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Non-Sparse Regularization and Efficient Training with
Multiple Kernels
Marius Kloft
Ulf Brefeld
Sören Sonnenburg
Alexander Zien
Electrical Engineering and Computer Sciences
University of California at Berkeley
Technical Report No. UCB/EECS-2010-21
http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-21.html
February 24, 2010
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Acknowledgement
The authors wish to thank Pavel Laskov, Motoaki Kawanabe, Vojtech
Franc, Peter Gehler, Gunnar Raetsch, Peter Bartlett and Klaus-Robert
Mueller for fruitful discussions and helpful comments. This work was
supported in part by the German Bundesministerium fuer Bildung und
Forschung (BMBF) under the project REMIND (FKZ 01-IS07007A), by the
German Academic Exchange Service, and by the FP7-ICT Programme of
the European Community, under the PASCAL2 Network of Excellence,
ICT-216886. Soeren Sonnenburg acknowledges financial support by the
German Research Foundation (DFG) under the grant MU 987/6-1 and RA
1894/1-1.
Non-Sparse Regularization and Efficient Training with
Multiple Kernels
Marius Kloft∗mkloft@cs.berkeley.edu
University of California
Computer Science Division
Berkeley, CA 94720-1758, USA
Ulf Brefeld brefeld@yahoo-inc.com
Yahoo! Research
Avinguda Diagonal 177
08018 Barcelona, Spain
S¨oren Sonnenburg∗soeren.sonnenburg@tuebingen.mpg.de
Friedrich Miescher Laboratory
Max Planck Society
Spemannstr. 39, 72076 T¨ubingen, Germany
Alexander Zien zien@lifebiosystems.com
LIFE Biosystems GmbH
Poststraße 34
69115 Heidelberg, Germany
Abstract
Learning linear combinations of multiple kernels is an appealing strategy when the right
choice of features is unknown. Previous approaches to multiple kernel learning (MKL)
promote sparse kernel combinations to support interpretability and scalability. Unfortu-
nately, this `1-norm MKL is rarely observed to outperform trivial baselines in practical
applications. To allow for robust kernel mixtures, we generalize MKL to arbitrary norms.
We devise new insights on the connection between several existing MKL formulations and
develop two efficient interleaved optimization strategies for arbitrary norms, like `p-norms
with p > 1. Empirically, we demonstrate that the interleaved optimization strategies are
much faster compared to the commonly used wrapper approaches. An experiment on con-
trolled artificial data experiment sheds light on the appropriateness of sparse, non-sparse
and `∞MKL in various scenarios. Application of `p-norm MKL to three hard real-world
problems from computational biology show that non-sparse MKL achieves accuracies that
go beyond the state-of-the-art. We conclude that our improvements finally made MKL fit
for deployment to practical applications: MKL now has a good chance of improving the
accuracy (over a plain sum kernel) at an affordable computational cost.
1. Introduction
Kernels allow to decouple machine learning from data. Finding an appropriate data rep-
resentation via a kernel function immediately opens the door to a vast world of powerful
∗. Also at Machine Learning Group, Technische Universit¨at Berlin, Franklinstr. 28/29, FR 6-9, 10587
Berlin, Germany.
1
machine learning models (e.g. Sch¨olkopf and Smola, 2002) with many efficient and reliable
off-the-shelf implementations. This has propelled the dissemination of machine learning
techniques to a wide range of diverse application domains.
Finding an appropriate data abstraction—or even engineering the best kernel—for the
problem at hand is not always trivial, though. Starting with cross-validation (Stone, 1974)
which is probably the most prominent approach to general model selection, a great many
approaches to selecting the right kernel(s) have been deployed in the literature.
Kernel target alignment (Cristianini et al., 2002) aims at learning the entries of a ker-
nel matrix by using the outer product of the label vector as the ground-truth. Chapelle
et al. (2002) and Bousquet and Herrmann (2002) minimize estimates of the generalization
error of support vector machines (SVMs) using a gradient descent algorithm over the set
of parameters. Ong et al. (2005) study hyperkernels on the space of kernels and alterna-
tive approaches include selecting kernels by DC programming (Argyriou et al., 2008) and
semi-infinite programming ( ¨
Oz¨og¨ur-Aky¨uz and Weber, 2008; Gehler and Nowozin, 2008).
Although finding non-linear kernel mixtures (Varma and Babu, 2009) generally results in
non-convex optimization problems, Cortes et al. (2009) show that convex relaxations may
be obtained for special cases.
However, learning arbitrary kernel combinations is a problem too general to allow for
a general optimal solution—by focusing on a restricted scenario, it is possible to achieve
guaranteed optimality. In their seminal work, Lanckriet et al. (2004) consider training an
SVM along with optimizing the linear combination of several positive semi-definite matrices,
K=PM
m=1 θmKm,subject to the trace constraint tr(K)≤cand requiring a valid combined
kernel K0. This spawned the new field of multiple kernel learning (MKL), the automatic
combination of several kernel functions. Lanckriet et al. (2004) show that their specific
version of the MKL task can be reduced to a convex optimization problem, namely a semi-
definite programming (SDP) optimization problem. Though convex, however, the SDP
approach is computationally too expensive for practical applications. Thus much of the
subsequent research focused on devising efficient optimization procedures for learning with
multiple kernels.
One conceptual milestone for developing MKL into a tool of practical utility is simply
to constrain the mixing coefficients θto be non-negative: by obviating the complex con-
straint K0, this small restriction allows one to transform the optimization problem into
a quadratically constrained program, hence drastically reducing the computational burden.
While the original MKL objective is stated and optimized in dual space, alternative formu-
lations have been studied. For instance, Bach et al. (2004) found a corresponding primal
problem, and Rubinstein (2005) decomposed the MKL problem into a min-max problem
that can be optimized by mirror-prox algorithms (Nemirovski, 2004).
The min-max formulation has been independently proposed by Sonnenburg et al. (2005).
They use it to recast MKL training as a semi-infinite linear program. Solving the latter
with column generation (e.g., Nash and Sofer, 1996) amounts to repeatedly training an SVM
on a mixture kernel while iteratively refining the mixture coefficients θ. This immediately
lends itself to a convenient implementation by a wrapper approach. These algorithms di-
rectly benefit from efficient SVM optimization routines (cf., e.g., Fan et al., 2005; Joachims,
1999) and are now commonly deployed in recent MKL solvers (e.g., Rakotomamonjy et al.,
2008; Xu et al., 2009), thereby allowing for large-scale multiple kernel learning training
2
(Sonnenburg et al., 2005, 2006a). However, the complete training of several SVMs can still
be prohibitive for large data sets. For this reason, Sonnenburg et al. (2005) also proposed
to interleave the SILP with the SVM training which reduced the training time drastically.
Alternative optimization schemes include level-set methods (Xu et al., 2009) and second
order approaches (Chapelle and Rakotomamonjy, 2008). Szafranski et al. (2008), Nath
et al. (2009), and Bach (2009) study composite and hierarchical kernel learning approaches.
Finally, Zien and Ong (2007) and Ji et al. (2009) provide extensions for multi-class and
multi-label settings, respectively.
Today, there exist two mayor families of multiple kernel learning models, characterized
either by Ivanov regularization (Ivanov et al., 2002) over the mixing coefficients (Rakotoma-
monjy et al., 2007; Zien and Ong, 2007), or as Tikhonov regularized optimization problem
(Tikhonov and Arsenin, 1977). In the both cases, there may be an additional parameter
controlling the regularization of the mixing coefficients (Varma and Ray, 2007).
All the above mentioned multiple kernel learning formulations promote sparse solutions
in terms of the mixing coefficients. The desire for sparse mixtures originates in practical
as well as theoretical reasons. First, sparse combinations are easier to interpret. Second,
irrelevant (and possibly expensive) kernels functions do not need to be evaluated at testing
time. Finally, sparseness appears to be handy also from a technical point of view, as
the additional simplex constraint kθk1≤1 simplifies derivations and turns the problem
into a linearly constrained program. Nevertheless, sparseness is not always beneficial in
practice. Sparse MKL is frequently observed to be outperformed by a regular SVM using
an unweighted-sum kernel K=PmKm.
Consequently, despite all the substantial progress in the field of MKL, there still remains
an unsatisfied need for an approach that is really useful for practical applications: a model
that has a good chance of improving the accuracy (over a plain sum kernel) together with
an implementation that matches today’s standards (i.e., that can be trained on 10,000s of
data points in a reasonable time). In addition, since the field has grown several competing
MKL formulations, it seems timely to consolidate the set of models.
In this article we argue that all of this is now achievable, at least when considering MKL
restricted to non-negative mixture coefficients. On the theoretical side, we cast multiple
kernel learning as a general regularized risk minimization problem for arbitrary convex loss
functions, Hilbertian regularizers, and arbitrary norm-penalties on θ. We first show that the
above mentioned Tikhonov and Ivanov regularized MKL variants are equivalent in the sense
that they yield the same set of hypotheses. Then we derive a generalized dual and show that
a variety of methods are special cases of our objective. Our detached optimization problem
subsumes state-of-the-art approaches to multiple kernel learning, covering sparse and non-
sparse MKL by arbitrary p-norm regularization (1 ≤p≤ ∞) on the mixing coefficients as
well as the incorporation of prior knowledge by allowing for non-isotropic regularizers. As
we demonstrate, the p-norm regularization includes both important special cases (sparse
1-norm and plain sum ∞-norm) and offers the potential to elevate predictive accuracy over
both of them.
With regard to the implementation, we introduce an appealing and efficient optimization
strategy which grounds on an exact update in closed-form in the θ-step; hence rendering
expensive semi-infinite and first- or second-order gradient methods unnecessary. By uti-
lizing proven working set optimization for SVMs, p-norm MKL can now be trained highly
3
efficiently for all p; in particular, we outpace other current 1-norm MKL implementations.
Moreover our implementation employs kernel caching techniques, which enables training
on ten thousands of data points or thousands of kernels respectively. In contrast, most
competing MKL software require all kernel matrices to be stored completely in memory,
which restricts these methods to small data sets with limited numbers of kernels. Our
implementation is freely available within the SHOGUN machine learning toolbox available
from http://www.shogun-toolbox.org/.
Our claims are backed up by experiments on artificial data and on a couple of real
world data sets representing diverse, relevant and challenging problems from the application
domain bioinformatics. The artificial data enables us to investigate the relationship between
properties of the true solution and the optimal choice of kernel mixture regularization. The
real world problems include the prediction of the subcellular localization of proteins, the
(transcription) starts of genes, and the function of enzymes. The results demonstrate (i)
that combining kernels is now tractable on large data sets, (ii) that it can provide cutting
edge classification accuracy, and (iii) that depending on the task at hand, different kernel
mixture regularizations are required for achieving optimal performance.
The remainder of this paper is structured as follows. We derive the generalized MKL in
Section 2 and discuss relations to existing approaches in Section 3. Section 4 introduces the
novel optimization strategy and shows the applicability of existing optimization techniques
to our generalized formulation. We report on our empirical results in Section 5. Section 6
concludes.
2. Generalized MKL
In this section we cast multiple kernel learning into a unified framework: we present a
regularized loss minimization formulation with additional norm constraints on the kernel
mixing coefficients. We show that it comprises many popular MKL variants currently
discussed in the literature, including seemingly different ones.
We derive generalized dual optimization problems without making specific assumptions
on the norm regularizers or the loss function, beside that the latter is convex. Our formu-
lation covers binary classification and regression tasks and can easily be extended to multi-
class classification and structural learning settings using appropriate convex loss functions
and joint kernel extensions. Prior knowledge on kernel mixtures and kernel asymmetries
can be incorporated by non-isotropic norm regularizers.
2.1 Preliminaries
We begin with reviewing the classical supervised learning setup. Given a labeled sample
D={(xi, yi)}i=1...,n, where the xilie in some input space Xand yi∈ Y ⊂ R, the goal is
to find a hypothesis f∈ H, that generalizes well on new and unseen data. Regularized risk
minimization returns a minimizer f∗,
f∗∈argminfRemp(f) + λΩ(f),
where Remp(f) = 1
nPn
i=1 V(f(xi), yi) is the empirical risk of hypothesis fw.r.t. to a convex
loss function V:R×Y → R, Ω : H → Ris a regularizer, and λ > 0 is a trade-off parameter.
4
We consider linear models of the form
f˜
w,b(x) = h˜
w, ψ(x)i+b, (1)
together with a (possibly non-linear) mapping ψ:X → H to a Hilbert space H(e.g.,
Sch¨olkopf et al., 1998; M¨uller et al., 2001) and constrain the regularization to be of the
form Ω(f) = 1
2||˜
w||2
2which allows to kernelize the resulting models and algorithms. We will
later make use of kernel functions K(x,x0) = hψ(x), ψ(x0)iHto compute inner products in
H.
2.2 Convex Risk Minimization with Multiple Kernels
When learning with multiple kernels, we are given Mdifferent feature mappings ψm:
X → Hm, m = 1,...M, each giving rise to a reproducing kernel Kmof Hm. Convex
approaches to multiple kernel learning consider linear kernel mixtures Kθ=PθmKm,
θm≥0. Compared to Eq. (1), the primal model for learning with multiple kernels is
extended to
f˜
w,b,θ(x) =
M
X
m=1 pθmh˜
wm, ψm(x)iHm+b=h˜
w, ψθ(x)iH+b(2)
where the parameter vector ˜
wand the composite feature map ψθhave a block structure
˜
w= ( ˜
w>
1,..., ˜
w>
M)>and ψθ=√θ1ψ1×. . . ×√θMψM, respectively.
In learning with multiple kernels we aim at minimizing the loss on the training data
w.r.t. to optimal kernel mixture PθmKmin addition to regularizing θto avoid overfitting.
Hence, in terms of regularized risk minimization, the optimization problem becomes
inf
˜
w,b,θ:θ≥0
1
n
n
X
i=1
V M
X
m=1 pθmh˜
wm, ψm(xi)iHm+b, yi!+λ
2
M
X
m=1 ||˜wm||2
Hm+ ˜µ˜
Ω[θ],(3)
for ˜µ > 0. Note that the objective value of Eq. (3) is an upper bound on the training error.
Previous approaches to multiple kernel learning employ regularizers of the form ˜
Ω(θ) = ||θ||1
to promote sparse kernel mixtures. By contrast, we propose to use convex regularizers of
the form ˜
Ω(θ) = ||θ||2, where || · ||2is an arbitrary norm in RM, possibly allowing for
non-sparse solutions and the incorporation of prior knowledge. The non-convexity arising
from the √θm˜
wmproduct in the loss term of Eq. (3) is not inherent and can be resolved by
substituting wm←√θm˜
wm. Furthermore, the regularization parameter and the sample
size can be decoupled by introducing ˜
C=1
nλ (and adjusting µ←˜µ
λ) which has favorable
scaling properties in practice. We obtain the following convex optimization problem (Boyd
and Vandenberghe, 2004) that has also been considered by (Varma and Ray, 2007) for hinge
loss and an `1-norm regularizer
inf
w,b,θ:θ≥0
˜
C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2
M
X
m=1
||wm||2
Hm
θm
+µ||θ||2,(4)
where we use the convention that t
0= 0 if t= 0 and ∞otherwise.
5
An alternative approach has been studied by Rakotomamonjy et al. (2007) and Zien
and Ong (2007), again using hinge loss and `1-norm. They upper bound the value of
the regularizer ||θ||1≤1 and incorporate the latter as an additional constraint into the
optimization problem. For C > 0, they arrive at the following problem which is the
primary object of investigation in this paper.
Primal MKL Optimization Problem
inf
w,b,θ:θ≥0C
n
X
i=1
VM
X
m=1hwm, ψm(xi)iHm+b, yi+1
2
M
X
m=1
||wm||2
Hm
θm
(P)
s.t. ||θ||2≤1.
Our first contribution shows that despite the additional regularization parameter the
Tikhonov regularization in (4) and the Ivanov regularization in Optimization Problem (P)
are equivalent, in the sense that they yield the same binary classification function.
Theorem 1 Let || · || be a norm on RM, be Va convex loss function. Suppose for the
optimal w∗in Optimization Problem (P) it holds w∗6=0. Then, for each pair (˜
C, µ)there
exists C > 0such that for each optimal solution (w, b, θ) of Eq. (4) using (˜
C, µ), we have
that (w, b, κ θ)is also an optimal solution of Optimization Problem (P) using C, and vice
versa, where κ > 0is a multiplicative constant.
For the proof we need Prop. 8, which justifies switching from Ivanov to Tikhonov
regularization, and back, if the regularizer is tight. We refer to Appendix A for formulation
and proof of the proposition.
Proof of Theorem 1 Let be ( ˜
C, µ)>0. In order to apply Prop. 8 to (4), we start by
showing that condition (35) in Prop. 8 is satisfied, i.e., that the regularizer is tight.
Suppose on the contrary, that Optimization Problem (P) yields the same infimum re-
gardless of whether we require
||θ||2≤1,(5)
or not. Then this implies that in the optimal point we have PM
m=1
||w∗
m||2
2
θ∗
m= 0, hence,
||w∗
m||2
2
θ∗
m
= 0 ∀m. (6)
Since all norms on RMare equivalent (cf., e.g., Rudin (1991)), there exists a L < ∞such
that ||θ∗||∞≤L||θ∗||. In particular, we have ||θ∗||∞<∞, from which we conclude by (6),
that wm= 0 holds for all m, which contradicts our assumption.
Hence, Prop. 8 can be applied and which yields that (4) is equivalent to
inf
w,b,θ
˜
C
n
X
i=1
VM
X
m=1hwm, ψm(x)i+b, yi+1
2
M
X
m=1
||wm||2
2
θm
s.t. ||θ||2≤τ,
6
for some τ > 0. Consider the optimal solution (w?, b?,θ?) corresponding to a given
parametrization ( ˜
C, τ ). For any λ > 0, the bijective transformation ( ˜
C, τ )7→ (λ−1/2˜
C, λτ )
will yield (w?, b?, λ1/2θ?) as optimal solution. Applying the transformation with λ:= 1/τ
and setting C=˜
Cτ 1
2as well as κ=τ−1/2yields Optimization Problem (P), which was to
be shown.
Zien and Ong (2007) also showed that the MKL optimization problems by Bach et al.
(2004), Sonnenburg et al. (2006a), and their own formulation are equivalent. As a main
implication of Theorem 1 and by using the result of Zien and Ong it follows that the
optimization problem of Varma and Ray (Varma and Ray, 2007) lies in the same equivalence
class as (Bach et al., 2004; Sonnenburg et al., 2006a; Rakotomamonjy et al., 2007; Zien and
Ong, 2007). In addition, our result shows the coupling between trade-off parameter C
and the regularization parameter µin Eq. (4): tweaking one also changes the other and
vice versa. Theorem 1 implies that optimizing Cin Optimization Problem (P) implicitly
searches the regularization path for the parameter µof Eq. (4). In the remainder, we will
therefore focus on the formulation in Optimization Problem (P), as a single parameter is
preferable in terms of model selection.
2.3 Convex MKL in Dual Space
In this section we study the generalized MKL approach of the previous section in the dual
space. Let us begin with rewriting Optimization Problem (P) by expanding the decision
values into slack variables as follows
inf
w,b,t,θ:θ≥0C
n
X
i=1
V(ti, yi) + 1
2
M
X
m=1
||wm||2
Hm
θm
(7)
s.t. ∀i:
M
X
m=1hwm, ψm(xi)iHm+b=ti;||θ||2≤1,
where ||·|| is an arbitrary norm in Rmand ||·||HMdenotes the Hilbertian norm of Hm. Ap-
plying Lagrange’s theorem re-incorporates the constraints into the objective by introducing
Lagrangian multipliers α∈Rnand β∈R+.1The Lagrangian saddle point problem is
then given by
sup
α,β:β≥0
inf
w,b,t,θ≥0C
n
X
i=1
V(ti, yi) + 1
2
M
X
m=1
||wm||2
Hm
θm
(8)
−
n
X
i=1
αi M
X
m=1hwm, ψm(xi)iHm+b−ti!+β1
2||θ||2−1
2.
1. Note that αis variable over the whole range of Rnsince it is incorporates an equality constraint.
7
Denoting the Lagrangian by Land setting its first partial derivatives with respect to wand
bto 0 reveals the optimality conditions
1>α= 0; (9a)
∀m= 1,··· M:wm=θm
n
X
i=1
αiψm(xi).(9b)
Resubstituting the above equations yields
sup
α:1>α=0, β:β≥0
inf
t,θ≥0C
n
X
i=1
(V(ti, yi) + αiti)−1
2
M
X
m=1
θmα>Kmα+β1
2||θ||2−1
2,
which can also be written in terms of unconstrained θbecause, without loss of generality,
a supremum with respect to θis trivially attained for arbitrary non-negative θ≥0. We
arrive at
sup
α:1>α=0, β≥0−C
n
X
i=1
sup
ti−αi
Cti−V(ti, yi)−βsup
θ 1
2β
M
X
m=1
θmα>Kmα−1
2||θ||2!−1
2β.
As a consequence, we now may express the Lagrangian as2
sup
α:1>α=0, β≥0−C
n
X
i=1
V∗−αi
C, yi−1
2β
1
2α>KmαM
m=1
2
∗−1
2β, (10)
where h∗(x) = supux>u−h(u) denotes the Fenchel-Legendre conjugate of a function h
and ||·||∗denotes the dual norm, i.e., the norm defined via the identity 1
2||·||2
∗:= 1
2|| · ||2∗.
In the following, we call V∗the dual loss. Eq. (10) now has to be maximized with respect
to the dual variables α, β, subject to 1>α= 0 and β≥0. Let us ignore for a moment
the non-negativity constraint on βand solve ∂L/∂β = 0 for the unbounded β. Setting the
partial derivative to zero allows to express the optimal βas
β=
1
2α>KmαM
m=1
∗
.(11)
Obviously, at optimality, we always have β≥0. We thus discard the corresponding
constraint from the optimization problem and plugging Eq. (11) into Eq. (10) results in
the following dual optimization problem which now solely depends on α:
Dual MKL Optimization Problem
sup
α:1>α=0 −C
n
X
i=1
V∗−αi
C, yi−1
2
α>KmαM
m=1
∗
.(D)
2. We employ the notation s= (s1,...,sM)>= (sm)M
m=1 for s∈RM.
8
The above dual generalizes multiple kernel learning to arbitrary convex loss functions
and norms. Note that if the loss function is continuous the supremum is also a maximum.
The threshold bcan be recovered from the solution by applying the KKT conditions.
The above dual can be characterized as follows. We start by noting that the expression
in Optimization Problem (D) is a composition of two terms, firstly, the left hand side
term, which depends on the conjugate loss function V∗, and, secondly, the right hand side
term which depends on the conjugate norm. The right hand side can be interpreted as
a regularizer on the quadratic terms that, according to the chosen norm, smoothens the
solutions. Hence we have a nice decomposition of the dual into a loss term (in terms of
the dual loss) and a regularizer (in terms of the dual norm). For a specific choice of a
pair (V, || ·||) we can immediately recover the corresponding dual by computing the pair of
conjugates (V∗,|| · ||∗). In the next section, this is illustrated by means of well-known loss
functions and regularizers.
3. Instantiations of the Model
In this section we show that existing MKL-based learners are subsumed by the generalized
formulation in Optimization Problem (D).
3.1 Support Vector Machines with Unweighted-Sum Kernels
First we note that the support vector machine with an unweighted-sum kernel can be
recovered as a special case of our model. To see this, we consider the RRM problem using
the hinge loss function V(t, y) = max(0,1−ty) and the regularizer ||θ||∞. We then can
obtain the corresponding dual in terms of Fenchel-Legendre conjugate functions as follows.
We first note that the dual loss of the hinge loss is V∗(t, y) = t
yif −1≤t
y≤0 and
∞elsewise (Rifkin and Lippert, 2007). Hence, for each ithe term V∗−αi
C, yiof the
generalized dual, i.e., Optimization Problem (D), translates to αi
Cyi, provided that 0 ≤αi
yi≤
C. Employing a variable substitution of the form αnew
i=αi
yi, Optimization Problem (D)
translates to
max
α1>α−1
2
α>Y KmYαM
m=1
∗
,s.t. y>α= 0 and 0≤α≤C1,(12)
where we denote Y= diag(y). The primal `∞-norm penalty ||θ||∞is dual to ||θ||1, hence,
via the identity || · ||∗=|| · ||1the right hand side of the last equation translates to
PM
m=1 α>Y KmYα. Combined with (12) this leads to the dual
sup
α
1>α−1
2
M
X
m=1
α>Y KmYα,s.t. y>α= 0 and 0≤α≤C1,
which is precisely an SVM with an unweighted-sum kernel.
3.2 QCQP MKL of Lanckriet et al. (2004)
A common approach in multiple kernel learning is to employ regularizers of the form
Ω = ||θ||1.(13)
9
This so-called `1-norm regularizers are specific instances of sparsity-inducing regularizers.
The obtained kernel mixtures are often sparse and hence equip the MKL problem by the
favor of interpretable solutions. Sparse MKL is a special case of our framework; to see
this, note that the conjugate of (13) is || · ||∞. Recalling the definition of an `p-norm, the
right hand side of Optimization Problem (D) translates to maxm∈{1,...,M}α>Y KmYα. The
maximum can subsequently be expanded into slack variables ξi, resulting in
sup
α,ξ
1>α−ξi
s.t. ∀m:1
2α>Y KmYα≤ξm;y>α= 0 ; 0≤α≤C1,
which is the original QCQP formulation of MKL, firstly given by Lanckriet et al. (2004).
3.3 `p-Norm MKL
The generalized MKL also allows for robust kernel mixtures by employing an `p-norm
constraint with p > 1, rather than an `1-norm constraint, on the mixing coefficients (Kloft
et al., 2009a). The following identity holds
1
2|| · ||2
p∗
=1
2|| · ||2
q,where 1
p+1
q= 1,
and we obtain for the dual norm of the `p-norm: || · ||∗=|| · ||q. This leads to the dual
problem
sup
α:1>α=0−C
n
X
i=1
V∗−αi
C, yi−1
2
α>KmαM
m=1
q
.
In the special case of hinge loss minimization, we obtain the optimization problem
sup
α
1>α−1
2
α>Y KmYαM
m=1
q
,s.t. y>α= 0 and 0≤α≤C1.
It is thereby worth mentioning that the optimality conditions yield the proportionality,
θ∗
m∼(α∗Kmα∗)2
p−1,
as we will show in Sect. 4.1.
3.4 A Smooth Variant of Group Lasso
Yuan and Lin (2006) studied the following optimization problem for the special case Hm=
Rdmand ψm= idRdm, also known as group lasso,
min
w,b
C
2
n
X
i=1 yi−
M
X
m=1hwm, ψm(xi)iHm!2
+1
2
M
X
m=1 ||wm||Hm.(14)
10
Above problem has been solved by active set methods in the primal (Roth and Fischer,
2008). We sketch an alternative approach based on dual optimization. First, we note that
Eq. (14) can be equivalently expressed as (Micchelli and Pontil, 2005a)
inf
w,b,θ:θ≥0
C
2
n
X
i=1 yi−
M
X
m=1hwm, ψm(xi)iHm!2
+1
2
M
X
m=1
||wm||2
Hm
θm
,s.t. ||θ||2
1≤1.
Thus, the dual of V(t, y) = 1
2(y−t)2is V∗(t, y) = 1
2t2+ty and the corresponding group
lasso dual can be written as,
max
αy>α−1
2C||α||2
2−1
2
α>Y KmYαM
m=1
∞
,(15)
which can be expanded into the following QCQP
sup
α,ξ
y>α−1
2C||α||2
2−ξi(16)
s.t. ∀m:1
2α>Y KmYα≤ξm.
For small n, the latter formulation can be handled efficiently by QCQP solvers. However,
the quadratic constraints caused by the non-smooth `∞-norm in the objective still are
computationally too demanding. As a remedy, we propose a smooth and unconstrained
variant based on `p-norms (p > 1), given by
max
αy>α−1
2C||α||2
2−1
2
α>Y KmYαM
m=1
p
,
which can be solved very efficiently by limited memory quasi-Newton descent methods (Liu
and Nocedal, 1989).
3.5 Density Level-Set Estimation
Density level-set estimators are frequently used for anomaly/novelty detection tasks
(Markou and Singh, 2003a,b). Kernel approaches, such as one-class SVMs (Sch¨olkopf
et al., 2001) and Support Vector Domain Descriptions (Tax and Duin, 1999) have been
extended to MKL settings by Sonnenburg et al. (2006a) and Kloft et al. (2008), respec-
tively. One-class MKL can be cast into our framework by employing loss functions of the
form V(t) = max(0,1−t). This gives rise to the primal
inf
w,b,θ:θ≥0C
n
X
i=1
max 0,
M
X
m=1hwm, ψm(xi)iHm!+1
2
M
X
m=1
||wm||2
Hm
θm
,s.t. ||θ||2≤1.
Noting that the dual loss is V∗(t) = tif −1≤t≤0 and ∞elsewise, we obtain the following
generalized dual
sup
α
1>α−1
2
α>KmαM
m=1
q
,s.t. 0≤α≤C1,
which has been studied by Sonnenburg et al. (2006a) for `1-norm and by Kloft et al. (2009b)
for `p-norms.
11
3.6 Non-Isotropic Norms
In practice, it is often desirable for an expert to incorporate prior knowledge about the
problem domain. For instance, an expert could have given an estimate of the interactions
within the set of kernels considered , e.g. in the form of an M×Mmatrix E. Alternatively,
it might be known in advance that a subset of the employed kernels is inferior to the
remaining kernels; for instance, such knowledge could result from previous experiments in
the considered application field. Those scenarios can be easily handled within our framework
by considering non-isotropic regularizers of the form
||θ||E=pθ>Eθwith E0.
The dual norm is again defined via 1
2|| · ||2
∗:= 1
2|| · ||2
E∗and the following easily-to-verify
identity,
1
2|| · ||2
E∗
=1
2|| · ||2
F,
with matrix inverse F=E−1, leads to the dual,
sup
α:1>α=0−C
n
X
i=1
V∗−αi
C, yi−1
2
α>KmαM
m=1
E−1
,
which is the desired non-isotropic MKL problem.
4. Efficient Optimization Strategies
The dual as given in Optimization Problem (D) does not lend itself to efficient large-scale
optimization in a straight-forward fashion, for instance by direct application of standard
appoaches like gradient descent. Instead, it is beneficial to exploit the structure of the MKL
cost function by alternating between optimizing w.r.t. to the mixings θand w.r.t. to the
remaining variables. Most recent MKL solvers (e.g., Rakotomamonjy et al., 2008; Xu et al.,
2009; Varma and Babu, 2009) do so by setting up a two-layer optimization procedure:
a master problem, which is parameterized only by θand independent of θ, is solved to
determine the kernel mixture; to solve this master problem, repeatedly a slave problem
is solved which amounts to training a standard SVM on a mixture kernel. Importantly,
for the slave problem, the mixture coefficients are fixed, such that convential, efficient
SVM optimizers can be recycled. Consequently these two-layer procedures are commonly
implemented as wrapper approaches. Albeit appearing advantageous, wrapper methods
suffer from a few shortcomings: (i) Due to kernel cache limitations, the kernel matrices
have to be pre-computed and stored or many kernel computations have to be carried out
repeatedly, inducing heavy wastage of either memory or time. (ii) The slave problem is
always optimized to the end (and many convergence proofs seem to require this), although
most of the computational time is spend on the non-optimal mixtures. Certainly suboptimal
slave solutions would already suffice to improve far-from-optimal θin the master problem.
Due to these problems, MKL is prohibitive when learning with a multitude of kernels and
on large-scale data sets as commonly encountered in many data-intense real world applica-
tions such as bioinformatics, web mining, databases, and computer security, etc. Therefore
12
all optimization approaches presented in this paper implement a true decomposition of the
MKL problem into smaller subproblems (Platt, 1999; Joachims, 1999; Fan et al., 2005) by
establishing a wrapper-like scheme within the decomposition algorithm.
Our algorithms are embedded into the large-scale framework of Sonnenburg et al. (2006a)
and extend them to optimization of non-sparse kernel mixtures induced by an `p-norm
penalty. Our first strategy alternates between minimizing the primal problem (7) w.r.t. θ
with incomplete optimization w.r.t. all other variables which, however, is performed in
terms of the dual variables α. For the second strategy, we devise a convex semi-infinite
program (SIP), which we solve by column generation with nested sequential quadratically
constrained linear programming (SQCLP). In both cases, optimization w.r.t. αis performed
by chunking optimization with minor iterations. The first, “direct” approach can be applied
without a common purpose QCQP solver. We show convergence of both algorithms: for
the “direct” algorithm in Prop. 5 and convergence of the SQCLP in Prop. 6. All algorithms
are implemented in the SHOGUN machine learning toolbox, which is freely available from
http://www.shogun-toolbox.org/.
4.1 An Analytical Method
In this section we present a simple and efficient optimization strategy for multiple kernel
learning. To derive the new algorithm, we first revisit the primal problem, i.e.
inf
w,b,θ:θ≥0C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2
M
X
m=1
||wm||2
Hm
θm
,s.t. ||θ||2≤1.(P)
In order to obtain an efficient optimization strategy, we divide the variables in the above OP
into two groups, (w, b) on one hand and θon the other. In the following we will derive an
algorithm, which alternatingly operates on those two groups via block coordinate descent
algorithm, also known as the non-linear Gauss-Seidel method. Thereby the optimization
w.r.t. θwill be carried out analytically and the (w, b)-step will be computed in the dual, if
needed.
The basic idea of our first approach is that for a given, fixed set of primal variables (w, b),
the optimal θin the primal problem (P) can be calculated analytically. In the subsequent
derivations we exemplarily employ non-sparse norms of the form ||θ||p= (PM
m=1 θp
m)1/p,
1< p < ∞, but the reasoning—including convergence guarantees—holds for arbitrary
continuously differentiable and strictly convex norms3.
The following proposition gives the an analytic update formula for the θgiven fixed
remaining variables (w, b) and will become the core of our proposed algorithm.
Proposition 2 Let Vbe a convex loss function, be p > 1. Given fixed (w, b), the optimal
solution of Optimization Problem (P) is attained for
θ∗
m=||w∗
m||
2
p+1
Hm
PM
m0=1 ||w∗
m0||
2p
p+1
Hm01/p ,∀m= 1, . . . , M. (17)
3. Lemma 26 in Micchelli and Pontil (2005b) indicates that the result could even be extended to an infinite
number of kernels.
13
Proof We start the derivation, by equivalently translating Optimization Problem (P) via
Theorem 1 into
inf
w,b,θ:θ≥0
˜
C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2
M
X
m=1
||wm||2
Hm
θm
+µ||θ||2
p.(18)
Setting the partial derivatives w.r.t. θto zero, we obtain the following condition on the
optimality of θ,
−||wm||2
Hm
2θ2
m
+β·∂1
2||θ||2
p
∂θm
= 0,∀m= 1, . . . , M, (19)
with non-zero β(it holds β > 0 by the strict convexity of || · ||). The first derivative of the
`p-norm with respect to the mixing coefficients can be expressed as
∂1
2||θ||2
p
∂θm
=θp−1
m||θ||2−p
p,
and hence Eq. (19) translates into the following optimality condition,
θ∗
m=ζ||w∗
m||
2
p+1
Hm,∀m= 1, . . . , M, (20)
with a suitable constant ζ. By the strict convexity of || · || the constraint ||θ||2
p≤1 in
Optimization Problem (P) is at the upper bound and hence we have that ||θ∗||p= 1 for an
optimal θ∗. Hence, ζcan be computed as ζ=PM
m=1 ||w∗
m||2p/p+1
Hm1/p. Combined with
(20), this results in the claimed formula (17).
In the more interesting case, we will perform the above update in the dual, thereby
operating on dual variables α:
Corollary 3 Let Vbe a convex loss function, be p > 1. Given fixed dual variable α, as
specified in Sect. (2.3), the optimal solution of Optimization Problem (P) is attained for
θ∗
m=(αKmα)2
p−1
PM
m0=1 (αKm0α)
2p
p−11/p ,∀m= 1, . . . , M. (21)
Note that if we deploy hinge loss, then we operate on variables αnew
i=αiyi(cf. Sect. 3.1).
Proof According to Eq. (9b) the dual variables αare specified in terms of wmby
w∗
m=θ∗
m
n
X
i=1
α∗
iψm(xi).
Plugging the above primal-dual relations into Eq. (20) and appropriately normalizing, we
obtain the desired dual update formula for θ.
14
Second we consider how to optimize Optimization Problem (P) w.r.t. the remaining
variables (wm, b) for a given, set of mixing coefficients θ. Since optimization often is con-
siderably easier in the dual space, we fix θand build the partial Lagrangian of Optimization
Problem (P) w.r.t. all other primal variables w,b. The resulting dual problem is of the
form
sup
α:1>α=0 −C
n
X
i=1
V∗−αi
C, yi−1
2
M
X
m=1
θmα>Kmα.(22)
We now have all ingredients for an efficient `p-norm algorithm, based on alternatingly
solving an SVM w.r.t. the actual mixture θand computing the analytical update according
to Eq. (17). A simple wrapper algorithm is stated in Alg. 1.
Algorithm 1 Simple `p>1-norm MKL wrapper-based training algorithm. The analytical
updates of θand the SVM computations are optimized alternatingly.
1: input feasible αand θ.
2: while optimality conditions are not satisfied do
3: solve Eq. (22), e.g., SVM, w.r.t. α
4: obtain updated θaccording to Eq. (21)
5: end while
A disadvantage of the above wrapper approach is that it deploys a full blown kernel
matrix. Instead, we propose to interleave the SVM optimization of SVMlight with the θ-
and α-steps at training time. We have implemented this so-called interleaved algorithm in
Shogun for hinge loss, thereby promoting sparse solutions in α. This allows us to solely
operate on a small number of active variables.4The resulting interleaved optimization
method is shown in Algorithm 2. Lines 3-5 are standard in chunking based SVM solvers
and carried out by SVMlight. Lines 6-8 compute (parts of) SVM-objective values for each
kernel independently. Finally lines 10 to 14 solve the analytical θ-step. The algorithm
terminates if the maximal KKT violation (c.f. Joachims, 1999) falls below a predetermined
precision εsvm and if the normalized maximal constraint violation |1−St
λ|< εmkl for the
MKL-step.
In the following, we exploit the primal view of the above algorithm as a non-linear
Gauss-Seidel method, to prove convergence. We first need the following useful result about
convergence of the non-linear Gauss-Seidel method in general.
Proposition 4 (Bertsekas, 1999) Let X=NM
m=1 Xmbe a the Cartesian product of
closed convex sets Xm∈Rdm, be f:X → Ra continuously differentiable function. Define
the non-linear Gauss-Seidel method recursively by letting x0∈ X be any feasible point, and
be
xk+1
m= argmin
ξ∈Xm
fxk+1
1,··· , xk+1
m−1, ξ, xk
m+1,··· , xk
M,∀m= 1, . . . , M. (23)
4. In practice, it turns out that the kernel matrix of active variables usually is about of the size 40 ×40
even when we deal with ten-thousands of examples.
15
Algorithm 2 `p-Norm MKL chunking-based training algorithm via analytical update. Ker-
nel weighting θand SVM αare optimized interleavingly. The accuracy parameter and the
subproblem size Qare assumed to be given to the algorithm.
1: gm,i = 0, ˆgi= 0, αi= 0, θm=p
p1/M for m= 1, . . . , M and i= 1, . . . , n
2: for t= 1,2, . . . and while SVM and MKL optimality conditions are not satisfied do
3: Select Q suboptimal variables αi1, . . . , αiQbased on the gradient ˆ
gand α; store αold =α
4: Solve SVM dual with respect to the selected variables and update α
5: Update gradient gm,i ←gm,i +PQ
q=1(αiq−αold
iq)yiqkm(xiq,xi) for all m= 1, . . . , M and
i= 1, . . . , n
6: for m= 1, . . . , M do
7: St
m=1
2Pigm,iαiyi
8: end for
9: if |1−St
λ| ≥
10: while MKL optimality conditions are not satisfied do
11: for m= 1, . . . , M
12: θm= (St
m)1/(p+1) /PM
m0=1 (St
m0)p/(p+1)1/p
13: end for
14: end while
15: end if
16: ˆgi=Pmθmgm,i for all i= 1, . . . , n
17: end for
Suppose that for each mand x∈ X, the minimum
min
ξ∈Xm
f(x1,··· , xm−1, ξ , xm+1,··· , xM) (24)
is uniquely attained. Then every limit point of the sequence {xk}k∈Nis a stationary point.
The proof can be found in Bertsekas (1999), p. 268-269. The next proposition basically
establishes convergence the proposed `p-norm MKL training algorithm.
Proposition 5 Let Vbe the hinge loss and be p > 1. Let the kernel matrices K1, . . . , KM
be positive definite. Then every limit point of Algorithm 1 is a globally optimal point of
Optimization Problem (P). Moreover, suppose that the SVM computation is solved exactly
in each iteration, then the same holds true for Algorithm 2.
Proof If we ignore the numerical speed-ups, then the Algorithms 1 and 2 coincidence for
the hinge loss. Hence, it suffices to show the wrapper algorithm converges.
To this aim, we have to transform Optimization Problem (P) into a form such that the
requirements for application of Prop. 4 are fulfilled. We start by expanding Optimization
Problem (P) into
min
w,b,ξ,θC
n
X
i=1
ξi+1
2
M
X
m=1
||wm||2
Hm
θm
,
s.t. ∀i:
M
X
m=1hwm, ψm(xi)iHm+b≥1−ξi;ξ≥0; ||θ||2
p≤1; θ≥0,
16
thereby extending the second block of variables, (w, b), into (w, b, ξ). Moreover, we note
that after an application of the representer theorem5(Kimeldorf and Wahba, 1971) we may
without loss of generality assume Hm=Rn.
In above problem’s current form, the possibility of θm= 0 while wm6= 0 renders the
objective function nondifferentiable, and it can take on infinite values. This hinders the
application of the Prop. 4. Fortunately, in the optimal point, we always have θm∗>0 for
all m, which can be verified by Eq. (21), where we use the positive definiteness of the kernel
matrices Km. We therefore can substitute the constraint θ≥1 by θ>1 for all m. In order
to maintain the closeness of the feasible set we subsequently apply a bijective coordinate
transformation φm:RM
+→RMwith φ(θm) = log(θm), resulting in the following equivalent
problem,
inf
w,b,ξ,θC
n
X
i=1
ξi+1
2
M
X
m=1
exp(−θm)||wm||2
Rn,
s.t. ∀i:
M
X
m=1hwm, ψm(xi)iRn+b≥1−ξi;ξ≥0; ||exp(θ)||2
p≤1,
where we employ the notation exp(θ) = (exp(θ1),··· ,exp(θM))>.
Applying the Gauss-Seidel method in Eq. (23) to the base problem (P) and to the
reparametrized problem yields the same sequence of solutions {(w, b, θ)k}k∈N0. Fortunately,
the above problem now allows to apply Prop. 4 for the two blocks of coordinates θ∈ X1
and (w, b, ξ)∈ X2: the objective is continuously differentiable and the sets X1are closed
and convex. To see the latter, note that ||·||2
p◦exp is a convex function since ||·||2
pis convex
and non-increasing in each argument (cf., e.g., Section 3.2.4 of Boyd and Vandenberghe,
2004). Moreover, the minima in Eq. (23) are uniquely attained: the (w, b)-step amounts
to solving an SVM on a positive definite kernel mixture, and the analytical θ-step clearly
yields unique solutions as well.
Hence, we conclude that every limit point of the sequence {(w, b, θ)k}k∈Nis a stationary
point of Optimization Problem (P). For a convex problem, this is equivalent to such a limit
point being globally optimal.
In practice, we are facing two problems. Firstly, the standard Hilbert space setup
necessarily implies that kwmk ≥ 0 for all m. However in practice this assumption may often
be violated, either due to numerical imprecision or because of using an indefinite “kernel”
function. However, for any kwmk ≤ 0 it also follows that θ?
m= 0 as long as at least one
strictly positive kwm0k>0 exists. This is because for any λ < 0 we have limh→0,h>0λ
h=
−∞. Thus, for any mwith kwmk ≤ 0, we can immediately set the corresponding mixing
coefficients θ?
mto zero. The remaining θare then computed according to Equation (2), and
convergence will be achieved as long as at least one strictly positive kwm0k>0 exists in
each iteration.
Secondly, in practice, the SVM problem will only be solved with finite precision, which
may lead to convergence problems. Moreover, we actually want to improve the αonly a
5. Note that the coordinate transformation into Rnis can be constructively given in terms of the empirical
kernel map (Sch¨olkopf et al., 1999).
17
little bit before recomputing θsince computing a high precision solution can be wasteful,
as indicated by the superior performance of the interleaved algorithms (cf. Sect. 5.5). This
helps to avoid spending a lot of α-optimization (SVM training) on a suboptimal mixture θ.
Fortunately, we can overcome the potential convergence problem by ensuring that the primal
objective decreases within each α-step. Then the alternating optimization is guaranteed
to converge. This is enforced in practice, by computing the SVM by a higher precision if
needed. However, in our computational experiments we find that this precaution is not even
necessary: even without it, the algorithm converges in all cases that we tried (cf. Section
5).
Finally, we would like to point out that the proposed block coordinate descent approach
lends itself more naturally to combination with primal SVM optimizers like (Chapelle, 2006),
LibLinear (Fan et al., 2008) or Ocas (Franc and Sonnenburg, 2008). Especially for linear
kernels this is extremely appealing.
4.2 Cutting Planes
In order to obtain an alternative optimization strategy, we fix θin the primal MKL opti-
mization problem (P) and build the partial Lagrangian w.r.t. all other primal variables w,
b. The resulting dual problem is a min-max problem of the form
inf
θ:θ≥0,||θ||2≤1sup
α:1>α=0 −C
n
X
i=1
V∗−αi
C, yi−1
2
M
X
m=1
θmα>Kmα(25)
We focus on the hinge loss, i.e., V(t, y) = max(0,1−ty), and non-sparse norms of the
form ||θ|| =PM
m=1 θp
m1/p (nevertheless, the following reasoning holds for every twice
differentiable norm). Thus, employing a variable substition of the form αnew
i=αiyi, Eq. (25)
translates into
min
θmax
α1>α−1
2α>
M
X
m=1
θmQmα
s.t. 0≤α≤C1;y>α= 0; θ≥0; kθk2
p≤1,
where Qj=Y KjYfor 1 ≤j≤mand Y= diag(y). The above optimization problem is a
saddle point problem and can be solved by alternating αand θoptimization step. While
the former can simply be carried out by a support vector machine for a fixed mixture θ, the
latter has been optimized for p= 1 by reduced gradients (Rakotomamonjy et al., 2007).
We take a different approach and translate the min-max problem into an equivalent
semi-infinite program (SIP) as follows. Denote the value of the target function by t(α,θ)
and suppose α∗is optimal. Then, according to the max-min inequality (Boyd and Vanden-
berghe, 2004, p. 115), we have t(α∗,θ)≥t(α,θ) for all αand θ. Hence, we can equivalently
minimize an upper bound ηon the optimal value and arrive at the following semi-infinite
18
Algorithm 3 Chunking-based `p-Norm MKL cutting plane training algorithm. It simul-
taneously optimizes the variables αand the kernel weighting θ.The accuracy parameter
and the subproblem size Qare assumed to be given to the algorithm. For simplicity, a few
speed-up tricks are not shown, e.g., hot-starts of the SVM and the QCQP solver.
1: gm,i = 0, ˆgi= 0, αi= 0, θm=p
p1/M for m= 1, . . . , M and i= 1, . . . , n
2: for t= 1,2, . . . and while SVM and MKL optimality conditions are not satisfied do
3: Select Q suboptimal variables αi1, . . . , αiQbased on the gradient ˆ
gand α; store αold =α
4: Solve SVM dual with respect to the selected variables and update α
5: Update gradient gm,i ←gm,i +PQ
q=1(αiq−αold
iq)yiqkm(xiq,xi) for all m= 1, . . . , M and
i= 1, . . . , n
6: for m= 1, . . . , M do
7: St
m=1
2Pigm,iαiyi
8: end for
9: Lt=Piαi,St=PmθmSt
m
10: if |1−St
λ| ≥
11: while MKL optimality conditions are not satisfied do
12: θold =θ
13: (θ, λ)←argmax λ
14: w.r.t. θ∈RM, λ ∈R
15: s.t. 0≤θ≤1,
16: p(p−1)
2Pm(θold
m)p−2θ2
m−Pmp(p−2)(θold
m)p−1θm≤p(3−p)
2and
17: PmθmSr
m−Lr≥λfor r= 1, . . . , t
18: θ←θ/||θ||p
19: Remove inactive constraints
20: end while
21: end if
22: ˆgi=Pmθmgm,i for all i= 1, . . . , n
23: end for
program,
min
ηη
s.t. ∀α∈ A :η≥1>α−1
2α>
M
X
m=1
θmQmα; (SIP)
θ≥0;kθk2
p≤1,
where A=α∈Rn|0≤α≤C1,y>α= 0.
Sonnenburg et al. (2006a) optimize the above SIP for p= 1 with interleaving cutting
plane algorithms. The solution of a quadratic program (here the regular SVM) generates
the most strongly violated constraint for the actual mixture θ. The optimal (θ∗, η) is then
identified by solving a linear program with respect to the set of active constraints. The
optimal mixture is then used for computing a new constraint and so on.
Unfortunately, for p > 1, a non-linearity is introduced by requiring kθk2
p≤1 and such
constraint is unlikely to be found in standard optimization toolboxes that often handle only
linear and quadratic constraints. As a remedy, we propose to approximate the constraint
19
kθkp
p≤1 by a sequence of second-order Taylor expansions6
||θ||p
p≈ ||˜
θ||p
p+p˜
θp−1>θ−˜
θ+p(p−1)
2θ−˜
θ>diag ˜
θp−2θ−˜
θ
= 1 + p(p−3)
2−
M
X
m=1
p(p−2)(˜
θm)p−1θm+p(p−1)
2
M
X
m=1
˜
θp−2
mθ2
m,
where θpis defined element-wise, that is θp:= (θp
1, . . . , θp
M). The sequence (θ0,θ1,···) is
initialized with a uniform mixture satisfying ||θ0||p
p= 1 as a starting point. Successively θt+1
is computed using ˜
θ=θt. Note that the Hessian of the quadratic term in the approximation
is diagonal, strictly positiv and very-well conditioned wherefore the resulting quadratically
constrained problem can be solved efficiently. In fact, since there is only one quadratic con-
straint, its complexity should rather be compared to that of a considerably easier quadratic
program. Moreover, in order to ensure convergence, we enhanced the resulting sequential
quadratically constrained quadratic programming by projection steps onto the boundary of
the feasible set, as given in Line 19. Finally note, that this approach can be further sped-up
by additional level-set projections in the θ-optimization phase similar to Xu et al. (2009).
In our case, the level-set projection is a convex quadratic problem with `p-norm constraints
and can again be approximated by a successive sequence of second-order Taylor expansions.
Algorithm 3 outlines the interleaved α,θMKL training algorithm. Lines 3-5 are stan-
dard in chunking based SVM solvers and carried out by SVMlight. Lines 6-9 compute (parts
of) SVM-objective values for each kernel independently. Finally lines 11 to 19 solve a
sequence of semi-infinite programs with the `p-norm constraint being approximated as a
sequence of second-order constraints. The algorithm terminates if the maximal KKT viola-
tion (see Joachims, 1999) falls below a predetermined precision εsvm and if the normalized
maximal constraint violation |1−St
λ|< εmkl for the MKL. The following proposition shows
the convergence of the semi-infinite programming loop in Algorithm 3.
Proposition 6 Let the kernel matrices K1, . . . , KMbe positive definite and be p > 1. Sup-
pose that the SVM computation is solved exactly in each iteration. Moreover, suppose there
exists an optimal limit point of nested sequence of QCCPs. Then the sequence generated by
Algorithm 3 has at least one point of accumulation that solves Optimization Problem (P).
Proof By assumption the SVM is solved to infinite precision in each MKL step which
simplifies our analysis in that the numerical details in Algorithm 3 can be ignored. We
conclude, that the outer loop of Alg. 3 amounts to a cutting-plane algorithm for solving
the semi-infinite program (SIP). It is well-known (Sonnenburg et al., 2006a), that this
algorithm converges, in the sense that there exists at least one point of accumulation,
which solves the primal problem (P). E.g. this can be seen by viewing the cutting plane
algorithm as a special instance of the class of so-called exchange methods and subsequently
applying Theorem 7.2 in Hettich and Kortanek (1993). A difference to the analysis in
Sonnenburg et al. (2006a) is the `p>1-norm constraint in our algorithm. However, according
to our assumption that the nonlinear subprogram is solved correctly, a quick inspection
6. We also tried out first-order Taylor expansions, whereby our algorithm basically boils down the renowned
sequential quadratic programming, but it empirically turned out to be inferior. Intuitively, second-order
expansions work best when the approximated function is almost quadratic, as given in our case.
20
of the preliminaries of the latter theorem clearly reveals, that they remain fulfilled when
introducing an `p-norm constraint.
In order to complete our convergence analysis, it remains to show that the inner loop
(lines 11-18), that is the sequence of QCQPs, converges against an optimal point. Ex-
isting analyses of this so-called sequential quadratically constrained quadratic programming
(SQCQP) can be divided into two classes. First, one class establishes local convergence, i.e.,
convergence in an open neighborhood of the optimal point, at a rate of O(n2), under rela-
tively mild smoothness and constraint qualification assumptions (Fern´andez and Solodov,
2008; Anitescu, 2002), whereas Anitescu (2002) additionally requires quadratic growth of
the nonlinear constraints. Those analyses basically guarantee local convergence the nested
sequences of QCQPs in our `p-norm training algorithm, for all p∈(1,∞) (Fern´andez and
Solodov, 2008) and p≥2 (Anitescu, 2002), respectively.
A second class of papers additionally establishes global convergence (e.g. Solodov, 2004;
Fukushima et al., 2002), so they need more restrictive assumptions. Moreover, in order to
ensure feasibility of the subproblems when the actual iterate is too far away from the true
solution, a modification of the algorithmic protocol is needed. This is usually dealt by per-
forming a subsequent line search and downweighting the quadratic term by a multiplicative
adaptive constant Di∈[0,1]. Unfortunately, the latter involves a complicated procedure
to tune Di(Fukushima et al., 2002, p. 7). Employing the above modifications, the analysis
in Fukushima et al. (2002) together with our Prop. 6 would guarantee the convergence of
our Alg. 3.
However, due to the special form of our SQCQP, we chose to discard the comfortable
convergence guarantees and to proceed with a much more simple and efficient strategy, which
renders both the expensive line search and the tuning of the constant Diunnecessary. The
idea of our method is that the projection of θonto the boundary of the feasible set, given
by line 18 in Alg. 3, can be performed analytically. This projection ensures the feasibility
of the QCQP subproblems. Note that in general, this projection can be as expensive as
performing a QCQP step, which is why projection-type algorithms for solving SQCQPs to
the best of our knowledge have not been studied yet by the optimization literature.
Although the projection procedure is appealingly simple and—as we found empirically—
seemingly shares nice convergence properties (the sequence of SQCQPs converged optimally
in all cases we tried, usually after 3-4 iterations), it unfortunately prohibits exploitation
of existing analyses for global convergence. However, the discussions in Fukushima et al.
(2002) and Solodov (2004) identify the reason of occasional divergence of the vanilla SQCQP
as the infeasibility of the subproblems. But in contrast, our projection algorithm always
ensures the feasibility of the subproblem. We therefore conjecture that based on the superior
empirical results and the discussions in Fukushima et al. (2002) and Solodov (2004), our
algorithm is designated to convergence. The theoretical analysis of this new class of so-called
SQCQP projection algorithms is beyond the scope of this paper.
21
4.3 Technical Considerations
4.3.1 Implementation Details
We have implemented the analytic and the cutting plane algorithm as well as a Newton
method (c.f. Kloft et al., 2009a) within the SHOGUN toolbox7for regression, one-class
classification, and two-class classification tasks. In addition one can choose the optimization
scheme, i.e., decide whether the interleaved optimization algorithm or the wrapper algorithm
should be applied. In all approaches any of the SVMs contained in SHOGUN can be used.
In the more conventional family of approaches, the so-called wrapper algorithms, an
optimization schme on θwraps around a single kernel SVM. Effectively this results in
alternatingly solving for αand θ. For the outer optimization (i.e., that on θ) SHOGUN
offers the three choices listed above. The semi-infinite program (SIP) uses a traditional
SVM to generate new violated constraints and thus requires a single kernel SVM. A linear
program (for p= 1) or a sequence of quadratically constrained linear programs (for p > 1)
is solved via GLPK8or IBM ILOG CPLEX9. Alternatively, either an analytic or a Newton
update (for `pnorms with p > 1) step can be performed, obviating the need for an additional
mathematical programming software.
The second, much faster approach performs interleaved optimization and thus re-
quires modification of the core SVM optimization algorithm. It is currently integrated
into the chunking-based SVRlight and SVMlight. To reduce the implementation effort,
we implement a single function perform mkl step(Pα, objm), that has the arguments
Pα=Pn
i=1 αiand objm=1
2αTKmα, i.e. the current linear α-term and the SVM objectives
for for each kernel. This function is either, in the interleaved optimization case, called as a
callback function (after each chunking step or a couple of SMO steps), or it is called by the
wrapper algorithm (after each SVM optimization to full precision).
Recovering Regression and One-Class Classification. It should be noted that one-
class classification is trivially implemented using Pα= 0 while support vector regression
(SVR) is typically performed by internally translating the SVR problem into a standard
SVM classification problem with twice the number of examples once positively and once
negatively labeled with corresponding αand α∗. Thus one needs direct access to α∗and
computes Pα=−Pn
i=1(αi+α∗
i)ε−Pn
i=1(αi−α∗
i)yi(cf. Sonnenburg et al., 2006a). Since
this requires modification of the core SVM solver we implemented SVR only for interleaved
optimization and SVMlight.
Efficiency Considerations and Kernel Caching. Note that the choice of the size of
the kernel cache becomes crucial when applying MKL to large scale learning applications.10
While for the wrapper algorithm only a single kernel SVM needs to be solved and thus a
single large kernel cache should be used, the story is different for interleaved optimization.
Since one must keep track of the several partial MKL objectives objm, requiring access to
individual kernel rows, the same cache size should be used for all sub-kernels.
7. http://www.shogun-toolbox.org.
8. http://www.gnu.org/software/glpk/.
9. http://www.ibm.com/software/integration/optimization/cplex/.
10. Large scale in the sense, that the data cannot be stored in memory or the computation reaches a
maintainable limit. In the case of MKL this can be due both a large sample size or a high number of
kernels.
22
4.3.2 Kernel Normalization
The normalization of kernels is as important for MKL as the normalization of features is
for training regularized linear or single-kernel models. This is owed to the bias introduced
by the regularization: optimal feature / kernel weights are requested to be small. This is
easier to achieve for features (or entire feature spaces, as implied by kernels) that are scaled
to be of large magnitude, while downscaling them would require a correspondingly upscaled
weight for representing the same predictive model. Upscaling (downscaling) features is thus
equivalent to modifying regularizers such that they penalize those features less (more). As is
common practice, we here use isotropic regularizers that, moreover, penalize all dimensions
uniformly. This implies that the kernels have to be normalized in a sensible way in order
to represent an “uninformative prior” as to which kernels are useful.
There exist several approaches to kernel normalization, of which we use two in the com-
putational experiments below. They are fundamentally different. The first one generalizes
the common practice of standardizing features to entire kernels, thereby directly implement-
ing the spirit of the discussion above. In contrast, the second normalization approach carries
the rescaling of data points to the world of kernels. Nevertheless it can have a beneficial
effect on the scaling of kernels, as we argue below.
Multiplicative Normalization. As done in Ong and Zien (2008), we multiplicatively
normalize the kernels to have uniform variance of data points in feature space. Formally, we
find a positive rescaling λmof the kernel, such that the rescaled kernel ˜
km(·,·) = λmkm(·,·)
and the corresponding feature map ˜
Φm(·) = √λmΦm(·) satisfy
1!
=1
n
n
X
i=1 ˜
Φm(xi)−˜
Φm(¯
x)2=1
n
n
X
i=1
˜
km(xi,xi)−1
n2
n
X
i=1
n
X
j=1
˜
km(xi,xj)
for each m= 1, . . . , M, where ˜
Φm(¯
x) := 1
nPn
i=1 ˜
Φm(xi) is the empirical mean of the data
in feature space. The final normalization rule is
k(x,¯
x)7−→ k(x,¯
x)
1
nPn
i=1 k(xi,xi)−1
n2Pn
i,j=1, k(xi¯
xj).(26)
Spherical Normalization. Frequently, kernels are normalized according to
k(x,¯
x)7−→ k(x,¯
x)
pk(x,x)k(¯
x,¯
x).(27)
After this operation, kxk=k(x,x) = 1 holds for each data point x; this means that each
data point is rescaled to lie on the unit sphere. Still, this also may have an effect on the
scale of the features: in case the kernel is centered (i.e. average of the data points lies on
the origin), the rescaled kernel satisfies the above goal that the points have unit variance
(around their mean). Thus the spherical normalization may be seen as an approximation
to the above multiplicative normalization and may be used as a substitute for it. Note,
however, that it changes the data points themselves by eliminating length information;
whether this is desired or not depends on the learning task at hand. Finally note that both
normalizations achieve that the optimal value of Cis not far from 1.
23
4.4 Relation to Block-Norm Formulation and Limitations of Our Framework
In this section we first show a connection of `p-norm MKL to a formulation based on block
norms and then point out a limitation of our framework. To this aim let us recall the primal
MKL problem (P) and consider the special case of `p-norm MKL given by
inf
w,b,θ:θ≥0C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2
M
X
m=1
||wm||2
Hm
θm
,s.t. ||θ||2
p≤1.
(28)
The subsequent proposition shows that Optimization Problem (P) equivalently can be trans-
lated into the following mixed-norm formulation,
inf
w,b,θ:θ≥0
˜
C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2
M
X
m=1 ||wm||q
Hm,(29)
where q=2p
p+1 , and ˜
Cis a constant. For q= 1 this has been studied by Bach et al. (2004).
Proposition 7 Let be p > 1and be Va convex loss function. Optimization Problem (28)
and (29) are equivalent, i.e., for each Cthere exists a ˜
C > 0, such that for each optimal
solution (w∗, b∗, θ∗) of OP (28) using C, we have that (w∗, b∗) is also optimal in OP (29)
using ˜
C, and vice versa.
Proof We begin by applying Theorem 1 to rephrase Optimization Problem (P) as
inf
w,b,θ:θ≥0
˜
C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2
M
X
m=1
||wm||2
Hm
θm
+µ||θ||2
p.
Setting the partial derivatives w.r.t. θto zero, we obtain the following equation at opti-
mality:
−||wm||2
Hm
2θ2
m
+β·θp−1
m||θ||2−p
p= 0,∀m= 1, . . . , M. (30)
Hence, Eq. (30) translates into the following optimality condition on wand θ:
θ∗
m=ζ||w∗
m||
2
p+1
Hm,∀m= 1, . . . , M,
with a suitable constant ζ. Plugging the above equation into Optimization Problem (P)
yields
inf
w,b,θ:θ≥0C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2ζ
M
X
m=1 ||wm||
2p
p+1
Hm.(31)
Defining q:= 2p
p+1 and ˜
C:= ζC results in (29) what was to show.
Now, let us take a closer look on the parameter range of q. It is easy to see that when
we vary pin the real interval [1,∞], then qis limited to range in [1,2]. This raises the
24
question whether we can derive an efficient wrapper-based optimization strategy for the
case of q > 2. A framework by Aflalo et al. (2010) covers the case q≥2, although their
method aims at hierarchical kernel learning. Note, that q≤2 and hence `p-norm MKL is
not covered by their approach.
We briefly sketch the analysis of Aflalo et al. (2010) and discuss a potential simplification
of their algorithm for the special case of `q>2block norm MKL. We start by noting that it
is possible to show that for q≥2, Eq. (29) is equivalent to
sup
θ:θ≥0,||θ||2
p≤1
inf
w,b
˜
C
n
X
i=1
V M
X
m=1hwm, ψm(xi)iHm+b, yi!+1
2
M
X
m=1
θm||wm||2
Hm,(32)
where p=q/2
q/2−1. Note that despite the similarity to `p-norm MKL, the above problem
significantly differs from `p-norm MKL for two reasons. Firstly, obvious differences such as
the mixing coefficients θappearing in the nominator and the consequential maximization
w.r.t. θ, render the above problem a min-max problem. Secondly, note that by varying p
in the interval [1,∞], the whole range of qin the interval [2,∞] can be obtained, which
explains why this method is complementary to ours, where qranges in [1,2].
Using the hinge loss, Eq. (32) can be partially dualized w.r.t. fixed θ, resulting in a
convex optimization problem (Boyd and Vandenberghe, 2004, p. 76)
max
α,θ1>α−1
2α>
M
X
m=1
Qm
θm
α(33)
s.t. 0≤α≤C1;y>α= 0; θ≥0; kθk2
p≤1,
where, as in the previous sections, we denote Qj=Y KjYand and Y= diag(y). Origi-
nally the authors aimed at hierarchical kernel learning and Aflalo et al. (2010) proposed to
optimize (33) by a mirror descent algorithm (Beck and Teboulle, 2003). However, for the
special case of q > 2 block norm MKL, which we consider here, a simple block gradient
procedure based on an analytical update of θ, similar to the one presented in Section 4.1, is
sufficient. We omit the derivations which are analogeous to those presented in Section 4.1.
5. Computational Experiments
In this section we study non-sparse MKL in terms of computational efficiency and predictive
accuracy. Throughout all our experiments both `p-norm MKL implementations, presented
in Sections 4.1 and 4.2, perform comparably. We apply the method of (Sonnenburg et al.,
2006a) in the case of p= 1, as it is recovered as a special case of our cutting plane strategy.
We write `∞-norm MKL for a regular SVM with the unweighted-sum kernel K=PmKm.
We first study a toy problem in Section 5.1 where we have full control over the distri-
bution of the relevant information in order to shed light on the appropriateness of sparse,
non-sparse, and `∞-MKL. We report on real-world problems from the Bioninformatics do-
main, namely protein subcellular localization (Section 5.2), finding transcription start sites
of RNA Polymerase II binding genes in genomic DNA sequences (Section 5.3), and recon-
structing metabolic gene networks (Section 5.4).
25
Complementarily, we would like to mention empirical results of other researchers which
have been experimenting with non-sparse MKL. Cortes et al. (2009) applies `2-norm MKL
to regression tasks on Reuters and various sentiment analysis datasets, and Yu et al. (2009)
studies `2-norm on two real-world genomic data sets for clinical decision support in cancer
diagnosis and disease relevant gene prioritization, respectively. Yan et al. (2009) apply `2-
norm MKL to image and video classification tasks. All those papers show an improvement
of `2-norm MKL over sparse MKL and the unweighted sum kernel SVM. Nakajima et al.
(2009) study `p-norm MKL for multi-label image categorization and show an improvement
of non-sparse MKL over `1/∞-norm MKL.
5.1 Measuring the Impact of Data Sparsity — Toy Experiment
The goal of this section is to study the relationship of the level of sparsity of the true
underlying function to be learnt to the chosen norm pin the model. It is suggestive that
the optimal choice of pdirectly corresponds to the true level of sparsity. Apart from verifying
this conjecture, we are also interested in the effects of suboptimal choice of p. To this aim
we constructed several artificial data sets in which we vary the degree of sparsity in the true
kernel mixture coefficients. We go from having all weight focussed on a single kernel (the
highest level of sparsity) to uniform weights (the least sparse scenario possible) in several
steps. We then study the statistical performance of `p-norm MKL for different values of p
that cover the entire range [0,∞].
We generate an n-elemental balanced sample D={(xi, yi)}n
i=1 from two d= 50-
dimensional isotropic Gaussian distributions with equal covariance matrices C=Id×d.
The two Gaussians are aligned at opposing means w.r.t. to the origin, µ1=ρ
||θ||2θand
µ2=−ρ
||θ||2θ. Thereby θis a binary vector, i.e., θi∈ {0,1}, encoding the true underlying
data sparsity as follows. Zero components θi= 0 clearly imply identical means of the two
classes distributions in the i-th feature set; hence the latter does not carry any discriminat-
ing information. In summary, the fraction of zero components, ν(θ) = 1 −1
dPd
i=1 θi, is a
measure for the feature sparsity of the learning problem.
For several values of νwe generate m= 250 data sets D1,...,Dmfixing ρ= 1.75. Then,
each feature is input to a linear kernel and the resulting kernel matrices are multiplicatively
normalized as described in Section 4.3.2. Hence, the ν