Distributed Gaussian Processesgpss.cc/gpss15/talks/marc.pdf · Table of Contents Gaussian Processes...

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 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

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

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

2 log |K| ` const

B log ppy|X, θq

Bθ“ 1

2 yJK´1 BKBθ

K´1y´ 12 tr

K´1 BKBθ

2 log |K| ` const

B log ppy|X, θq

Bθ“ 1

2 yJK´1 BKBθ

K´1y´ 12 tr

K´1 BKBθ

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

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

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)

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

Table of Contents

Gaussian Processes

GP Factor Graph

f(x1) f(xN ) f∗1 f∗

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˜«

K f f K f˚

K˚ f K˚˚

GP Factor Graph

f(x1) f(xN ) f∗1 f∗

f(u1) f(uM )

§ 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˜«

K f f K f˚

K˚ f K˚˚

Inducing Variables

f(x1) f(xN ) f∗1 f∗

f(u1) f(uM )

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

Inducing Variables

f(x1) f(xN ) f∗1 f∗

f(u1) f(uM )

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∗

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

K f f Q f˚

Q˚ f K˚˚

Q˚ f :“ K˚ fu K´1fu fu

K fu f Nystrom approximation

Central Approximation Scheme

f(x1) f(xN ) f∗1 f∗

f(u1) f(uM )

§ 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

K f f Q f˚

Q˚ f K˚˚

Q˚ f :“ K˚ fu K´1fu fu

K fu f Nystrom approximation

FI(T)C Sparse Approximation

f(x1) f(xN ) f∗1 f∗

f(u1) f(uM )

§ 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

Q f f ´ diagpQ f f ´K f f q Q f˚

Q˚ f K˚˚

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

§ Training: OpNM2q, Prediction: OpM2q

f(x1) f(xN ) f∗1 f∗

f(u1) f(uM )

qp f , f˚q “ N

Q˚ f K˚˚

f(x1) f(xN ) f∗1 f∗

f(u1) f(uM )

qp f , f˚q “ N

Q˚ f K˚˚

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

Inducing Inputs

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

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

Table of Contents

Gaussian Processes

Joint work with Jun Wei Ng

An Orthogonal Approximation: Distributed GPs

−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

−1 −1−1−1−1

Standard GP

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

−1 −1−1−1−1

Standard GP

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

−1 −1−1−1−1

Standard GP

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

−1 −1−1−1−1

Standard GP

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

−1 −1−1−1−1

Standard GP

Distributed GP

GP GP GP GP

O(N 3)

O(MP 3)

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]

Empirical Training Time

103 104 105 106 107 108

Training data set size

tion t

and its

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

Empirical Training Time

103 104 105 106 107 108

Training data set size

tion t

and its

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)

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)

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

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

Objectives

µ, σ

µ1, σ1 µ2, σ2

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

cluster, ...)

Objectives

µ, σ

µ1, σ1 µ2, σ2

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

cluster, ...)

Objectives

µ, σ

µ1, σ1 µ2, σ2

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

cluster, ...)

Running Example

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

Product of GP Experts

§ Prediction model (independent predictors):

pp f˚|x˚,Dq9Mź

pkp f˚|x˚,Dpkqq ,

pkp f˚|x˚,Dpkqq “ N`

f˚ | µkpx˚q, σ2k px˚q

§ Predictive precision (inverse variance) and mean:

pσpoe˚ q´2 “

kσ´2

k px˚q

µpoe˚ “ pσ

poe˚ q2

kσ´2

k px˚qµkpx˚q

pp f˚|x˚,Dq9Mź

pkp f˚|x˚,Dpkqq ,

§ Predictive precision (inverse variance) and mean:

pσpoe˚ q´2 “

kσ´2

k px˚q

µpoe˚ “ pσ

poe˚ q2

kσ´2

k px˚qµkpx˚q

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9Mź

pkp f˚|x˚,Dpkqq

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

pkp f˚|Dpkqq “Lź

pkip f˚|Dpkiqq ,ÿ

Lk “ M

Independent of computational graph 3

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9Mź

pkp f˚|x˚,Dpkqq

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

pkp f˚|Dpkqq “Lź

pkip f˚|Dpkiqq ,ÿ

Lk “ M

§ Unreasonable variances for M ą 1:

pσpoe˚ q´2 “

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

§ Unreasonable variances for M ą 1:

pσpoe˚ q´2 “

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

Generalized Product of GP Experts

§ Weight the responsiblity of each expert in PoE with βk

pp f˚|x˚,Dq9Mź

k p f˚|x˚,Dpkqq

§ Predictive precision and mean:

pσgpoe˚ q´2 “

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

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)

pp f˚|x˚,Dq9Mź

k p f˚|x˚,Dpkqq

pσgpoe˚ q´2 “

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

kβkσ´2

k px˚q µkpx˚q

§ Withř

pp f˚|x˚,Dq9Mź

k p f˚|x˚,Dpkqq

pσgpoe˚ q´2 “

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

kβkσ´2

k px˚q µkpx˚q

§ Withř

pp f˚|x˚,Dq9Mź

k p f˚|x˚,Dpkqq

pσgpoe˚ q´2 “

kβkσ´2

k px˚q

µgpoe˚ “ pσ

gpoe˚ q2

kβkσ´2

k px˚q µkpx˚q

§ Withř

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9Mź

k p f˚|x˚,Dpkqq “Lź

kip f˚|Dpkiqq ,

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)

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9Mź

kip f˚|Dpkiqq ,

βki “ 1

k,i βki “ 1 3

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9Mź

kip f˚|Dpkiqq ,

βki “ 1

k,i βki “ 1 3

§ 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

pσbcm˚ q´2 “

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

pp f˚|x˚,Dq9śM

pσbcm˚ q´2 “

k“1σ´2

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

pp f˚|x˚,Dq9śM

pσbcm˚ q´2 “

k“1σ´2

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

pp f˚|x˚,Dq9śM

pσbcm˚ q´2 “

k“1σ´2

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

pp f˚|x˚,Dq9śM

pσbcm˚ q´2 “

k“1σ´2

µbcm˚ “ pσbcm

˚ q2ÿM

k“1σ´2

k px˚qµkpx˚q

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9śM

śMk“1 pkp f˚|Dpkqq

pM´1p f˚q“

śLk“1

śLki“1 pkip f˚|Dpkiqq

pM´1p f˚q

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9śM

pM´1p f˚q“

śLk“1

pM´1p f˚q

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9śM

pM´1p f˚q“

śLk“1

pM´1p f˚q

§ Variance estimates are about right 3

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

Robustify

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

pσrbcm˚ q´2 “

k“1βkσ´2

k px˚q `p1´řM

k“1 βkqσ´2˚˚ ,

µrbcm˚ “ pσrbcm

˚ q2ÿ

kβkσ´2

k px˚q µkpx˚q

pp f˚|x˚,Dq9śM

k“1 pβk

k βk´1p f˚q

pσrbcm˚ q´2 “

k“1βkσ´2

k px˚q `p1´řM

k“1 βkqσ´2˚˚ ,

˚ q2ÿ

kβkσ´2

k px˚q µkpx˚q

pp f˚|x˚,Dq9śM

k“1 pβk

k βk´1p f˚q

pσrbcm˚ q´2 “

k“1βkσ´2

k px˚q `p1´řM

k“1 βkqσ´2˚˚ ,

˚ q2ÿ

kβkσ´2

k px˚q µkpx˚q

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9śM

k“1 pβk

k βk´1p f˚q“

śLk“1

śLki“1 p

kip f˚|Dpkiqq

kiβki´1p f˚q

Independent of computational graph, even with arbitrary βki 3

Computational Graph

GP Experts

Prediction:

pp f˚|x˚,Dq9śM

k“1 pβk

k βk´1p f˚q“

śLk“1

śLki“1 p

kip f˚|Dpkiqq

kiβki´1p f˚q

Independent of computational graph, even with arbitrary βki 3

§ 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

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

Empirical Approximation Error (2)

100 101 102 103

Gradient time in sec

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

Large Data Sets: Airline Data (700K)

rBCM BCM gPoE PoE Dist−VGP SVI−GP

34DGP, random data assignment

Sparse GP

Airline Delay, 700K

§ (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

Large Data Sets: Airline Data (700K)

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

34DGP, KD- tree data assignment

DGP, random data assignment

Sparse GP

Airline Delay, 700K

§ (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

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)

m.deisenroth@imperial.ac.uk

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.

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Gaussian Processes: An Introduction · Gaussian Processes De nition. A Gaussian process fZ tg, (t 2T) is a stochastic process, each subset of fZ tgforming a (multivariate) Gaussian

Kernel, RKHS, and Gaussian Processes

Distributed Gaussian Processesmpd37/talks/2015-09-15-gpss.pdf · Distributed Gaussian Processes Marc Deisenroth @GPSS, 15th September 2015 10. Inducing Variables f(x 1) f(x N) f f

CARMA Processes driven by Non-Gaussian Noise · paper where Gaussian CARMA processes appeared under the name of Gaussian processes with rational spectral density. 2 Levy processes´

Band-Limited Gaussian Processes: The Sinc Kernelpapers.nips.cc/paper/9436-band-limited-gaussian...Band-Limited Gaussian Processes: The Sinc Kernel Felipe Tobar Center for Mathematical

Gaussian Processes for Big Data

Additive Gaussian Processes

Gaussian Random Processes

Gaussian and non-Gaussian processes of zero power variation, and

Another introduction to Gaussian Processes

Gaussian Processes and Empirical Bayes

Learning with Uncertainty { Gaussian Processes and ...€¦ · pleteness, in Chap. 3 we study a particular family of approximations to Gaussian Processes, Reduced Rank Gaussian Processes

Gaussian Processes · Gaussian Processes. Forecasting Processes Agenda Gaussian Processes Profit Code Speed. Forecasting Processes. Project Structure & Docs. Packaging & Testing

Gaussian Processes and Office Buildings