (26) to hold we must have ui,r = 0. Thus,
using Eq. (25) we get that,
Fi,r = vi .
We now replace the single equality constraint with the following two inequalities,
Fi,r ≥ vi and Fi,r ≤ vi .

(29)

To remind the reader, the constraints on τ̄ from the optimization problem given by Eq. (15)
imply that for all i, τ̄i ≤ 1̄yi and τ̄i · 1̄ = 0. Therefore, if these constraints are satisﬁed there
must exist at least one label r for which τi,r < δyi ,r . We thus get that vi = maxr Fi,r . Notealso that if τ̄i = 0 then Fi,yi = vi = maxr Fi,r and Fi,yi is the unique maximum. We now
combine the set of constraints from Eqs. (28) and (29) into a single inequality,
max Fi,r ≤ vi ≤r

minr : τi,r <δyi ,rFi,r .

(30)

Finally, dropping vi we obtain,
max Fi,r ≤rminr : τi,r <δyi ,rFi,r .

(31)

We now deﬁne,ψi = max Fi,r −rminr : τi,r <δyi ,r275

Fi,r .

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Crammer and Singer

Since maxr Fi,r ≥ minr : τi,r <δyi ,r Fi,r then the necessary and suﬃcient condition for a feasible vector τ̄i to be an optimum for Eq. (15) is that, ψi = 0. In the actual numerical
implementation it is suﬃcient to ﬁnd τ̄i such that ψi ≤  where  is a predeﬁned accuracy
parameter. We therefore keep performing the main loop of Figure 2 so long as there areexamples (x̄i , yi ) whose values ψi are greater than .
The variables ψi also serve as our means for choosing an example for an update. In
our implementation we try to keep the memory requirements as small as possible and thus
manipulate a single example on each loop. We choose the example index p for which ψp is
maximal. We then ﬁnd the vector τ̄p which is the (approximate) solution of the reduced
optimization problem given by Eq. (15). Due to the change in τ̄p we need to update Fi,rand ψi for all i and r. The pseudo-code describing this process is deferred to the next
section in which we describe a simple and eﬃcient algorithm for ﬁnding an approximate
solution for the optimization problem of Eq. (15). Lin (2001) showed that this scheme
does converge to the solution in a ﬁnite number of steps. Finally, we would like to notethat some of the underlying ideas described in this section have been also explored by
Keerthi and Gilbert (2000).

6. Solving the reduced optimization problem
The core of our algorithm relies on an eﬃcient method for solving the reduced optimization
given by Eq. (15) or the equivalent problem as deﬁned by Eq. (20). In this section we
describe an eﬃcient ﬁxed-point algorithm that ﬁnds an approximate solution to Eq. (20).
We would like to note that an exact solution can also be derived. In (Crammer and Singer,
2000) we described a closely related algorithm for solving a similar quadratic optimization
problem in the context of output coding. A simple modiﬁcation of the algorithm can beused here. However, the algorithm needs to sort k values on each iteration and thus might
be slow when k is large. Furthermore, as we discuss in the next section, we found empiricallythat the quality of the solution is quite insensitive to how well we fulﬁll the Karush-KuhnTucker condition by bounding ψi . Therefore, it is enough to ﬁnd a vector τ̄p that decreases
signiﬁcantly the value of Q(τ̄ ) but is not necessarily the optimal solution.
We start by rewriting Q(τ̄ ) from Eq. (20) using a completion to quadratic form anddropping the pattern index p,
1
Q(τ̄ ) = − A(τ̄ · τ̄ ) − B̄ · τ̄
2
1
B̄
B̄
B̄ · B̄= − A[(τ̄ + ) · (τ̄ + )] +
.
2
A
A
2A
We now perform the following change of variables,B̄B̄+ 1̄y .
D̄ =
(33)
A
A
At this point, we omit additive constants and the multiplicative factor A since they do notaﬀect the value of the optimal solution. Using the above variable the optimization problem
from Eq. (20) now becomes,ν̄ = τ̄ +minν

Q(ν̄) = ν̄2subject to : ν̄ ≤ D̄ and ν̄ · 1̄ = D̄ · 1̄ − 1 .
276

(34)

Multiclass Kernel-based Vector Machines

We would like to note that sinceFi,r = Bi,r + Ai τi,r ,we can compute ψi from B̄i and thus need to store either Bi,r or Fi,r .
Let us denote by θ and αi the variables of the dual problem of Eq. (34). Then, theKarush-Kuhn-Tucker conditions imply that,∀r νr ≤ Dr ; αr (νr − Dr ) = 0 ; νr + αr − θ = 0 .

(35)

Note that since αr ≥ 0 the above conditions imply that νr ≤ θ for all r. Combining thisinequality with the constraint that νr ≤ Dr we get that the solution satisﬁesνr ≤ min{θ, Dr } .

(36)

If αr = 0 we get that νr = θ and if αr > 0 we must have that νr = Dr . Thus, Eq. (36)holds with equality, namely, the solution is of the form, νr = min{θ, Dr }. Now, sinceν̄ · 1̄ = D̄ · 1̄ − 1 we get that θ satisﬁes the following constraint,k
r=1

min{θ, Dr } =rkDr − 1 .

(37)

r=1


The above equation uniquely deﬁnes θ since the sum kr=1 minr {θ, Dr } is a strictly mono
tone and continuous function in θ. For θ = maxr Dr we have that kr=1 minr {θ, Dr } >kk
r=1 Dr − 1 while
r=1 minr {θ, Dr } → −∞ as θ → −∞. Therefore, there always exists aunique value θ that satisﬁes Eq. (37). The following theorem shows that θ is indeed the
optimal solution of the quadratic optimization problem.
Theorem 1 Let νr∗ = min{θ , Dr } where θ is the solution of
1. Then, for every point ν̄ we have that ν̄2 > ν̄ ∗ 2 .

k

r=1 minr {θ, Dr } =k

r=1 Dr −¯ whichProof Assume by contradiction that there is another feasible point ν̄ = ν̄ ∗ + ∆¯ = 0. Both ν̄ and ν̄ ∗ satisfy
minimizes the objective function. Since ν̄ = ν̄ ∗ we know that ∆the equality constraint of Eq. (34), thus r ∆r = 0. Also, both points satisfy the inequality
constraint of Eq. (34) thus ∆r ≤ 0 when νr∗ = Dr . Combining the last two equations with
¯ = 0 we get that ∆s > 0 for some s with νs∗ = θ. Using again thethe assumptionthat ∆
equality r ∆r = 0 we have that there exists an index u with ∆u < 0. Let us denote by = min{|∆s |, |∆u |}. We now deﬁne a new feasible point ν̄  as follows. Let νs = νs − ,νu = νu + , and νr = νr otherwise. We now show that the norm of ν̄  is smaller than thenorm of ν̄. Since ν̄ and ν̄  diﬀer only in their s and u coordinates, we have that,ν̄  2 − ν̄2 = (νs )2 + (νu )2 − (νs )2 − (νu )2 .
Writing the values of ν̄  in terms of ν̄ and  we get,ν̄  2 − ν̄2 = 2 ( − νs + νu ) .
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Crammer and Singer

From our construction of ν̄  we have that either νu −  = θ > νs or νu −  > θ ≥ νs andtherefore we get νu −  > νs . This implies thatν̄  2 − ν̄2 < 0 ,which is clearly a contradiction.
We now use the above characterization of the solution to derive a simple ﬁxed-point
algorithm that ﬁnds θ . We use the simple identity min{θ, Dr } + max{θ, Dr } = θ + Dr andreplace the minimum function with the above sum in Eq. (37) and get,k[θ + Dr − max{θ, Dr }] =r=1

which amounts to,kDr − 1 ,

r=1


k
1 
1
∗
max{θ , Dr } − .
θ =
k
k
∗

(38)

r=1

Let us deﬁne,


k
1
1 
max{θ, Dr } − .
F (θ) =
k
k

(39)

r=1

Then, the optimal value θ∗ satisﬁesθ∗ = F (θ∗ ) .

(40)

Eq. (40) can be used for the following iterative algorithm. The algorithm starts with an
initial value of θ and then computes the next value using Eq. (39). It continues iteratingby substituting each new value of θ in F (·) and producing a series of values for θ. The
algorithm halts when a required accuracy is met, that is, when two successive values of θ
are close enough. A pseudo-code of the algorithm is given in Figure 3. The input to thealgorithm is the vector D̄, an initial suggestion for θ, and a required accuracy . We next
show that if θ1 ≤ maxr Dr then the algorithm does converge to the correct value of θ∗ .
Theorem 2 Let θ∗ be the ﬁxed point of Eq. (40) (θ ∗ = F (θ∗ )). Assume that θ1 ≤ maxr Dr
and let θl+1 = F (θl ). Then for l ≥ 1|θl+1 − θ∗ |
1
≤1− ,
∗
|θl − θ |kwhere k is the number of classes.
def

Proof Assume without loss of generality that maxr Dr = D1 ≥ D2 ≥ . . . ≥ Dk ≥ Dk+1 =−∞. Also assume that θ∗ ∈ (Ds+1 , Ds ) and θl ∈ (Du+1 , Du ) where u, s ∈ {1, 2, . . . , k}.
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Multiclass Kernel-based Vector Machines

FixedPointAlgorithm(D̄, θ, )Input D̄, θ1 , .
Initialize l = 0.
Repeat• l ← l + 1.

k
max{θ
,
D
}
− k1 .
• θl+1 ← k1
r
l
r=1


 θl − θl+1  ≤ .
Until θlAssign for r = 1, . . . , k: νr = min{θl+1 , Dr }Return: τ̄ = ν̄ − B̄A.
Figure 3: The ﬁxed-point algorithm for solving the reduced quadratic program.
Thus,θl+1 = F (θl )


k
1
1 
max{θl , Dr } −
=
k
k
r=1


 k
 u

1 
1
1
θl +
Dr −
=
k
k
k
r=u+1
r=1

 u

u
1 
=
1−
Dr − 1 .
θl +
k
k

(41)

r=1

Note that if θl ≤ maxr Dr then θl+1 ≤ maxr Dr . Similarly,
 s




s
1
θ∗ = F (θ∗ ) = 1 −Dr − 1
θ∗ +
k
k
r=1
 s

1 
Dr − 1 .
⇒ θ∗ =
s

(42)

r=1

We now need to consider three cases depending on the relative order of s and u. The ﬁrst
case is when u = s. In this case we get that,
| 1 − ks θl + k1 ( sr=1 Dr − 1) − θ∗ ||θl+1 − θ∗ |=|θl − θ∗ ||θl − θ∗ |
| 1 − ks θl + ks θ∗ − θ∗ |=|θl − θ∗ |1s= 1− ≤1− .
kkwhere the second equality follows from Eq. (42). The second case is where u > s. In thiscase we get that for all r = s + 1, . . . , u :θ l ≤ Dr ≤ θ ∗ .
279

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Crammer and Singer

Using Eq. (41) and Eq. (42) we get,
θl+1 =




=

=

u
1
θl +
1−
k
k

 u


s1
u
θl +
1−
k
ks


Dr − 1

r=1
 s

r=1

s
1
u
1−
θl + θ ∗ +kkkDr − 1



u




1
+
k


Dr

u



Dr

r=s+1

.

r=s+1

Applying Eq. (43) we obtain,


u
s
1
θl + θ∗ + (u − s)θ∗
k
k
k

u
u ∗
θl + θ .
=1−kkθl+1 ≤

1−

Since θl+1 is bounded by a convex combination of θl and θ∗ , and θ∗ is larger than θl , thenθ∗ ≥ θl+1 . We therefore ﬁnally get that,|θl+1 − θ∗ ||θl − θ∗ |

θ∗ − θl+1
θ ∗ − θlθ∗ − 1 − uk θl − uk θ∗≤θ ∗ − θlu
1= 1− ≤1− .
kk=

The last case, where u < s, is derived analogously to the second case, interchanging theroles of u and s.
From the proof we see that the best convergence rate is obtained for large values of u. Thus,
a good feasible initialization for θ1 can be minr Dr . In this case
 k

1
1 
Dr − .
θ2 = F (θ1 ) =kk
r=1This gives a simple initialization of the algorithm which ensures that the initial rate ofconvergence will be fast.
We are now ready to describe the complete implementation of the algorithm for learning
multiclass kernel machine. The algorithm gets a required accuracy parameter and the valueof β. It is initialized with τ̄i = 0 for all indices 1 ≤ i ≤ m. This value yields a simple
initialization of the variables Fi,r . On each iteration we compute from Fi,r the value ψi foreach example and choose the example index p for which ψp is the largest. We then call theﬁxed-point algorithm which in turn ﬁnds an approximate solution to the reduced quadratic
optimization problem for the example indexed p. The ﬁxed-point algorithm returns a setof new values for τ̄p which triggers the update of Fi,r . This process is repeated until thevalue ψi is smaller than  for all 1 ≤ i ≤ m. The pseudo-code of the algorithm is given in
Figure 4.
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Multiclass Kernel-based Vector Machines

Input {(x̄1 , y1 ), . . . , (x̄m , ym )}.
Initialize for i = 1, . . . , m:
• τ̄i = 0̄
• Fi,r = −βδr,yi (r = 1 . . . k)
• Ai = K(x̄i , x̄i )Repeat:ψi = max Fi,r −• Calculate for i = 1 . . . m:
• Set:rmin

r : τi,r <δyi ,rFi,rp = arg max{ψi }F• Call:τ̄p =FixedPointAlgorithm(D̄, θ, /2).

(See Figure 3)• Set:∆τ̄p = τ̄p − τ̄p− τp,r + δr,yp and θ = k1
• Set for r = 1 . . . k : Dr = Ap,rpk
r=1 Dr− k1• Update for i = 1 . . . m and r = 1 . . . k: Fi,r ← Fi,r + ∆τp,r K (x̄p , x̄i )• Update: τ̄p ← τ̄p
Until ψp < βOutput : H(x̄) = arg maxrτi,r K (x̄, x̄i ) .

iFigure 4: Basic algorithm for learning a multiclass, kernel-based, support vector machineusing KKT conditions for example selection.

7. Implementation details
We have discussed so far the underlying principal and algorithmic issues that arise in thedesign of multiclass kernel-based vector machines. However, to make the learning algorithm
practical for large datasets we had to make several technical improvements to the baseline
implementation. While these improvements do not change the underlying design principals
they lead to a signiﬁcant improvement in running time. We therefore devote this section toa description of the implementation details. To compare the performance of the diﬀerent
versions presented in this section we used the MNIST OCR dataset1 . The MNIST dataset
contains 60, 000 training examples and 10, 000 test examples and thus can underscore significant implementation improvements. Before diving into the technical details we would liketo note that many of the techniques are by no means new and have been used in prior implementation of two-class support vector machines (see for instance Platt, 1998, Joachims,
1998, Collobert and Bengio, 2001). However, a few of our implementation improvementsbuild on the speciﬁc algorithmic design of multiclass kernel machines.
1. Available at http://www.research.att.com/˜yann/exdb/mnist/index.html

281

Crammer and Singer

4

x 10

1.90

1.25

1.85

1.20

1.80

1.15

1.75

1.10

1.70

Test Error

Run Time (sec)

1.30

1.05
1.00

1.65
1.60

0.95

1.55

0.90

1.50

0.85

1.45

0.80

−4

10

−3

10

−2

10
epsilon

−1

10

1.40

0

10

−4

10

−3

10

−2

10
epsilon

−1

10

0

10

Figure 5: The run time (left) and test error (right) as a function of required accuracy .
Our starting point and base-line implementation is the algorithm described in Figure 2
combined with the ﬁxed-point algorithm for solving the reduced quadratic optimization
problem. In the base-line implementation we simply cycle through the examples for aﬁxed number of iterations, solving the reduced optimization problem, ignoring the KKT
conditions for the examples. This scheme is very simple to implement and use. However, it
spends unnecessary time in the optimization of patterns which are correctly classiﬁed witha large conﬁdence. We use this scheme to illustrate the importance of the eﬃcient example
selection described in the previous section. We now describe the diﬀerent steps we took,
starting from the version described in Section 5.
Using KKT for example selection This is the algorithm described in Figure 4. For
each example i and label r we compute Fi,r . These variables are used to compute ψi as
described in Section 5. On each round we choose the example p for which ψp is the largest
and iterate the process until the value of ψi is smaller for a predeﬁned accuracy denotedby . It turns out that the choice of  is not crucial and a large range of values yield goodresults. The larger  is the sooner we terminate the main loop of the algorithm. Therefore,
we would like to set  to a large value as long as the generalization performance is not
eﬀected. In Figure 5 we show the running time and the test error as a function of . The
results show that a moderate value of  of 0.1 already yields good generalization. The
increase in running time when using smaller values for  is between %20 to %30. Thus, thealgorithm is rather robust to the actual choice of the accuracy parameter  so long as it is
not set to a value which is evidently too large.
Maintaining an active set The standard implementation described above scans theentire training set and computes ψi for each example x̄i in the set. However, if only a few
support patterns constitute the multiclass machine then the vector τ̄ is the zero vector for
many example. We thus partition the set of examples into two sets. The ﬁrst, denotedby A and called the active set, is composed of the set of examples that contribute to thesolution, that is, A = {i|τ̄i = 0̄}. The second set is simply its complement, Ac = {i|τ̄i = 0̄}.
During the course of the main loop we ﬁrst search for an example to update from the set A.
Only if such an example does not exist, which can happen iﬀ ∀i ∈ A, ψi < , we scan the282

Multiclass Kernel-based Vector Machines−4

Objective function value

−10

−3

−10

−2

−10

w/o coolingwith cooling

1

2

10

10

Iteration

Figure 6: The value of the objective function Q as a function of the number of iteration fora ﬁxed and variable scheduling of the accuracy parameter .

set Ac for an example p with ψp > . If such an example exists we remove it from Ac , add
it to A, and call the ﬁxed-point algorithm with that example. This procedure spends mostof it time adjusting the weights of examples that constitute the active set and adds a new
example only when the active set is exhausted. A natural implication of this procedure isthat the support patterns can come only from the active set.
Cooling of the accuracy parameter The employment of an active set yields signiﬁcantreduction in running time. However, the scheme also forces the algorithm to keep updating
the vectors τ̄i for i ∈ A as long there is even a single example i for which ψi > . This may
result in minuscule changes and a slow decrease in Q once most examples in A have been
updated. In Figure 6 we plot in bold line the value of Q as a function of the number of
iterations when  is kept ﬁxed. The line has a staircase-like shape. Careful examination of
the iterations in which there was a signiﬁcant drop in Q revealed that these are the iterations
on which new examples were added to the active set. After each addition of a new example
numerous iterations are spent in adjust the weights τ̄i . To accelerate the process, especiallyon early iterations during which we mostly add new examples to the active set, we use avariable accuracy parameter, rather than a ﬁxed accuracy. On early iterations the accuracy
value is set to a high value so that the algorithm will mostly add new examples to the active
set and spend only a small time on adjusting the weights of the support patterns. As thenumber of iterations increases we decrease  and spend more time on adjusting the weights
of support patterns. The result is a smoother and more rapid decrease in Q which leads
to faster convergence of the algorithm. We refer to this process of gradually decreasing
283

Crammer and Singer

5

10

4

10

3

Run Time

10

2

10

1

10

1
2
3
4
5

0

10

2

10

3

10
Size of Training Set

4

10

Figure 7: Comparison of the run-time on the MNIST dataset of the diﬀerent versions as afunction of the training-set size. Version 1 is the baseline implementation. Version
2 uses KKT conditions for selecting an example to update. Version 3 adds theusage of an active set and cooling of . Version 4 adds caching of inner-products.
Finally, version 5 uses data structures for representing and using sparse inputs.

 as cooling. We tested the following cooling schemes (for t = 0, 1, . . .): (a) exponential:
(t) = 0 exp(−t); (b) linear: (t) = 0 /(t + 1) (c) logarithmic: (t) = 0 / log10 (t + 10). The
initial accuracy 0 was set to 0.999. We found that all of these cooling schemes improve therate of decrease in Q, especially the logarithmic scheme for which (t) is relatively large fora long period and than decreases moderately. The dashed line in Figure 6 designate thevalue of Q as a function of the number of iterations using a logarithmic cooling scheme for. In the particular setting of the ﬁgure, cooling reduces the number of iterations, and thus
the running time, by an order of magnitude.
Caching Previous implementations of algorithms for support vector machines employ
a cache for saving expensive kernel-based inner-products (see for instance Platt, 1998,
Joachims, 1998, Collobert and Bengio, 2001). Indeed, one of the most expensive steps
in the algorithm is the evaluation of the kernel. Our scheme for maintaining a cache is asfollows. For small datasets we store in the cache all the kernel evaluations between eachexample in the active set and all the examples in the training set. For large problems withmany support patterns (and thus a large active set) we use a least-recently-used (LRU)
scheme as a caching strategy. In this scheme, when the cache is full we replace least used
inner-products of an example with the inner-products of a new example. LRU caching is
also used in SVMlight (Joachims, 1998).
284

Multiclass Kernel-based Vector MachinesNamesatimage
shuttlemnist
isoletlettervowel
glass

No. ofTraining Examples
4435
5000
5000
6238
5000
528
214

No. of
Test Examples
2000
9000
10000
1559
4000
462
5-fold cval

No. ofClasses
6
7
10
26
26
11
7

No. ofAttributes
36
9
784
617
16
10
9

Table 1: Description of the small databases used in the experiments.

Data-structures for sparse input instances Platt (1998) and others have observedthat when many components of the input vectors are zero, a signiﬁcant saving of space
and time can be achieved using data-structures for sparse vectors and computing only the
products of the non-zero components in kernel evaluations. Our implementation uses linked
list for sparse vectors. Experimental evaluation we performed indicate that it is enough tohave 20% sparseness of the input instances to achieve a speedup in time and reduction inmemory over a non-sparse implementation.
To conclude this section we give in Figure 7a comparison of the run-time of the various technical improvements we outlined above. Each version that we plot include all of
the previous improvements. The running-time of the version that includes all the algorithmic and implementation improvements is two orders of magnitude faster than the baseline
implementation. It took the fastest version 3 hours and 25 minutes to train a multiclasskernel-machine on the MNIST dataset using a Pentium III computer running at 600MHz
with 2Gb of physical memory. (The fastest version included two more technical improvements which are not discussed here but will be documented in the code that we will shortly
make available.) In comparison, Platt (1998) reports a training time of 8 hours and 10
minutes for learning a single classiﬁer that discriminates one digit from the rest. Therefore,
it takes over 80 hours to train a set of 10 classiﬁers that constitute a multiclass predictor.
Platt’s results were obtained using a Pentium II computer running at 266MHz. While there
are many diﬀerent factors that inﬂuence the running time, the running time results above
give an indication of the power of our approach for multiclass problems. We believe thatthe advantage of our direct approach to multiclass problems will become even more evident
in problems with a large number of classes such as Kanji character recognition. We would
also like to note that the improved running time is not achieved at the expense of deteriorated classiﬁcation accuracy. In the next section we describe experiments that show that
our approach is comparable to, and sometimes better than, the standard support vector
technology.

8. Experiments
In this section we report the results of experiments we conducted in order to evaluate
our implementation of the multiclass support vector machine described in this paper. We
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Crammer and Singer

7

Decrease in error rate

6

5

4

3

2

1

0

satimage

shuttlemnist

isoletletter

vowel

glass

Figure 8: Comparison of a multiclass SVM build using the one-against-rest approach withthe multiclass support vector machines studied in this paper.

selected seven datasets, six from UCI repository 2 and the MNIST dataset. All but one of thedatasets (glass), contain a separate test set. For the remaining dataset we used ﬁve foldcross validation to compute the error rate. A description of the datasets we used is given in
Table 1. Note that on MNIST, Letter and Shuttle we used only a subset of the training
set. On each data set we ran two algorithms. The ﬁrst algorithm uses the multiclass SVMof this paper in a binary mode by training one classiﬁer for each class. Each such classiﬁer istrained to distinguish between the class to the rest of the classes. To classify a new instancewe compute the output of each of the binary classiﬁers and predict the label which attains
the highest conﬁdence value. The second algorithm we compared is the multiclass SVM
described in this paper. We used our multiclass SVM as the basis for the two algorithms inorder to have a common framework for comparison. In both algorithms we used Gaussian
kernels. To determine the value of β and σ we used cross validation on the training set. We
used 5-fold cross validation for the large datasets and 10-fold cross validation for the small
datasets. In all the experiments we set the value of  to 0.001.
A summary of the results is depicted in Figure 8. Each bar in the ﬁgure is proportional
to the diﬀerence in the test error between the two algorithms. Positive value means that thealgorithm proposed in this paper achieved a lower error rate than the strawman algorithm
based on the ‘one-vs-rest’ approach. In general the multiclass support vector machine
achieved lower error rate than the ‘one-vs-rest’ method where for the datasets with a large
example per class ratio this improvement is signiﬁcant.
2. Available at http://www.ics.uci.edu/˜mlearn/MLRepository.html

286

Multiclass Kernel-based Vector Machines

1 ( 3:0.16 7:0.26 9:0.57 )8 ( 2:0.46 3:0.00 6:0.54 )7 ( 2:0.33 4:0.32 9:0.35 )1 ( 7:1.00 )4 ( 9:1.00 )4 ( 1:0.79 9:0.21 )4 ( 7:1.00 ) E

1 ( 7:1.00 )7 ( 1:1.00 )9 ( 0:0.22 3:0.54 8:0.24 )4 ( 9:1.00 )4 ( 1:0.66 9:0.34 )

4 ( 1:0.97 5:0.03 )3 ( 8:0.74 9:0.26 )1 ( 7:1.00 )4 ( 1:0.21 9:0.79 )7 ( 1:0.48 3:0.04 8:0.09 9:0.39 )4 ( 9:1.00 )8 ( 1:0.22 7:0.10 9:0.68 )4 ( 1:0.67 2:0.06 8:0.27 )9 ( 4:1.00 )7 ( 9:1.00 )8 ( 9:1.00 )9 ( 7:1.00 )7 ( 1:0.21 9:0.79 )9 ( 1:0.83 3:0.17 )1 ( 6:1.00 )9 ( 1:1.00 )9 ( 7:1.00 )4 ( 7:0.21 9:0.79 )4 ( 1:1.00 )7 ( 3:0.17 4:0.03 9:0.80 )9 ( 4:1.00 )9 ( 4:1.00 )1 ( 4:0.62 7:0.14 9:0.24 )5 ( 6:0.80 8:0.20 )5 ( 6:1.00 )1 ( 6:1.00 )4 ( 5:0.13 8:0.12 9:0.75 )9 ( 4:1.00 )9 ( 7:0.22 8:0.78 )7 ( 1:1.00 )4 ( 9:1.00 ) E

9 ( 4:1.00 )7 ( 9:1.00 )9 ( 7:1.00 )6 ( 2:0.21 4:0.79 )4 ( 9:1.00 )8 ( 1:1.00 )7 ( 1:1.00 )9 ( 4:1.00 )8 ( 0:0.02 7:0.65 9:0.34 )9 ( 0:0.12 7:0.88 )9 ( 1:0.12 4:0.70 7:0.18 )Figure 9: Examples of images which are support patterns with a large norm of τ̄ (MNIST
dataset).

287

Crammer and Singer

Name
mnist
USPS
shuttleletterTest error(%)
1.42
4.38
0.12
1.95

Table 2: Summary of experiments with large datasets.
We also ran the multiclass SVM on the complete training set of MNIST, Letter and
Shuttle and on the USPS 3 dataset. The results are summarized in Table 2. Schölkopf etal. (1997) achieved a test error of 1.4% on the MNIST dataset and a test error of 4.2% on
the USPS (Schölkopf et al., 1996) using a set of binary SVMs training by the ‘one-vs-rest’
method. In another work, Dietterich (2000) achieved a test error of 2.71% on the Letter
dataset and a test error of 0.01% on the Shuttle dataset by boosting C4.5. Note thatin the ﬁrst three cases our algorithm is comparable to the other algorithms, while it is 10
times worse on the Shuttle dataset. This gap can be explained by the fact that most of theinstances in the dataset belong to the same class, which is a phenomena that is easier forboosting algorithms to cope with. Finally, we would like to note that using the polynomial
kernel of degree 9 and the normalization described by DeCoste and Schölkopf (2001) we
achieve an error rate of 1.24% on the test set of MNIST.
In Figure 9 we give training images taken from the MNIST dataset with high norm ofτ̄ . All the images we plot attain a margin value of less than β. For each image we giveits correct label and in parentheses the probability vector η̄ using a sparse representation:
each entry of the vector is of the form label:probability. If there is a training error on animage we added a trailing E in the text. As discussed in the previous sections, the vectorη̄ is a probability vector that is concentrated on the corrected label if it is not a support
pattern. Therefore, all the above images are support pattern and the vectors η̄ designate theset of confusable labels – the labels for which the corresponding entries in η̄ are non-zero.
Examination of the diﬀerent images reveal that the confusion is indeed correlated with theactual digit each training example can be confused with. For instance, the top image on theleft hand-side from Figure 9 can be mistakenly classiﬁed as 3, 7, or 9. In many cases, theexamples with a large norm of τ̄ correspond to mislabeled examples or corrupted images.
Therefore, the norm of τ̄ can be used as an aid for data cleaning.

9. Summary
In this paper we presented a novel approach for building multiclass support vector machines.
We described in detail an eﬃcient learning algorithm and discussed various implementation
issues required for making it practical. Our methods achieve state of the art results witha running time that is competitive with previous methods, and is one order of magnitude
faster in some problems. An interesting questions that stem from our research is whether
the approach taken in this paper can also be used for other machine learning tasks withkernels, such as ranking problems and regression. We leave this for future research. Another
3. Available at ftp.kyb.tuebingen.mpg.de288

Multiclass Kernel-based Vector Machinesinteresting direction that we are currently pursuing is analogous online algorithms that use
multiple prototypes for multiclass problems.

Acknowledgments
We would like to thank Elisheva Bonchek for useful comments and carefully reading themanuscript. Thanks also to Chih-Jen Lin for fruitful email exchange.

Appendix A. Derivation of the dual optimization problem
We develop the Lagrangian using only the dual variables. For simplicity we use the identity
Kernel K(xi , xj ) = xi · xj . Substituting Eq. (9) to Eq. (8) we obtain,
Q(η) =

m


ξi +



i=1

ηi,r xi · M̄r −i,rηi,r xi · M̄yi −



i,r

ξi



i

ηi,r +

r


i,r

1 
ηi,r bi,r + βM̄r 22
2 r

=1
def

=



def

= S3

def

= S1

ηi,r xi · M̄r −

i,r


i,r

= S2


1 
ηi,r xi · M̄yi + βM̄r 22 +ηi,r bi,r .
2 r

(44)

i,r

We substitute M̄r in the above equation using Eq. (10) and get,S1 =ηi,r xi · M̄r
i,r

=


i,r

= β −1S2 =

ηi,r xi · β −1


=

i,r

= β −1xi · xjηi,r (δyj ,r − ηj,r ) ,ηi,r xi · M̄yi
ηi,r xi · β −1xj (δyj ,yi − ηj,yi )jxi · xjηi,r (δyj ,yi − ηj,yi )rxi · xj (δyj ,yi − ηj,yi )

= β



i,j

S3 =

ηi,r

=1

xi · xj (δyj ,yi − ηj,yi )i,j= β −1


r

i,j
−1

(45)

r

i,j

i,j

= β −1xj (δyj ,r − ηj,r )

j

i,r





xi · xj1 βM̄r · M̄r
2 r

δyi ,r (δyj ,r − ηj,r ) ,

r

289

(46)

Crammer and Singer
1  −1 β[β
xi (δyi ,r − ηi,r )][β −1xj (δyj ,r − ηj,r )]
2 r
i
j

1 −1 
βxi · xj(δyi ,r − ηi,r )(δyj ,r − ηj,r ) .
2
r

=
=

(47)

i,j

Taking the diﬀerence S1 − S2 while using Eqs. (45) and (46) we get,
S1 − S2 = β −1xi · xj
ηi,r (δyj ,r − ηj,r ) − β −1xi · xj
δyi ,r (δyj ,r − ηj,r )ri,j= −β −1xi · xj



i,jr(δyi ,r − ηi,r )(δyj ,r − ηj,r ) .

(48)

r

i,j

Finally, plugging the values for S1 ,S2 and S3 from Eqs. (47) and (48) into Eq. (44) we get,Q(η) = −β −1
xi · xj(δyi ,r − ηi,r )(δyj ,r − ηj,r )
i,j

r




1
xi · xj(δyi ,r − ηi,r )(δyj ,r − ηj,r ) +ηi,r bi,r+ β −1
2
r
i,j
i,r



1
= − β −1xi · xj(δyi ,r − ηi,r )(δyj ,r − ηj,r ) +ηi,r bi,r .
2
r
i,j

i,r

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