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Distributed Gaussian Processes

Marc Deisenroth

Department of ComputingImperial College London

http://wp.doc.ic.ac.uk/sml/marc-deisenroth

Gaussian Process Summer School, University of Sheffield15th September 2015

Table of Contents

Gaussian Processes

Sparse Gaussian Processes

Distributed Gaussian Processes

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 2

Problem Setting

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

−2

0

2

x

f(x)

Objective

For a set of N observations yi “ f pxiq ` ε, ε „ N`

0, σ2ε

˘

, find adistribution over functions pp f |X, yq that explains the data

GP is a good solution to this probabilistic regression problem

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 3

GP Training via Marginal Likelihood Maximization

GP TrainingMaximize the evidence/marginal likelihood ppy|X, θqwith respect tothe hyper-parameters θ: θ˚ P arg maxθ log ppy|X, θq

log ppy|X, θq “ ´12 yJK´1y ´ 1

2 log |K| ` const

§ Automatic trade-off between data fit and model complexity

§ Gradient-based optimization possible:

B log ppy|X, θq

Bθ“ 1

2 yJK´1 BKBθ

K´1y´ 12 tr

`

K´1 BKBθ

˘

§ Computational complexity: OpN3q for |K| and K´1

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 4

GP Training via Marginal Likelihood Maximization

GP TrainingMaximize the evidence/marginal likelihood ppy|X, θqwith respect tothe hyper-parameters θ: θ˚ P arg maxθ log ppy|X, θq

log ppy|X, θq “ ´12 yJK´1y ´ 1

2 log |K| ` const

§ Automatic trade-off between data fit and model complexity

§ Gradient-based optimization possible:

B log ppy|X, θq

Bθ“ 1

2 yJK´1 BKBθ

K´1y´ 12 tr

`

K´1 BKBθ

˘

§ Computational complexity: OpN3q for |K| and K´1

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 4

GP Training via Marginal Likelihood Maximization

GP TrainingMaximize the evidence/marginal likelihood ppy|X, θqwith respect tothe hyper-parameters θ: θ˚ P arg maxθ log ppy|X, θq

log ppy|X, θq “ ´12 yJK´1y ´ 1

2 log |K| ` const

§ Automatic trade-off between data fit and model complexity

§ Gradient-based optimization possible:

B log ppy|X, θq

Bθ“ 1

2 yJK´1 BKBθ

K´1y´ 12 tr

`

K´1 BKBθ

˘

§ Computational complexity: OpN3q for |K| and K´1

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 4

GP Predictions

At a test point x˚ the predictive (posterior) distribution is Gaussian:

pp f px˚q|x˚, X, y, θq “ N`

f˚ |m˚, σ2˚

˘

m˚ “ kpX, x˚qJK´1y

σ2˚ “ kpx˚, x˚q ´ kpX, x˚qJK´1kpX, x˚q

When you cache K´1 and K´1y after training, then

§ The mean prediction can be computed in OpNq§ The variance prediction can be computed in OpN2q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 5

GP Predictions

At a test point x˚ the predictive (posterior) distribution is Gaussian:

pp f px˚q|x˚, X, y, θq “ N`

f˚ |m˚, σ2˚

˘

m˚ “ kpX, x˚qJK´1y

σ2˚ “ kpx˚, x˚q ´ kpX, x˚qJK´1kpX, x˚q

When you cache K´1 and K´1y after training, then

§ The mean prediction can be computed in OpNq§ The variance prediction can be computed in OpN2q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 5

Application Areas

§ Bayesian Optimization (Experimental Design)Model unknown utility functions with GPs

§ Reinforcement Learning and RoboticsModel value functions and/or dynamics with GPs

§ Data visualizationNonlinear dimensionality reduction (GP-LVM)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 6

Limitations of Gaussian Processes

Computational and memory complexity

§ Training scales in OpN3q

§ Prediction (variances) scales in OpN2q

§ Memory requirement: OpND` N2q

Practical limit N « 10, 000

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 7

Table of Contents

Gaussian Processes

Sparse Gaussian Processes

Distributed Gaussian Processes

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 8

GP Factor Graph

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values Training function values fi

§ Probabilistic graphical model (factor graph) of a GP§ All function values are jointly Gaussian distributed (e.g., training

and test function values)

§ GP prior

pp f , f˚q “ N˜«

00

ff

,

«

K f f K f˚

K˚ f K˚˚

ff¸

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 9

GP Factor Graph

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values Training function values fi

§ Probabilistic graphical model (factor graph) of a GP§ All function values are jointly Gaussian distributed (e.g., training

and test function values)§ GP prior

pp f , f˚q “ N˜«

00

ff

,

«

K f f K f˚

K˚ f K˚˚

ff¸

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 9

Inducing Variables

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values Training function values fi

Hypothetical function values uj

§ Introduce inducing function values fu

“Hypothetical” function values

§ All function values are still jointly Gaussian distributed (e.g.,training, test and inducing function values)

§ Approach: “Compress” real function values into inducingfunction values

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 10

Inducing Variables

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values Training function values fi

Hypothetical function values uj

§ Introduce inducing function values fu

“Hypothetical” function values§ All function values are still jointly Gaussian distributed (e.g.,

training, test and inducing function values)§ Approach: “Compress” real function values into inducing

function valuesDistributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 10

Central Approximation Scheme

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values

§ Approximation: Training and test set are conditionallyindependent given the inducing function values: f KK f˚| fu

§ Then, the effective GP prior is

qp f , f˚q “ż

pp f | fuqpp f˚| fuqpp fuqd fu “ N

¨

˝

«

00

ff

,

»

–

K f f Q f˚

Q˚ f K˚˚

fi

fl

˛

‚ ,

Q˚ f :“ K˚ fu K´1fu fu

K fu f Nystrom approximation

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 11

Central Approximation Scheme

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values

§ Approximation: Training and test set are conditionallyindependent given the inducing function values: f KK f˚| fu

§ Then, the effective GP prior is

qp f , f˚q “ż

pp f | fuqpp f˚| fuqpp fuqd fu “ N

¨

˝

«

00

ff

,

»

–

K f f Q f˚

Q˚ f K˚˚

fi

fl

˛

‚ ,

Q˚ f :“ K˚ fu K´1fu fu

K fu f Nystrom approximation

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 11

FI(T)C Sparse Approximation

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values

§ Assume that training (and test sets) arefully independent given the inducingvariables (Snelson & Ghahramani, 2006)

§ Effective GP prior with this approximation

qp f , f˚q “ N

¨

˝

«

00

ff

,

»

–

Q f f ´ diagpQ f f ´K f f q Q f˚

Q˚ f K˚˚

fi

fl

˛

‚

§ Q˚˚ ´ diagpQ˚˚ ´K˚˚q can be used instead of K˚˚ FIC

§ Training: OpNM2q, Prediction: OpM2q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 12

FI(T)C Sparse Approximation

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values

§ Assume that training (and test sets) arefully independent given the inducingvariables (Snelson & Ghahramani, 2006)

§ Effective GP prior with this approximation

qp f , f˚q “ N

¨

˝

«

00

ff

,

»

–

Q f f ´ diagpQ f f ´K f f q Q f˚

Q˚ f K˚˚

fi

fl

˛

‚

§ Q˚˚ ´ diagpQ˚˚ ´K˚˚q can be used instead of K˚˚ FIC

§ Training: OpNM2q, Prediction: OpM2q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 12

FI(T)C Sparse Approximation

f(x1) f(xN ) f∗1 f∗

L

Training data Test data

f(u1) f(uM )

Inducing function values

§ Assume that training (and test sets) arefully independent given the inducingvariables (Snelson & Ghahramani, 2006)

§ Effective GP prior with this approximation

qp f , f˚q “ N

¨

˝

«

00

ff

,

»

–

Q f f ´ diagpQ f f ´K f f q Q f˚

Q˚ f K˚˚

fi

fl

˛

‚

§ Q˚˚ ´ diagpQ˚˚ ´K˚˚q can be used instead of K˚˚ FIC

§ Training: OpNM2q, Prediction: OpM2q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 12

Inducing Inputs

§ FI(T)C sparse approximation exploits inducing function valuesf puiq, where ui are the corresponding inputs

§ These inputs are unknown a priori Find “optimal” ones

§ Find them by maximizing the FI(T)C marginal likelihood withrespect to the inducing inputs (and the standardhyper-parameters):

u˚1:M P arg maxu1:M

qFITCpy|X, u1:M, θq

§ Intuitively: The marginal likelihood is not only parameterized bythe hyper-parameters θ, but also by the inducing inputs u1:M.

§ End up with a high-dimensional non-convex optimizationproblem with MD additional parameters

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 13

Inducing Inputs

§ FI(T)C sparse approximation exploits inducing function valuesf puiq, where ui are the corresponding inputs

§ These inputs are unknown a priori Find “optimal” ones

§ Find them by maximizing the FI(T)C marginal likelihood withrespect to the inducing inputs (and the standardhyper-parameters):

u˚1:M P arg maxu1:M

qFITCpy|X, u1:M, θq

§ Intuitively: The marginal likelihood is not only parameterized bythe hyper-parameters θ, but also by the inducing inputs u1:M.

§ End up with a high-dimensional non-convex optimizationproblem with MD additional parameters

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 13

Inducing Inputs

§ FI(T)C sparse approximation exploits inducing function valuesf puiq, where ui are the corresponding inputs

§ These inputs are unknown a priori Find “optimal” ones

§ Find them by maximizing the FI(T)C marginal likelihood withrespect to the inducing inputs (and the standardhyper-parameters):

u˚1:M P arg maxu1:M

qFITCpy|X, u1:M, θq

§ Intuitively: The marginal likelihood is not only parameterized bythe hyper-parameters θ, but also by the inducing inputs u1:M.

§ End up with a high-dimensional non-convex optimizationproblem with MD additional parameters

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 13

Inducing Inputs

§ These inputs are unknown a priori Find “optimal” ones

u˚1:M P arg maxu1:M

qFITCpy|X, u1:M, θq

§ End up with a high-dimensional non-convex optimizationproblem with MD additional parameters

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 13

Inducing Inputs

§ These inputs are unknown a priori Find “optimal” ones

u˚1:M P arg maxu1:M

qFITCpy|X, u1:M, θq

§ End up with a high-dimensional non-convex optimizationproblem with MD additional parameters

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 13

FITC Example

Figure from Ed Snelson

§ Pink: Original data

§ Red crosses: Initialization ofinducing inputs

§ Blue crosses: Location of inducinginputs after optimization

§ Efficient compression of the original data set

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 14

Summary Sparse Gaussian Processes

§ Sparse approximations typically approximate a GP with N datapoints by a model with M ! N data points

§ Selection of these M data points can be tricky and may involvenon-trivial computations (e.g., optimizing inducing inputs)

§ Simple (random) subset selection is fast and generally robust(Chalupka et al., 2013)

§ Computational complexity: OpM3q or OpNM2q for training

§ Practical limit M ď 104. Often: M P Op102q in the case ofinducing variables

§ If we set M “ N{100, i.e., each inducing function valuesummarizes 100 real function values, our practical limit isN P Op106q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 15

Summary Sparse Gaussian Processes

§ Sparse approximations typically approximate a GP with N datapoints by a model with M ! N data points

§ Selection of these M data points can be tricky and may involvenon-trivial computations (e.g., optimizing inducing inputs)

§ Simple (random) subset selection is fast and generally robust(Chalupka et al., 2013)

§ Computational complexity: OpM3q or OpNM2q for training

§ Practical limit M ď 104. Often: M P Op102q in the case ofinducing variables

§ If we set M “ N{100, i.e., each inducing function valuesummarizes 100 real function values, our practical limit isN P Op106q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 15

Summary Sparse Gaussian Processes

§ Sparse approximations typically approximate a GP with N datapoints by a model with M ! N data points

§ Selection of these M data points can be tricky and may involvenon-trivial computations (e.g., optimizing inducing inputs)

§ Simple (random) subset selection is fast and generally robust(Chalupka et al., 2013)

§ Computational complexity: OpM3q or OpNM2q for training

§ Practical limit M ď 104. Often: M P Op102q in the case ofinducing variables

§ If we set M “ N{100, i.e., each inducing function valuesummarizes 100 real function values, our practical limit isN P Op106q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 15

Summary Sparse Gaussian Processes

§ Simple (random) subset selection is fast and generally robust(Chalupka et al., 2013)

§ Computational complexity: OpM3q or OpNM2q for training

§ Practical limit M ď 104. Often: M P Op102q in the case ofinducing variables

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 15

Summary Sparse Gaussian Processes

§ Simple (random) subset selection is fast and generally robust(Chalupka et al., 2013)

§ Computational complexity: OpM3q or OpNM2q for training

§ Practical limit M ď 104. Often: M P Op102q in the case ofinducing variables

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 15

Summary Sparse Gaussian Processes

§ Simple (random) subset selection is fast and generally robust(Chalupka et al., 2013)

§ Computational complexity: OpM3q or OpNM2q for training

§ Practical limit M ď 104. Often: M P Op102q in the case ofinducing variables

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 15

Table of Contents

Gaussian Processes

Sparse Gaussian Processes

Distributed Gaussian Processes

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 16

Distributed Gaussian Processes

Joint work with Jun Wei Ng

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 17

An Orthogonal Approximation: Distributed GPs

−1

−1 −1−1−1−1

=

Standard GP

Data set Kernelmatrix

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

§ Randomly split the full data set into M chunks

§ Place M independent GP models (experts) on these small chunks

§ Independent computations can be distributed

§ Block-diagonal approximation of kernel matrix K

§ Combine independent computations to an overall result

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 18

An Orthogonal Approximation: Distributed GPs

−1

−1 −1−1−1−1

=

Standard GP

Data set Kernelmatrix

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

§ Randomly split the full data set into M chunks

§ Place M independent GP models (experts) on these small chunks

§ Independent computations can be distributed

§ Block-diagonal approximation of kernel matrix K

§ Combine independent computations to an overall result

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 18

An Orthogonal Approximation: Distributed GPs

−1

−1 −1−1−1−1

=

Standard GP

Data set Kernelmatrix

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

§ Randomly split the full data set into M chunks

§ Place M independent GP models (experts) on these small chunks

§ Independent computations can be distributed

§ Block-diagonal approximation of kernel matrix K

§ Combine independent computations to an overall result

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 18

An Orthogonal Approximation: Distributed GPs

−1

−1 −1−1−1−1

=

Standard GP

Data set Kernelmatrix

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

§ Randomly split the full data set into M chunks

§ Place M independent GP models (experts) on these small chunks

§ Independent computations can be distributed

§ Block-diagonal approximation of kernel matrix K

§ Combine independent computations to an overall result

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 18

An Orthogonal Approximation: Distributed GPs

−1

−1 −1−1−1−1

=

Standard GP

Data set Kernelmatrix

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

§ Randomly split the full data set into M chunks

§ Place M independent GP models (experts) on these small chunks

§ Independent computations can be distributed

§ Block-diagonal approximation of kernel matrix K

§ Combine independent computations to an overall result

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 18

An Orthogonal Approximation: Distributed GPs

−1

−1 −1−1−1−1

=

Standard GP

Data set Kernelmatrix

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

§ Randomly split the full data set into M chunks

§ Place M independent GP models (experts) on these small chunks

§ Independent computations can be distributed

§ Block-diagonal approximation of kernel matrix K

§ Combine independent computations to an overall result

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 18

Training the Distributed GP

§ Split data set of size N into M chunks of size P

§ Independence of experts Factorization of marginal likelihood:

log ppy|X, θq «ÿM

k“1log pkpypkq|Xpkq, θq

§ Distributed optimization and training straightforward

§ Computational complexity: OpMP3q [instead of OpN3q]But distributed over many machines

§ Memory footprint: OpMP2 ` NDq [instead of OpN2 ` NDq]

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 19

Training the Distributed GP

§ Split data set of size N into M chunks of size P

§ Independence of experts Factorization of marginal likelihood:

log ppy|X, θq «ÿM

k“1log pkpypkq|Xpkq, θq

§ Distributed optimization and training straightforward

§ Computational complexity: OpMP3q [instead of OpN3q]But distributed over many machines

§ Memory footprint: OpMP2 ` NDq [instead of OpN2 ` NDq]

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 19

Training the Distributed GP

§ Split data set of size N into M chunks of size P

§ Independence of experts Factorization of marginal likelihood:

log ppy|X, θq «ÿM

k“1log pkpypkq|Xpkq, θq

§ Distributed optimization and training straightforward

§ Computational complexity: OpMP3q [instead of OpN3q]But distributed over many machines

§ Memory footprint: OpMP2 ` NDq [instead of OpN2 ` NDq]

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 19

Empirical Training Time

103 104 105 106 107 108

Training data set size

10-2

10-1

100

101

102

103

Com

puta

tion t

ime o

f N

LML

and its

gra

die

nts

(s)

101

102

103

104

105

106

107

Num

ber

of

GP e

xpert

s (D

GP)

Computation time (DGP)

Computation time (Full GP)

Computation time (FITC)

Number of GP experts (DGP)

§ NLML is proportional to training time

§ Full GP (16K training points) « sparse GP (50K training points)« distributed GP (16M training points)

Push practical limit by order(s) of magnitude

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 20

Empirical Training Time

103 104 105 106 107 108

Training data set size

10-2

10-1

100

101

102

103

Com

puta

tion t

ime o

f N

LML

and its

gra

die

nts

(s)

101

102

103

104

105

106

107

Num

ber

of

GP e

xpert

s (D

GP)

Computation time (DGP)

Computation time (Full GP)

Computation time (FITC)

Number of GP experts (DGP)

§ NLML is proportional to training time

§ Full GP (16K training points) « sparse GP (50K training points)« distributed GP (16M training points)

Push practical limit by order(s) of magnitudeDistributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 20

Practical Training Times

§ Training* with N “ 106, D “ 1 on a laptop: « 10–30 min

§ Training* with N “ 5ˆ 106, D “ 8 on a workstation: « 4 hours

*: Maximize the marginal likelihood, stop when converged**

**: Convergence often after 30–80 line searches***

***: Line search « 2–3 evaluations of marginal likelihood andits gradient (usually OpN3q)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 21

Practical Training Times

§ Training* with N “ 106, D “ 1 on a laptop: « 10–30 min

§ Training* with N “ 5ˆ 106, D “ 8 on a workstation: « 4 hours

*: Maximize the marginal likelihood, stop when converged**

**: Convergence often after 30–80 line searches***

***: Line search « 2–3 evaluations of marginal likelihood andits gradient (usually OpN3q)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 21

Practical Training Times

§ Training* with N “ 106, D “ 1 on a laptop: « 10–30 min

§ Training* with N “ 5ˆ 106, D “ 8 on a workstation: « 4 hours

*: Maximize the marginal likelihood, stop when converged**

**: Convergence often after 30–80 line searches***

***: Line search « 2–3 evaluations of marginal likelihood andits gradient (usually OpN3q)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 21

Practical Training Times

§ Training* with N “ 106, D “ 1 on a laptop: « 10–30 min

§ Training* with N “ 5ˆ 106, D “ 8 on a workstation: « 4 hours

*: Maximize the marginal likelihood, stop when converged**

**: Convergence often after 30–80 line searches***

***: Line search « 2–3 evaluations of marginal likelihood andits gradient (usually OpN3q)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 21

Predictions with the Distributed GP

µ, σ

§ Prediction of each GP expert is Gaussian N`

µi, σ2i

˘

§ How to combine them to an overall prediction N`

µ, σ2˘

?

Product-of-GP-experts

§ PoE (product of experts; Ng & Deisenroth, 2014)

§ gPoE (generalized product of experts; Cao & Fleet, 2014)

§ BCM (Bayesian Committee Machine; Tresp, 2000)

§ rBCM (robust BCM; Deisenroth & Ng, 2015)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 22

Predictions with the Distributed GP

µ, σ

§ Prediction of each GP expert is Gaussian N`

µi, σ2i

˘

§ How to combine them to an overall prediction N`

µ, σ2˘

?

Product-of-GP-experts

§ PoE (product of experts; Ng & Deisenroth, 2014)

§ gPoE (generalized product of experts; Cao & Fleet, 2014)

§ BCM (Bayesian Committee Machine; Tresp, 2000)

§ rBCM (robust BCM; Deisenroth & Ng, 2015)Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 22

Objectives

µ, σ

µ, σ

µ1, σ1 µ2, σ2

µ11, σ11 µ12, σ12 µ13, σ13 µ21, σ21 µ22, σ22 µ23, σ23

Figure: Two computational graphs

§ Scale to large data sets 3

§ Good approximation of full GP (“ground truth”)

§ Predictions independent of computational graphRuns on heterogeneous computing infrastructures (laptop,

cluster, ...)

§ Reasonable predictive variances

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 23

Objectives

µ, σ

µ, σ

µ1, σ1 µ2, σ2

µ11, σ11 µ12, σ12 µ13, σ13 µ21, σ21 µ22, σ22 µ23, σ23

Figure: Two computational graphs

§ Scale to large data sets 3

§ Good approximation of full GP (“ground truth”)

§ Predictions independent of computational graphRuns on heterogeneous computing infrastructures (laptop,

cluster, ...)

§ Reasonable predictive variances

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 23

Objectives

µ, σ

µ, σ

µ1, σ1 µ2, σ2

µ11, σ11 µ12, σ12 µ13, σ13 µ21, σ21 µ22, σ22 µ23, σ23

Figure: Two computational graphs

§ Scale to large data sets 3

§ Good approximation of full GP (“ground truth”)

§ Predictions independent of computational graphRuns on heterogeneous computing infrastructures (laptop,

cluster, ...)

§ Reasonable predictive variances

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 23

Objectives

µ, σ

µ, σ

µ1, σ1 µ2, σ2

µ11, σ11 µ12, σ12 µ13, σ13 µ21, σ21 µ22, σ22 µ23, σ23

Figure: Two computational graphs

§ Scale to large data sets 3

§ Good approximation of full GP (“ground truth”)

cluster, ...)

§ Reasonable predictive variances

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 23

Running Example

Investigate various product-of-experts modelsSame training procedure, but different mechanisms for predictions

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 24

Product of GP Experts

§ Prediction model (independent predictors):

pp f˚|x˚,Dq9Mź

k“1

pkp f˚|x˚,Dpkqq ,

pkp f˚|x˚,Dpkqq “ N`

f˚ | µkpx˚q, σ2k px˚q

˘

§ Predictive precision (inverse variance) and mean:

pσpoe˚ q´2 “

ÿ

kσ´2

k px˚q

µpoe˚ “ pσ

poe˚ q2

ÿ

kσ´2

k px˚qµkpx˚q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 25

Product of GP Experts

§ Prediction model (independent predictors):

pp f˚|x˚,Dq9Mź

k“1

pkp f˚|x˚,Dpkqq ,

pkp f˚|x˚,Dpkqq “ N`

f˚ | µkpx˚q, σ2k px˚q

˘

§ Predictive precision (inverse variance) and mean:

pσpoe˚ q´2 “

ÿ

kσ´2

k px˚q

µpoe˚ “ pσ

poe˚ q2

ÿ

kσ´2

k px˚qµkpx˚q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 25

Computational Graph

GP Experts

PoE

PoE

Prediction:

pp f˚|x˚,Dq9Mź

k“1

pkp f˚|x˚,Dpkqq

Multiplication is associative: a ˚ b ˚ c ˚ d “ pa ˚ bq ˚ pc ˚ dq

Mź

k“1

pkp f˚|Dpkqq “Lź

k“1

Lkź

i“1

pkip f˚|Dpkiqq ,ÿ

k

Lk “ M

Independent of computational graph 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 26

Computational Graph

GP Experts

PoE

PoE

Prediction:

pp f˚|x˚,Dq9Mź

k“1

pkp f˚|x˚,Dpkqq

Multiplication is associative: a ˚ b ˚ c ˚ d “ pa ˚ bq ˚ pc ˚ dq

Mź

k“1

pkp f˚|Dpkqq “Lź

k“1

Lkź

i“1

pkip f˚|Dpkiqq ,ÿ

k

Lk “ M

Independent of computational graph 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 26

Product of GP Experts

§ Unreasonable variances for M ą 1:

pσpoe˚ q´2 “

ÿ

kσ´2

k px˚q

§ The more experts the more certain the prediction, even if everyexpert itself is very uncertain 7 Cannot fall back to the prior

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 27

Product of GP Experts

§ Unreasonable variances for M ą 1:

pσpoe˚ q´2 “

ÿ

kσ´2

k px˚q

§ The more experts the more certain the prediction, even if everyexpert itself is very uncertain 7 Cannot fall back to the prior

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 27

Generalized Product of GP Experts

§ Weight the responsiblity of each expert in PoE with βk

§ Prediction model (independent predictors):

pp f˚|x˚,Dq9Mź

k“1

pβk

k p f˚|x˚,Dpkqq

pkp f˚|x˚,Dpkqq “ N`

f˚ | µkpx˚q, σ2k px˚q

˘

§ Predictive precision and mean:

pσgpoe˚ q´2 “

ÿ

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

ÿ

kβkσ´2

k px˚q µkpx˚q

§ Withř

k βk “ 1, the model can fall back to the prior 3

“Log-opinion pool” model (Heskes, 1998)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 28

Generalized Product of GP Experts

§ Weight the responsiblity of each expert in PoE with βk

§ Prediction model (independent predictors):

pp f˚|x˚,Dq9Mź

k“1

pβk

k p f˚|x˚,Dpkqq

pkp f˚|x˚,Dpkqq “ N`

f˚ | µkpx˚q, σ2k px˚q

˘

§ Predictive precision and mean:

pσgpoe˚ q´2 “

ÿ

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

ÿ

kβkσ´2

k px˚q µkpx˚q

§ Withř

k βk “ 1, the model can fall back to the prior 3

“Log-opinion pool” model (Heskes, 1998)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 28

Generalized Product of GP Experts

§ Weight the responsiblity of each expert in PoE with βk

§ Prediction model (independent predictors):

pp f˚|x˚,Dq9Mź

k“1

pβk

k p f˚|x˚,Dpkqq

pkp f˚|x˚,Dpkqq “ N`

f˚ | µkpx˚q, σ2k px˚q

˘

§ Predictive precision and mean:

pσgpoe˚ q´2 “

ÿ

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

ÿ

kβkσ´2

k px˚q µkpx˚q

§ Withř

k βk “ 1, the model can fall back to the prior 3

“Log-opinion pool” model (Heskes, 1998)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 28

Generalized Product of GP Experts

§ Weight the responsiblity of each expert in PoE with βk

§ Prediction model (independent predictors):

pp f˚|x˚,Dq9Mź

k“1

pβk

k p f˚|x˚,Dpkqq

pkp f˚|x˚,Dpkqq “ N`

f˚ | µkpx˚q, σ2k px˚q

˘

§ Predictive precision and mean:

pσgpoe˚ q´2 “

ÿ

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

ÿ

kβkσ´2

k px˚q µkpx˚q

§ Withř

k βk “ 1, the model can fall back to the prior 3

“Log-opinion pool” model (Heskes, 1998)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 28

Computational Graph

GP Experts

gPoE

PoE

PoE

Prediction:

pp f˚|x˚,Dq9Mź

k“1

pβk

k p f˚|x˚,Dpkqq “Lź

k“1

Lkź

i“1

pβki

kip f˚|Dpkiqq ,

ÿ

k,i

βki “ 1

§ Independent of computational graph ifř

k,i βki “ 1 3

§ A priori setting of βki required 7

βki “ 1{M a priori (3)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 29

Computational Graph

GP Experts

gPoE

PoE

PoE

Prediction:

pp f˚|x˚,Dq9Mź

k“1

pβk

k p f˚|x˚,Dpkqq “Lź

k“1

Lkź

i“1

pβki

kip f˚|Dpkiqq ,

ÿ

k,i

βki “ 1

§ Independent of computational graph ifř

k,i βki “ 1 3

§ A priori setting of βki required 7

βki “ 1{M a priori (3)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 29

Computational Graph

GP Experts

gPoE

PoE

PoE

Prediction:

pp f˚|x˚,Dq9Mź

k“1

pβk

k p f˚|x˚,Dpkqq “Lź

k“1

Lkź

i“1

pβki

kip f˚|Dpkiqq ,

ÿ

k,i

βki “ 1

§ Independent of computational graph ifř

k,i βki “ 1 3

§ A priori setting of βki required 7

βki “ 1{M a priori (3)

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 29

Generalized Product of GP Experts

§ Same mean as PoE§ Model no longer overconfident and falls back to prior 3

§ Very conservative variances 7Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 30

Bayesian Committee Machine

§ Apply Bayes’ theorem when combining predictions (and not onlyfor computing predictions)

§ Prediction model (Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

§ Predictive precision and mean:

pσbcm˚ q´2 “

ÿM

k“1σ´2

k px˚q ´pM´ 1qσ´2˚˚

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

§ Product of GP experts, divided by M´ 1 times the prior

§ Guaranteed to fall back to the prior outside data regime 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 31

Bayesian Committee Machine

§ Apply Bayes’ theorem when combining predictions (and not onlyfor computing predictions)

§ Prediction model (Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

§ Predictive precision and mean:

pσbcm˚ q´2 “

ÿM

k“1σ´2

k px˚q ´pM´ 1qσ´2˚˚

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

§ Product of GP experts, divided by M´ 1 times the prior

§ Guaranteed to fall back to the prior outside data regime 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 31

Bayesian Committee Machine

§ Apply Bayes’ theorem when combining predictions (and not onlyfor computing predictions)

§ Prediction model (Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

§ Predictive precision and mean:

pσbcm˚ q´2 “

ÿM

k“1σ´2

k px˚q ´pM´ 1qσ´2˚˚

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

§ Product of GP experts, divided by M´ 1 times the prior

§ Guaranteed to fall back to the prior outside data regime 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 31

Bayesian Committee Machine

§ Apply Bayes’ theorem when combining predictions (and not onlyfor computing predictions)

§ Prediction model (Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

§ Predictive precision and mean:

pσbcm˚ q´2 “

ÿM

k“1σ´2

k px˚q ´pM´ 1qσ´2˚˚

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

§ Product of GP experts, divided by M´ 1 times the prior

§ Guaranteed to fall back to the prior outside data regime 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 31

Bayesian Committee Machine

§ Apply Bayes’ theorem when combining predictions (and not onlyfor computing predictions)

§ Prediction model (Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

§ Predictive precision and mean:

pσbcm˚ q´2 “

ÿM

k“1σ´2

k px˚q ´pM´ 1qσ´2˚˚

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

§ Product of GP experts, divided by M´ 1 times the prior

§ Guaranteed to fall back to the prior outside data regime 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 31

Computational Graph

GP Experts

PoE

PoE

Prior

Prediction:

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

śMk“1 pkp f˚|Dpkqq

pM´1p f˚q“

śLk“1

śLki“1 pkip f˚|Dpkiqq

pM´1p f˚q

Independent of computational graph 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 32

Computational Graph

GP Experts

PoE

PoE

Prior

Prediction:

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

śMk“1 pkp f˚|Dpkqq

pM´1p f˚q“

śLk“1

śLki“1 pkip f˚|Dpkiqq

pM´1p f˚q

Independent of computational graph 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 32

Computational Graph

GP Experts

PoE

PoE

Prior

Prediction:

pp f˚|x˚,Dq9śM

k“1 pkp f˚|x˚,DpkqqpM´1p f˚q

śMk“1 pkp f˚|Dpkqq

pM´1p f˚q“

śLk“1

śLki“1 pkip f˚|Dpkiqq

pM´1p f˚q

Independent of computational graph 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 32

Bayesian Committee Machine

§ Variance estimates are about right 3

§ When leaving the data regime, the BCM can produce junk 7

Robustify

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 33

Robust Bayesian Committee Machine

§ Merge gPoE (weighting of experts) with the BCM (Bayes’theorem when combining predictions)

§ Prediction model (conditional independence Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pβk

k p f˚|x˚,Dpkqqpř

k βk´1p f˚q

§ Predictive precision and mean:

pσrbcm˚ q´2 “

ÿM

k“1βkσ´2

k px˚q `p1´řM

k“1 βkqσ´2˚˚ ,

µrbcm˚ “ pσrbcm

˚ q2ÿ

kβkσ´2

k px˚q µkpx˚q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 34

Robust Bayesian Committee Machine

§ Merge gPoE (weighting of experts) with the BCM (Bayes’theorem when combining predictions)

§ Prediction model (conditional independence Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pβk

k p f˚|x˚,Dpkqqpř

k βk´1p f˚q

§ Predictive precision and mean:

pσrbcm˚ q´2 “

ÿM

k“1βkσ´2

k px˚q `p1´řM

k“1 βkqσ´2˚˚ ,

µrbcm˚ “ pσrbcm

˚ q2ÿ

kβkσ´2

k px˚q µkpx˚q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 34

Robust Bayesian Committee Machine

§ Merge gPoE (weighting of experts) with the BCM (Bayes’theorem when combining predictions)

§ Prediction model (conditional independence Dpjq KK Dpkq| f˚):

pp f˚|x˚,Dq9śM

k“1 pβk

k p f˚|x˚,Dpkqqpř

k βk´1p f˚q

§ Predictive precision and mean:

pσrbcm˚ q´2 “

ÿM

k“1βkσ´2

k px˚q `p1´řM

k“1 βkqσ´2˚˚ ,

µrbcm˚ “ pσrbcm

˚ q2ÿ

kβkσ´2

k px˚q µkpx˚q

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 34

Computational Graph

GP Experts

gPoE

PoE

PoE

Prior

Prediction:

pp f˚|x˚,Dq9śM

k“1 pβk

k p f˚|x˚,Dpkqqpř

k βk´1p f˚q“

śLk“1

śLki“1 p

βki

kip f˚|Dpkiqq

př

kiβki´1p f˚q

Independent of computational graph, even with arbitrary βki 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 35

Computational Graph

GP Experts

gPoE

PoE

PoE

Prior

Prediction:

pp f˚|x˚,Dq9śM

k“1 pβk

k p f˚|x˚,Dpkqqpř

k βk´1p f˚q“

śLk“1

śLki“1 p

βki

kip f˚|Dpkiqq

př

kiβki´1p f˚q

Independent of computational graph, even with arbitrary βki 3

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 35

Robust Bayesian Committee Machine

§ Does not break down in case of weak experts Robustified 3

§ Robust version of BCM Reasonable predictions 3

§ Independent of computational graph (for all choices of βk) 3Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 36

Empirical Approximation Error

100 101 102 103

Gradient time in sec

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

RM

SE

rBCM

BCM

gPoE

PoE

SOD

GP

39 156 625 2500 10000#Points/Expert

§ Simulated robot arm data (10K training, 10K test)§ Hyper-parameters of ground-truth full GP§ RMSE as a function of the training time§ Sparse GP (SOR) performs worse than any distributed GP§ rBCM performs best with “weak” GP experts

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 37

Empirical Approximation Error (2)

100 101 102 103

Gradient time in sec

−3

−2

−1

0

1

2

NL

PD

rBCM

BCM

gPoE

PoE

SOD

GP

39 156 625 2500 10000#Points/Expert

§ NLPD as a function of the training time Mean and variance§ BCM and PoE are not robust for weak experts§ gPoE suffers from too conservative variances§ rBCM consistently outperforms other methods

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 38

Large Data Sets: Airline Data (700K)

rBCM BCM gPoE PoE Dist−VGP SVI−GP

24

26

28

30

32

34DGP, random data assignment

Sparse GP

Airline Delay, 700K

RMSE

§ (r)BCM and (g)PoE with4096 GP experts

§ Gradient time: 13 seconds(12 cores)

§ Inducing inputs:Dist-VGP (Gal et al., 2014),SVI-GP (Hensman et al.,2013)

§ rBCM performs best

§ (g)PoE and BCM perform not worse than sparse GPs

§ KD-tree data assignment clearly helps (r)BCM

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 39

Large Data Sets: Airline Data (700K)

rBCM rBCM−KD BCM BCM−KD gPoE gPoE−KD PoE PoE−KDDist−VGP SVI−GP

24

26

28

30

32

34DGP, KD- tree data assignment

DGP, random data assignment

Sparse GP

Airline Delay, 700K

RMSE

§ (r)BCM and (g)PoE with4096 GP experts

§ Gradient time: 13 seconds(12 cores)

§ Inducing inputs:Dist-VGP (Gal et al., 2014),SVI-GP (Hensman et al.,2013)

§ rBCM performs best

§ (g)PoE and BCM perform not worse than sparse GPs

§ KD-tree data assignment clearly helps (r)BCM

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 39

Summary: Distributed Gaussian Processes

µ, σ

µ1, σ1 µ2, σ2

µ11, σ11 µ12, σ12 µ13, σ13 µ21, σ21 µ22, σ22 µ23, σ23

§ Scale Gaussian processes to large data (beyond 106)§ Model conceptually straightforward and easy to train§ Key: Distributed computation§ Currently tested with N P Op107q

§ Scales to arbitrarily large data sets (with enough computingpower)

Thank you for your attentionDistributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 40

References

[1] Y. Cao and D. J. Fleet. Generalized Product of Experts for Automatic and Principled Fusion of Gaussian ProcessPredictions. http://arxiv.org/abs/1410.7827, October 2014.

[2] K. Chalupka, C. K. I. Williams, and I. Murray. A Framework for Evaluating Approximate Methods for Gaussian ProcessRegression. Journal of Machine Learning Research, 14:333–350, February 2013.

[3] M. P. Deisenroth and J. W. Ng. Distributed Gaussian Processes. In Proceedings of the International Conference on MachineLearning, 2015.

[4] M. P. Deisenroth, C. E. Rasmussen, and D. Fox. Learning to Control a Low-Cost Manipulator using Data-EfficientReinforcement Learning. In Proceedings of Robotics: Science and Systems, Los Angeles, CA, USA, June 2011.

[5] Y. Gal, M. van der Wilk, and C. E. Rasmussen. Distributed Variational Inference in Sparse Gaussian Process Regressionand Latent Variable Models. In Advances in Neural Information Processing Systems. 2014.

[6] J. Hensman, N. Fusi, and N. D. Lawrence. Gaussian Processes for Big Data. In A. Nicholson and P. Smyth, editors,Proceedings of the Conference on Uncertainty in Artificial Intelligence. AUAI Press, 2013.

[7] T. Heskes. Selecting Weighting Factors in Logarithmic Opinion Pools. In Advances in Neural Information Processing Systems,pages 266–272. Morgan Kaufman, 1998.

[8] N. Lawrence. Probabilistic Non-linear Principal Component Analysis with Gaussian Process Latent Variable Models.Journal of Machine Learning Research, 6:1783–1816, November 2005.

[9] J. Ng and M. P. Deisenroth. Hierarchical Mixture-of-Experts Model for Large-Scale Gaussian Process Regression.http://arxiv.org/abs/1412.3078, December 2014.

[10] J. Quinonero-Candela and C. E. Rasmussen. A Unifying View of Sparse Approximate Gaussian Process Regression.Journal of Machine Learning Research, 6(2):1939–1960, 2005.

[11] E. Snelson and Z. Ghahramani. Sparse Gaussian Processes using Pseudo-inputs. In Y. Weiss, B. Scholkopf, and J. C. Platt,editors, Advances in Neural Information Processing Systems 18, pages 1257–1264. The MIT Press, Cambridge, MA, USA, 2006.

[12] M. K. Titsias. Variational Learning of Inducing Variables in Sparse Gaussian Processes. In Proceedings of the InternationalConference on Artificial Intelligence and Statistics, 2009.

[13] V. Tresp. A Bayesian Committee Machine. Neural Computation, 12(11):2719–2741, 2000.

Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 41

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