Efficient Multioutput Gaussian Processes through Variational Inducing Kernels.
ABSTRACT Interest in multioutput kernel methods is increas- ing, whether under the guise of multitask learn- ing, multisensor networks or structured output data. From the Gaussian process perspective a multioutput Mercer kernel is a covariance func- tion over correlated output functions. One way of constructing such kernels is based on convolution processes (CP). A key problem for this approach is efficient inference. ´ Alvarez and Lawrence re- cently presented a sparse approximation for CPs that enabled efficient inference. In this paper, we extend this work in two directions: we in- troduce the concept of variational inducing func- tions to handle potential non-smooth functions involved in the kernel CP construction and we consider an alternative approach to approximate inference based on variational methods, extend- ing the work by Titsias (2009) to the multiple output case. We demonstrate our approaches on prediction of school marks, compiler perfor- mance and financial time series.
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ABSTRACT: A central task of Bayesian machine learning is to infer the posterior distribution of hidden random variables given observations and calculate expectations with respect to this distribution. However, this is often computationally intractable so that people have to seek approximation schemes. Deterministic approximate inference techniques are an alternative of the stochastic approximate inference methods based on numerical sampling, namely Monte Carlo techniques, and during the last 15 years, many advancements in this field have been made. This paper reviews typical deterministic approximate inference techniques, some of which are very recent and need further explorations. With an aim to promote research in deterministic approximate inference, we also attempt to identify open problems that may be helpful for future investigations in this field.Neural Computing and Applications 01/2013; · 1.76 Impact Factor - SourceAvailable from: Roni Khardon
Conference Paper: Sparse gaussian processes for multi-task learning
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ABSTRACT: Multi-task learning models using Gaussian processes (GP) have been recently developed and successfully applied in various applications. The main difficulty with this approach is the computational cost of inference using the union of examples from all tasks. The paper investigates this problem for the grouped mixed-effect GP model where each individual response is given by a fixed-effect, taken from one of a set of unknown groups, plus a random individual effect function that captures variations among individuals. Such models have been widely used in previous work but no sparse solutions have been developed. The paper presents the first sparse solution for such problems, showing how the sparse approximation can be obtained by maximizing a variational lower bound on the marginal likelihood, generalizing ideas from single-task Gaussian processes to handle the mixed-effect model as well as grouping. Experiments using artificial and real data validate the approach showing that it can recover the performance of inference with the full sample, that it outperforms baseline methods, and that it outperforms state of the art sparse solutions for other multi-task GP formulations.Proceedings of the 2012 European conference on Machine Learning and Knowledge Discovery in Databases - Volume Part I; 09/2012 - SourceAvailable from: Achim Schweikard[Show abstract] [Hide abstract]
ABSTRACT: In extracranial robotic radiotherapy, tumour motion is compensated by tracking external and internal surrogates. To compensate system specific time delays, time series prediction of the external optical surrogates is used. We investigate whether the prediction accuracy can be increased by expanding the current clinical setup by an accelerometer, a strain belt and a flow sensor. Four previously published prediction algorithms are adapted to multivariate inputs-normalized least mean squares (nLMS), wavelet-based least mean squares (wLMS), support vector regression (SVR) and relevance vector machines (RVM)-and evaluated for three different prediction horizons. The measurement involves 18 subjects and consists of two phases, focusing on long term trends (M1) and breathing artefacts (M2). To select the most relevant and least redundant sensors, a sequential forward selection (SFS) method is proposed. Using a multivariate setting, the results show that the clinically used nLMS algorithm is susceptible to large outliers. In the case of irregular breathing (M2), the mean root mean square error (RMSE) of a univariate nLMS algorithm is 0.66 mm and can be decreased to 0.46 mm by a multivariate RVM model (best algorithm on average). To investigate the full potential of this approach, the optimal sensor combination was also estimated on the complete test set. The results indicate that a further decrease in RMSE is possible for RVM (to 0.42 mm). This motivates further research about sensor selection methods. Besides the optical surrogates, the sensors most frequently selected by the algorithms are the accelerometer and the strain belt. These sensors could be easily integrated in the current clinical setup and would allow a more precise motion compensation.Physics in Medicine and Biology 09/2014; 59(20):6043-6060. · 2.70 Impact Factor
Page 1
Efficient Multioutput Gaussian Processes through Variational Inducing Kernels
Mauricio A.´Alvarez
School of Computer Science
University of Manchester
Manchester, UK, M13 9PL
alvarezm@cs.man.ac.uk
David LuengoMichalis K. Titsias, Neil D. Lawrence
School of Computer Science
University of Manchester
Manchester, UK, M13 9PL
{mtitsias,neill}@cs.man.ac.uk
Depto. Teor´ ıa de Se˜ nal y Comunicaciones
Universidad Carlos III de Madrid
28911 Legan´ es, Spain
luengod@ieee.org
Abstract
Interest in multioutput kernel methods is increas-
ing, whether under the guise of multitask learn-
ing, multisensor networks or structured output
data. From the Gaussian process perspective a
multioutput Mercer kernel is a covariance func-
tion over correlated output functions. One way of
constructingsuchkernelsisbasedonconvolution
processes (CP). A key problem for this approach
is efficient inference.´Alvarez and Lawrence re-
cently presented a sparse approximation for CPs
that enabled efficient inference. In this paper,
we extend this work in two directions: we in-
troduce the concept of variational inducing func-
tions to handle potential non-smooth functions
involved in the kernel CP construction and we
consider an alternative approach to approximate
inference based on variational methods, extend-
ing the work by Titsias (2009) to the multiple
output case.We demonstrate our approaches
on prediction of school marks, compiler perfor-
mance and financial time series.
1Introduction
In this paper we are interested in developing priors over
multiple functions in a Gaussian processes (GP) frame-
work. While such priors can be trivially specified by con-
sidering the functions to be independent, our focus is on
priors which specify correlations between the functions.
Mostattemptstoapplysuchpriors(Tehetal.,2005;Rogers
et al., 2008; Bonilla et al., 2008) have focused on what
is known in the geostatistics community as “linear model
of coregionalization” (LMC) (Goovaerts, 1997). In these
Appearing in Proceedings of the 13thInternational Conference
on Artificial Intelligence and Statistics (AISTATS) 2010, Chia La-
guna Resort, Sardinia, Italy. Volume 9 of JMLR: W&CP 9. Copy-
right 2010 by the authors.
models the different outputs are assumed to be linear com-
binations of a set of one or more “latent functions”. GP pri-
ors are placed, independently, over each of the latent func-
tions inducing a correlated covariance function over the D
outputs {fd(x)}D
We wish to go beyond the LMC framework, in particu-
lar, our focus is on convolution processes (CPs). Using
CPs for multi-output GPs was proposed by Higdon (2002)
and introduced to the machine learning audience by Boyle
and Frean (2005). Convolution processes allow the inte-
gration of prior information from physical models, such as
ordinary differential equations, into the covariance func-
tion. ´Alvarez et al. (2009a), inspired by Lawrence et al.
(2007), have demonstrated how first and second order dif-
ferential equations, as well as partial differential equations,
can be accommodated in a covariance function. They in-
terpret the set of latent functions as a set of latent forces,
and they term the resulting models “latent force models”
(LFM). The covariance functions for these models are de-
rived through convolution processes. In the CP framework,
output functions are generated by convolving R indepen-
dent latent processes {ur}R
tions Gd,r(x), for each output d and latent force r,
?
The LMC can be seen as a particular case of the CP, in
which the kernel functions Gd,r(x) correspond to a scaled
Dirac δ-function Gd,r(x − z) = ad,rδ(x − z).
A practical problem associated with the CP framework is
that in these models inference has computational complex-
ity O(N3D3) and storage requirements O(N2D2). Re-
cently´AlvarezandLawrence(2009)introduced anefficient
approximation for inference in this multi-output GP model.
Their idea was to exploit a conditional independence as-
sumption over the output functions {fd(x)}D
tent functions are fully observed then the output functions
are conditionally independent of one another (as can be
seen in (1)). Furthermore, if the latent processes are suf-
ficiently smooth, the conditional independence assumption
d=1.
r=1with smoothing kernel func-
fd(x) =
R
?
r=1
Z
Gd,r(x − z)ur(z)dz.
(1)
d=1: if the la-
Page 2
Efficient Multioutput Gaussian Processes through Variational Inducing Kernels
will hold approximately even for a finite number of obser-
vations of the latent functions
the variables {zk}K
ing inputs. These assumptions led to approximations that
were very similar in spirit to the PITC and FITC approx-
imations of Snelson and Ghahramani (2006); Qui˜ nonero
Candela and Rasmussen (2005).
Inthispaperwebuildontheworkof´AlvarezandLawrence
and extend it in two ways. First, we notice that if the
locations of the inducing points are close relative to the
length scale of the latent function, the PITC approxima-
tion will be accurate enough. However, if the length scale
becomes small the approximation requires very many in-
ducing points. In the worst case, the latent process could
be white noise (as suggested by Higdon (2002) and imple-
mented by Boyle and Frean (2005)). In this case the ap-
proximation will fail completely. To deal with such type
of latent functions, we develop the concept of an inducing
function, a generalization of the traditional concept of in-
ducing variable commonly employed in several sparse GP
methods. As we shall see, an inducing function is an arti-
ficial construction generated from a convolution operation
between a smoothing kernel or inducing kernel and the la-
tent functions ur. The artificial nature of the inducing func-
tion is based on the fact that its construction is immersed in
a variational-like inference procedure that does not modify
the marginal likelihood of the true model. This leads us
to the second extension of the paper: a problem with the
FITC and PITC approximations can be their tendency to
overfit when inducing inputs are optimized. A solution to
this problem was given in a recent work by Titsias (2009)
who provided a sparse GP approximation that has an as-
sociated variational bound. In this paper we show how the
ideas of Titsias can be extended to the multiple output case.
Our variational approximation is developed through the in-
ducing functions and the quality of the approximation can
be controlled through the inducing kernels and the num-
ber and location of the inducing inputs. Our approxima-
tion allows us to consider latent force models with a large
number of states, D, and data points N. The use of induc-
ing kernels also allows us to extend the inducing variable
approximation of the latent force model framework to sys-
tems of stochastic differential equations (SDEs). We apply
the approximation to different real world datasets, includ-
ing a multivariate financial time series example.
?
{ur(zk)}K
k=1
?R
r=1, where
k=1are usually referred to as the induc-
A similar idea to the inducing function one introduced
in this paper, was simultaneously proposed by L´ azaro-
Gredilla and Figueiras-Vidal (2010). L´ azaro-Gredilla and
Figueiras-Vidal (2010) introduced the concept of inducing
feature to improve performance over the pseudo-inputs ap-
proach of Snelson and Ghahramani (2006) in sparse GP
models. Our use of inducing functions and inducing ker-
nels is motivated by the necessity to deal with non-smooth
latent functions in the CP model of multiple outputs.
2Multioutput GPs (MOGPs)
Let yd ∈ RN, where d = 1,...,D, be the observed
data associated with the output function yd(x). For sim-
plicity, we assume that all the observations associated with
different outputs are evaluated at the same inputs X (al-
though this assumption is easily relaxed).
ten use the stacked vector y = (y1,...,yD) to collec-
tively denote the data of all the outputs. Each observed
vector ydis assumed to be obtained by adding indepen-
dent Gaussian noise to a vector of function values fdso
that the likelihood is p(yd|fd) = N(yd|fd,σ2
fdis defined via (1). More precisely, the assumption in
(1) is that a function value fd(x) (the noise-free version
of yd(x)) is generated from a common pool of R inde-
pendent latent functions {ur(x)}R
variance function (Mercer kernel) given by kr(x,x?). No-
tice that the outputs share the same latent functions, but
they also have their own set of parameters ({αdr}R
where αdr are the parameters of the smoothing kernel
Gd,r(·). Because convolution is a linear operation, the co-
variance between any pair of function values fd(x) and
fd?(x?) is given by kfd,fd?(x,x?) = Cov[fd(x),fd?(x?)] =
?R
GP prior p(f1,...,fD) over all the function values associ-
ated with the different outputs. The joint probability dis-
tribution of the multioutput GP model can be written as
p({yd,fd}D
prior p(f1,...,fD) has a zero mean vector and a (ND) ×
(ND) covariance matrix Kf,f, where f = (f1,...,fD),
which consists of N × N blocks of the form Kfd,fd?. Ele-
ments of each block are given by kfd,fd?(x,x?) for all pos-
sible values of x. Each such block is a cross-covariance (or
covariance) matrix of pairs of outputs.
We will of-
dI), where
r=1, each having a co-
r=1,σ2
d)
r=1
?
ZGd,r(x − z)?
ZGd?,r(x?− z?)kr(z,z?)dzdz?.
This covariance function is used to define a fully-coupled
d=1) =
?D
d=1p(yd|fd)p(f1,...,fD). The GP
Prediction using the above GP model, as well as the maxi-
mization of the marginal likelihood p(y) = N(y|0,Kf,f+
Σ), where Σ = diag(σ2
time and O(N2D2) storage which rapidly becomes infea-
sible even when only a few hundred outputs and data points
are considered. Efficient approximations are needed in or-
der to make the above multioutput GP model more practi-
cal.
1I,...,σ2
DI), requires O(N3D3)
3PITC-like approximation for MOGPs
Before we propose our variational sparse inference method
for multioutput GP regression in Section 4, we review
the sparse method proposed by ´Alvarez and Lawrence
(2009). This method is based on a likelihood approxima-
tion. More precisely, each output function yd(x) is in-
dependent from the other output functions given the full-
length of each latent function ur(x). This means, that the
likelihood of the data factorizes according to p(y|u) =
Page 3
Mauricio A.´Alvarez, David Luengo, Michalis K. Titsias, Neil D. Lawrence
?D
Lawrence (2009) makes use of this factorization by as-
suming that it remains valid even when we are only al-
lowed to exploit the information provided by a finite set
of function values, ur, instead of the full-length function
ur(x) (which involves uncountably many points). Let ur,
for r = 1,...,R, be a K-dimensional vector of values
from the function ur(x) which are evaluated at the in-
puts Z = {zk}K
notes all these variables. The sparse method approximates
the exact likelihood function p(y|u) with the likelihood
p(y|u) =?D
Kfd,uK−1
the conditional GP priors p(fd|u). The matrix Ku,u is
a block diagonal covariance matrix where the rth block
Kur,uris obtained by evaluating kr(z,z?) at the inducing
inputs Z. Further, the matrix Kfd,uis defined by the cross-
covariance function Cov[fd(x),ur(z)] =
z?)kr(z?,z)dz?.The variables u follow the GP prior
p(u) = N(u|0,Ku,u) and can be integrated out to give the
following approximation to the exact marginal likelihood:
d=1p(yd|u) =?D
d=1p(yd|fd), with u = {ur}R
r=1the
set of latent functions. The sparse method in´Alvarez and
k=1. The vector u = (u1,...,uR) de-
d=1p(yd|u) =?D
u,uKu,fdare the mean and covariance matrices of
d=1N(yd|µfd|u,Σfd|u+
u,uu and Σfd|u= Kfd,fd−
σ2
dI), where µfd|u= Kfd,uK−1
?
ZGd,r(x −
p(y|θ) = N(y|0,D + Kf,uK−1
Here, D is a block-diagonal matrix, where each block is
given by Kfd,fd− Kfd,uK−1
proximate marginal likelihood represents exactly each di-
agonal (output-specific) block Kfd,fdwhile each off diag-
onal (cross-output) block Kfd,fd?is approximated by the
Nystr¨ om matrix Kfd,uK−1
The above sparse method has a similar structure to the
PITC approximation introduced for single-output regres-
sion (Qui˜ nonero Candela and Rasmussen, 2005). Because
of this similarity,´Alvarez and Lawrence (2009) call their
multioutput sparse approximation PITC as well. Two of the
properties of this PITC approximation (which may some-
times be seen as limitations) are:
u,uKu,f+ Σ).
(2)
u,uKu,fdfor all d. This ap-
u,uKu,fd?.
1. It assumes that all latent functions u are smooth.
2. It is based on a modification of the initial full GP
model. This implies that the inducing inputs Z are
extra kernel hyparameters in the modified GP model.
Because of point 1, the method is not applicable when
the latent functions are white noise processes. An impor-
tant class of problems where we have to deal with white
noise processes arise in linear SDEs where the above sparse
method is currently not applicable there. Because of 2, the
maximization of the marginal likelihood in eq. (2) with re-
spect to (Z,θ), where θ are model hyperparameters, may
be prone to overfitting especially when the number of vari-
ables in Z is large. Moreover, fitting a modified sparse GP
model implies that the full GP model is not approximated
in a systematic and rigorous way since there is no distance
or divergence between the two models that is minimized.
In the next section, we address point 1 above by introduc-
ing the concept of variational inducing kernels that allow us
to efficiently sparsify multioutput GP models having white
noise latent functions. Further, these inducing kernels are
incorporated into the variational inference method of Tit-
sias (2009) (thus addressing point 2) that treats the induc-
ing inputs Z as well as other quantities associated with the
inducing kernels as variational parameters. The whole vari-
ational approach provides us with a very flexible, robust to
overfitting, approximation framework that overcomes the
limitations of the PITC approximation.
4Sparse variational approximation
In this section, we introduce the concept of variational in-
ducing kernels (VIKs). VIKs give us a way to define more
general inducing variables that have larger approximation
capacity than the u inducing variables used earlier and im-
portantly allow us to deal with white noise latent functions.
To motivate the idea, we first explain why the u variables
can work when the latent functions are smooth and fail
when these functions become white noises.
In PITC, we assume each latent function ur(x) is smooth
and we sparsify the GP model through introducing, ur, in-
ducing variables which are direct observations of the latent
function, ur(x), at particular input points. Because of the
latent function’s smoothness, the ur variables also carry
information about other points in the function through the
imposed prior over the latent function. So, having observed
urwe can reduce the uncertainty of the whole function.
With the vector of inducing variables u, if chosen to be
sufficiently large relative to the length scales of the la-
tent functions, we can efficiently represent the functions
{ur(x)}R
convolved versions of the latent functions.1When the re-
construction of f from u is perfect, the conditional prior
p(f|u) becomes a delta function and the sparse PITC ap-
proximation becomes exact. Figure 1(a) shows a cartoon
description of a summarization of ur(x) by ur.
r=1and subsequently variables f which are just
In contrast, when some of the latent functions are white
noise processes the sparse approximation will fail. If ur(z)
is white noise2it has a covariance function δ(z−z?). Such
processesnaturallyariseintheapplicationofstochasticdif-
ferential equations (see section 6) and are the ultimate non-
1Thisideaislikea“softversion”oftheNyquist-Shannonsam-
pling theorem. If the latent functions were bandlimited, we could
compute exact results given a high enough number of inducing
points. In general they won’t be bandlimited, but for smooth func-
tions low frequency components will dominate over high frequen-
cies, which will quickly fade away.
2Such a process can be thought as the “time derivative” of the
Wiener process.
Page 4
Efficient Multioutput Gaussian Processes through Variational Inducing Kernels
(a) Latent function is smooth(b) Latent function is noise
Figure 1: With a smooth latent function as in (a), we can use some inducing variables ur (red dots) from the complete latent process
ur(x) (in black) to generate smoothed versions (for example the one in blue), with uncertainty described by p(ur|ur). However, with a
white noise latent function as in (b), choosing inducing variables ur(red dots) from the latent process (in black) does not give us a clue
about other points (for example the blue dots).
smooth processes where two values ur(z) and ur(z?) are
uncorrelated when z ?= z?. When we apply the sparse ap-
proximation a vector of “white-noise” inducing variables
ur does not carry information about ur(z) at any input
z that differs from all inducing inputs Z. In other words
there is no additional information in the conditional prior
p(ur(z)|ur) over the unconditional prior p(ur(z)). Figure
1(b) shows a pictorial representation. The lack of structure
makes it impossible to exploit the correlations in the stan-
dard sparse methods like PITC.3
Our solution to this problem is the following. We will de-
fine a more powerful form of inducing variable, one based
not around the latent function at a point, but one given by
the convolution of the latent function with a smoothing ker-
nel. More precisely, let us replace each inducing vector ur
with variables λrwhich are evaluated at the inputs Z and
are defined according to
?
where Tr(x) is a smoothing kernel (e.g. Gaussian) which
we call the inducing kernel (IK). This kernel is not nec-
essarily related to the model’s smoothing kernels. These
newly defined inducing variables can carry information
about ur(z) not only at a single input location but from
the entire input space. We can even allow a separate IK
for each inducing point, this is, if the set of inducing points
is Z = {zk}K
with the advantage of associating to each inducing point zk
its own set of adaptive parameters in Tr,k. For the PITC
approximation, this adds more hyperparameters to the like-
lihood, perhaps leading to overfitting. However, in the vari-
ational approximation we define all these new parameters
as variational parameters and therefore they do not cause
the model to overfit.
λr(z) =
Tr(z − v)ur(v)dv,
(3)
k=1, then λr(zk) =?Tr,k(zk− v)ur(v)dv,
Ifur(z)hasawhitenoise4GPpriorthecovariancefunction
3Returning to our sampling theorem analogy, the white noise
process has infinite bandwidth. It is therefore impossible to rep-
resent it by observations at a few fixed inducing points.
4It is straightforward to generalize the method for rough latent
for λr(x) is
Cov[λr(x),λr(x?)] =
?
Tr(x − z)Tr(x?− z)dz
(4)
and the cross-covariance between fd(x) and λr(x?) is
?
Notice that this cross-covariance function, unlike the case
of u inducing variables, maintains a weighted integration
over the whole input space. This implies that a single in-
ducing variable λr(x) can properly propagate information
from the full-length process ur(x) into f.
Cov[fd(x),λr(x?)] =
Gd,r(x − z)Tr(x?− z)dz. (5)
It is possible to combine the IKs defined above with the
PITC approximation of´Alvarez and Lawrence (2009), but
in this paper our focus will be on applying them within the
variational framework of Titsias (2009). We therefore refer
to the kernels as variational inducing kernels (VIKs).
Variational inference
We now extend the variational inference method of Titsias
(2009) to deal with multiple outputs and incorporate them
into the VIK framework.
We
p({yd,fd}D
step of the variational method is to augment this model
with inducing variables. For our purpose, suitable inducing
variables are defined through VIKs.
let λ = (λ1,...,λR) be the whole vector of inducing
variables where each λr is a K-dimensional vector of
values obtained according to eq. (3). λr’s role is to carry
information about the latent function ur(z). Each λr is
evaluated at the inputs Z and has its own VIK, Tr(x), that
depends on parameters θTr. The λ variables augment the
GP model according to p(y,f,λ) = p(y|f)p(f|λ)p(λ).
Here, p(λ) = N(λ|0,Kλ,λ) and Kλ,λis a block diagonal
functions that are not white noise or to combine smooth latent
functions with white noise.
compactly
d=1) as p(y,f)
write thejointprobability
p(y|f)p(f).
model
The first
=
More precisely,
Page 5
Mauricio A.´Alvarez, David Luengo, Michalis K. Titsias, Neil D. Lawrence
matrix where each block Kλr,λris obtained by evaluating
the covariance function in eq. (4) at the inputs Z. Addition-
ally, p(f|λ) = N(f|Kf,λK−1
where the cross-covariance Kf,λ is computed through
eq.(5).Becauseof
?p(f|λ)p(λ)dλ
exact inference in the initial GP model. Crucially, this
holds for any values of the augmentation parameters
(Z,{θTr}R
to turn these augmentation parameters into variational
parameters by applying approximate sparse inference.
λ,λλ,Kf,f− Kf,λK−1
λ,λKλ,f)
theconsistencycondition
=
p(f), performing exact inference
in the above augmented model is equivalent to performing
r=1). This is the key property that allows us
Our method now proceeds along the lines of Titsias
(2009). We introduce the variational distribution q(f,λ) =
p(f|λ)φ(λ), where p(f|λ) is the conditional GP prior de-
fined earlier and φ(λ) is an arbitrary variational distribu-
tion. By minimizing the KL divergence between q(f,λ)
and the true posterior p(f,λ|y), we can compute the fol-
lowing Jensen’s lower bound on the true log marginal like-
lihood (a detailed derivation of the bound is available in
´Alvarez et al. (2009b)):
FV = logN
?
y|0,Kf,λK−1
λ,λKλ,f+ Σ
?
−1
2tr
?
Σ−1?K
?
,
where Σ is the covariance function associated with the ad-
ditivenoiseprocessand?K = Kf,f−Kf,λK−1
log of a GP prior with the only difference that now the co-
variance matrix has a particular low rank form. This form
allows the inversion of the covariance matrix to take place
in O(NDK2) time rather than O(N3D3). The second part
can be seen as a penalization term that regularizes the es-
timation of the parameters. Notice also that only the diag-
onal of the exact covariance matrix Kf,fneeds to be com-
puted. Overall, the computation of the bound can be done
efficiently in O(NDK2) time.
λ,λKλ,f. Note
that this bound consists of two parts. The first part is the
The bound can be maximized with respect to all parameters
ofthecovariancefunction; bothmodelparametersandvari-
ational parameters. The variational parameters are the in-
ducing inputs Z and the parameters θTrof each VIK which
are rigorously selected so that the KL divergence is mini-
mized. In fact each VIK is also a variational quantity and
one could try different forms of VIKs in order to choose
the one that gives the best lower bound.
The form of the bound is very similar to the projected pro-
cess approximation, also known as DTC (Csat´ o and Op-
per, 2001; Seeger et al., 2003; Rasmussen and Williams,
2006). However, the bound has an additional trace term
that penalizes the movement of inducing inputs away from
the data. This term converts the DTC approximation to a
lower bound and prevents overfitting. In what follows, we
refer to this approximation as DTCVAR, where the VAR
suffix refers to the variational framework.
5Experiments
We present results of applying the method proposed for
two real-world datasets that will be described in short.
We compare the results obtained using PITC, the intrin-
sic coregionalization model (ICM)5employed in Bonilla et
al. (2008) and the method using variational inducing ker-
nels. For PITC we estimate the parameters through the
maximization of the approximated marginal likelihood of
equation (2) using the scaled-conjugate gradient method.
We use one latent function and both the covariance func-
tion of the latent process, kr(x,x?), and the kernel smooth-
ing function, Gd,r(x), follow a Gaussian form, this is
k(x,x?) = N(x − x?|0,C), where C is a diagonal ma-
trix. For the DTCVAR approximations, we maximize the
variational bound FV. Optimization is also performed us-
ing scaled conjugate gradient. We use one white noise la-
tent function and a corresponding inducing function. The
inducing kernels and the model kernels follow the same
Gaussian form. Using this form for the covariance or ker-
nel, all convolution integrals are solved analytically.
5.1 Exam score prediction
In this experiment the goal is to predict the exam score
obtained by a particular student belonging to a particular
school. The data comes from the Inner London Education
Authority (ILEA).6It consists of examination records from
139 secondary schools in years 1985, 1986 and 1987. It is a
random 50% sample with 15362 students. The input space
consists of features related to each student and features re-
lated to each school. From the multiple output point of
view, each school represents one output and the exam score
of each student a particular instantiation of that output.
We follow the same preprocessing steps employed in
Bonilla et al. (2008). The only features used are the
student-dependent ones (year in which each student took
the exam, gender, VR band and ethnic group), which are
categorical variables. Each of them is transformed to a bi-
nary representation. For example, the possible values that
the variable year of the exam can take are 1985, 1986 or
1987 and are represented as 100, 010 or 001. The trans-
formation is also applied to the variables gender (two bi-
nary variables), VR band (four binary variables) and ethnic
group (eleven binary variables), ending up with an input
space with dimension 20. The categorical nature of data
restricts the input space to 202 unique input feature vec-
tors. However, two students represented by the same in-
put vector x and belonging both to the same school d, can
obtain different exam scores. To reduce this noise in the
5The ICM is a particular case of the LMC with one latent func-
tion (Goovaerts, 1997).
6Data is available at http://www.cmm.bristol.ac.
uk/learning-training/multilevel-m-support/
datasets.shtml
Page 6
Efficient Multioutput Gaussian Processes through Variational Inducing Kernels
data, we follow Bonilla et al. (2008) in taking the mean of
the observations that, within a school, share the same in-
put vector and use a simple heteroskedastic noise model in
which the variance for each of these means is divided by
the number of observations used to compute it. The perfor-
mance measure employed is the percentage of unexplained
variance defined as the sum-squared error on the test set as
a percentage of the total data variance.7The performance
measure is computed for ten repetitions with 75% of the
data in the training set and 25% of the data in the test set.
Figure 5.1 shows results using PITC, DTCVAR with one
smoothing kernel and DTCVAR with as many inducing
kernels as inducing points (DTCVARS in the figure). For
50 inducing points all three alternatives lead to approx-
imately the same results. PITC keeps a relatively con-
stant performance for all values of inducing points, while
the DTCVAR approximations increase their performance
as the number of inducing points increases. This is consis-
tent with the expected behaviour of the DTCVAR methods,
since the trace term penalizes the model for a reduced num-
ber of inducing points. Notice that all the approximations
outperform independent GPs and the best result of the ICM
presented in Bonilla et al. (2008).
SM 5SM 20SM 50 ICM IND
40
45
50
55
60
65
70
75
Percentage of unexplained variance
Method
PITC
DTCVAR
DTCVARS
Figure2: Examscorepredictionresultsfortheschooldataset. Re-
sults include the mean of the percentage of unexplained variance
of ten repetitions of the experiment, together with one standard
deviation. In the bottom, SM X stands for sparse method with X
inducing points, DTCVAR refers to the DTC variational approx-
imation with one smoothing kernel and DTCVARS to the same
approximation using as many inducing kernels as inducing points.
Results using the ICM model and independent GPs (appearing as
IND in the figure) have also been included.
5.2Compiler prediction performance
In this dataset the outputs correspond to the speed-up of 11
C programs after some transformation sequence has been
applied to them. The speed-up is defined as the execution
time of the original program divided by the execution time
of the transformed program. The input space consists of
13-dimensional binary feature vectors, where the presence
7In Bonilla et al. (2008), results are reported in terms of ex-
plained variance.
of a one in these vectors indicates that the program has re-
ceived that particular transformation. The dataset contains
88214observations foreach outputand thesamenumber of
input vectors. All the outputs share the same input space.
Due to technical requirements, it is important that the pre-
diction of the speed-up for the particular program is made
using few observations in the training set. We compare our
results to the ones presented in Bonilla et al. (2008) and use
N = 16, 32, 64 and 128 for the training set. The remaining
88214−N observations are used for testing, employing as
performance measure the mean absolute error. The experi-
ment is repeated ten times and standard deviations are also
reported. We only include results for the average perfor-
mance over the 11 outputs.
Figure 3 shows the results of applying independent GPs
(IND in the figure), the intrinsic coregionalization model
(ICM in the figure), PITC, DTCVAR with one inducing
kernel(DTCVARinthefigure)andDTCVARwithasmany
inducing kernels as inducing points (DTCVARS in the fig-
ure). Since the training sets are small enough, we also in-
cluderesultsofapplyingtheGPgeneratedusingthefullco-
variance matrix of the convolution construction (see FULL
GP in the figure). We repeated the experiment for different
values of K, but show results only for K = N/2. Re-
sults for ICM and IND were obtained from Bonilla et al.
(2008). In general, the DTCVAR variants outperform the
16 3264128
0.02
0.04
0.06
0.08
0.1
0.12
Mean Absolute Error
Number of training points
IND
ICM
PITC
DTCVAR
DTCVARS
FULL GP
Figure 3: Mean absolute error and standard deviation over ten
repetitions of the compiler experiment as a function of the train-
ing points. IND stands for independent GPs, ICM stands for in-
trinsic coregionalization model, DTCVAR refers to the DTCVAR
approximation using one inducing kernel, DTCVARS refers to
the DTCVAR approximation using as many inducing kernels as
inducing points and FULL GP stands for the GP for the multiple
outputs without any approximation.
ICMmethod, andtheindependentGPsforN = 16, 32and
64. Inthiscase, usingasmanyinducingkernelsasinducing
points improves on average the performance. All methods,
including the independent GPs are better than PITC. The
size of the test set encourages the application of the sparse
methods: for N = 128, making the prediction of the whole
dataset using the full GP takes in average 22 minutes while
the prediction with DTCVAR takes 0.65 minutes. Using
more inducing kernels improves the performance, but also
Page 7
Mauricio A.´Alvarez, David Luengo, Michalis K. Titsias, Neil D. Lawrence
makes the evaluation of the test set more expensive. For
DTCVARS, the evaluation takes in average 6.8 minutes.
Time results are average results over the ten repetitions.
6 Stochastic Latent Force Models
The starting point of stochastic differential equations is
a stochastic version of the equation of motion, which is
called Langevin’s equation:
df(t)
dt
where f(t) is the velocity of the particle, −Cf(t) is a sys-
tematic friction term, u(t) is a random fluctuation external
force, i.e. white noise, and S indicates the sensitivity of the
ouputtotherandomfluctuations. Inthemathematicalprob-
ability literature, the above is written more rigorously as
df(t) = −Cf(t)dt + SdW(t) where W(t) is the Wiener
process (standard Brownian motion). Since u(t) is a GP
and the equation is linear, f(t) must be also a GP which
turns out to be the Ornstein-Uhlenbeck (OU) process.
= −Cf(t) + Su(t),
(6)
Here, we are interested in extending the Langevin equation
to model multivariate time series. The way that the model
in (6) is extended is by adding more output signals and
more external forces. The forces can be either smooth (sys-
tematic or drift-type) forces or white noise forces. Thus,
dfd(t)
dt
= −Ddfd(t) +
R
?
r=1
Sd,rur(t),
(7)
where fd(t) is the dth output signal. Each ur(t) can be ei-
ther a smooth latent force that is assigned a GP prior with
covariance function kr(t,t?) or a white noise force that has
a GP prior with covariance function δ(t − t?). That is, we
have a composition of R latent forces, where Rsof them
correspond to smooth latent forces and Rocorrespond to
white noise processes. The intuition behind this combi-
nation of input forces is that the smooth part can be used
to represent medium/long term trends that cause a depar-
ture from the mean of the output processes, whereas the
stochastic part is related to short term fluctuations around
the mean. A model with Rs= 1 and Ro= 0 was proposed
by Lawrence et al. (2007) to describe protein transcription
regulation in a single input motif (SIM) gene network.
Solving the differential equation (7), we obtain
fd(t) = e−Ddtfd0+
R
?
r=1
Sd,r
?t
0
e−Dd(t−z)ur(z)dz,
where fd0arises from the initial condition. This model now
is a special case of the multioutput regression model dis-
cussed in sections 1 and 2 where each output signal yd(t) =
fd(t) + ? has a mean function e−Ddtfd0and each model
kernel Gd,r(x) is equal to Sd,re−Dd(t−z).
model can be viewed as a stochastic latent force model
(SLFM) following the work of´Alvarez et al. (2009a).
The above
Latent market forces
The application considered is the inference of missing data
in a multivariate financial data set: the foreign exchange
rate w.r.t. the dollar of 10 of the top international curren-
cies (Canadian Dollar [CAD], Euro [EUR], Japanese Yen
[JPY], Great British Pound [GBP], Swiss Franc [CHF],
Australian Dollar [AUD], Hong Kong Dollar [HKD], New
Zealand Dollar [NZD], South Korean Won [KRW] and
Mexican Peso [MXN]) and 3 precious metals (gold [XAU],
silver [XAG] and platinum [XPT]).8We considered all the
data available for the calendar year of 2007 (251 working
days). In this data there are several missing values: XAU,
XAG and XPT have 9, 8 and 42 days of missing values re-
spectively. On top of this, we also introduced artificially
long sequences of missing data. Our objective is to model
the data and test the effectiveness of the model by imputing
these missing points. We removed a test set from the data
by extracting contiguous sections from 3 currencies asso-
ciated with very different geographic locations: we took
days 50–100 from CAD, days 100–150 from JPY and days
150–200 from AUD. The remainder of the data comprised
the training set, which consisted of 3051 points, with the
test data containing 153 points. For preprocessing we re-
moved the mean from each output and scaled them so that
they all had unit variance.
It seems reasonable to suggest that the fluctuations of the
13 correlated financial time-series are driven by a smaller
number of latent market forces. We therefore modelled
the data with up to six latent forces which could be noise
or smooth GPs.The GP priors for the smooth latent
forces are assumed to have a Gaussian covariance function,
kurur(t,t?) = (1/?2π?2
We present an example with R = 4. For this value of R, we
consider all the possible combinations of Roand Rs. The
training was performed in all cases by maximizing the vari-
ational bound using the scale conjugate gradient algorithm
until convergence was achieved. The best performance in
terms of achieving the highest value for FV was obtained
for Rs= 1 and Ro= 3. We compared against the LMC
model for different values of the latent functions in that
framework. While our best model resulted in an standard-
izedmeansquareerrorof0.2795, thebestLMC(withR=2)
resulted in 0.3927. We plotted predictions from the latent
market force model to characterize the performance when
filling in missing data. In figure 4 we show the output
signals obtained using the model with the highest bound
(Rs = 1 and Ro = 3) for CAD, JPY and AUD. Note
that the model performs better at capturing the deep drop
in AUD than it does for the fluctuations in CAD and JPY.
r)exp(−((t − t?)2)/2?2
r), where
the hyperparameter ?ris known as the lengthscale.
8Data is available at http://fx.sauder.ubc.ca/
data.html.
Page 8
Efficient Multioutput Gaussian Processes through Variational Inducing Kernels
50100 150200250
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
(a) CAD: Real data and prediction
50 100150 200250
7.8
8
8.2
8.4
8.6
8.8
9
9.2
9.4x 10
−3
(b) JPY: Real data and prediction
50100 150200 250
0.7
0.75
0.8
0.85
0.9
0.95
(c) AUD: Real data and prediction
Figure 4: Predictions from the model with Rs = 1 and Ro = 3 are shown as solid lines for the mean and grey bars for error bars at 2
standard deviations. For CAD, JPY and AUD the data was artificially held out. The true values are shown as a dotted line. Crosses on
the x-axes of all plots show the locations of the inducing inputs.
7 Conclusions
We have presented a variational approach to sparse approx-
imations in convolution processes. Our main focus was to
provide efficient mechanisms for learning in multiple out-
put Gaussian processes when the latent function is fluctuat-
ing rapidly. In order to do so, we have introduced the con-
cept of inducing function, which generalizes the idea of in-
ducing point, traditionally employed in sparse GP methods.
The approach extends the variational approximation of Tit-
sias (2009) to the multiple output case. Using our approach
we can perform efficient inference on latent force models
which are based around SDEs, but also contain a smooth
driving force. Our approximation is flexible enough and
has been shown to be applicable to a wide range of data
sets, including high-dimensional ones.
Acknowledgements
We would like to thank Edwin Bonilla for his valuable feedback
with respect to the datasets in section 5. Also to the authors of
Bonilla et al. (2008) who kindly made the compiler dataset avail-
able. DL has been partly financed by the Spanish government
through CICYT projects TEC2006-13514-C02-01 and TEC2009-
14504-C02-01, and the CONSOLIDER-INGENIO 2010 Program
(Project CSD2008-00010). MA and NL have been financed by a
Google Research Award “Mechanistically Inspired Convolution
Processes for Learning” and MA, NL and MT have been financed
by EPSRC Grant No EP/F005687/1 “Gaussian Processes for Sys-
tems Identification with Applications in Systems Biology”.
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