Distributed Gaussian Processes

Transcription

1 Distributed Gaussian Processes Marc Deisenroth Department of Computing Imperial College London Gaussian Process Summer School, University of Sheffield 15th September 2015

2 Table of Contents Gaussian Processes Sparse Gaussian Processes Distributed Gaussian Processes Distributed Gaussian Processes Marc 15th September

3 Problem Setting 2 f(x) x Objective For a set of N observations y i f px i q ` ε, ε N `0, σε 2, find a distribution over functions pp f X, yq that explains the data GP is a good solution to this probabilistic regression problem Distributed Gaussian Processes Marc 15th September

4 GP Training via Marginal Likelihood Maximization GP Training Maximize the evidence/marginal likelihood ppy X, θq with respect to the hyper-parameters θ: θ P arg max θ log ppy X, θq log ppy X, θq 1 2 yjk 1 y 1 2 log K ` const Distributed Gaussian Processes Marc 15th September

5 GP Training via Marginal Likelihood Maximization GP Training Maximize the evidence/marginal likelihood ppy X, θq with respect to the hyper-parameters θ: θ P arg max θ log ppy X, θq log ppy X, θq 1 2 yjk 1 y 1 2 log K ` const Automatic trade-off between data fit and model complexity Distributed Gaussian Processes Marc 15th September

6 GP Training via Marginal Likelihood Maximization GP Training Maximize the evidence/marginal likelihood ppy X, θq with respect to the hyper-parameters θ: θ P arg max θ log ppy X, θq log ppy X, θq 1 2 yjk 1 y 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 BK 2yJK 1 K 1 y 1 Bθ 2 BK tr`k 1 Bθ Computational complexity: OpN 3 q for K and K 1 Distributed Gaussian Processes Marc 15th September

7 GP Predictions At a test point x the predictive (posterior) distribution is Gaussian: pp f px q x, X, y, θq N ` f m, σ 2 m kpx, x q J K 1 y σ 2 kpx, x q kpx, x q J K 1 kpx, x q Distributed Gaussian Processes Marc 15th September

8 GP Predictions At a test point x the predictive (posterior) distribution is Gaussian: pp f px q x, X, y, θq N ` f m, σ 2 m kpx, x q J K 1 y σ 2 kpx, x q kpx, x q J K 1 kpx, x q When you cache K 1 and K 1 y after training, then The mean prediction can be computed in OpNq The variance prediction can be computed in OpN 2 q Distributed Gaussian Processes Marc 15th September

9 Application Areas Bayesian Optimization (Experimental Design) Model unknown utility functions with GPs Reinforcement Learning and Robotics Model value functions and/or dynamics with GPs Data visualization Nonlinear dimensionality reduction (GP-LVM) Distributed Gaussian Processes Marc 15th September

10 Limitations of Gaussian Processes Computational and memory complexity Training scales in OpN 3 q Prediction (variances) scales in OpN 2 q Memory requirement: OpND ` N 2 q Practical limit N «10, 000 Distributed Gaussian Processes Marc 15th September

11 Table of Contents Gaussian Processes Sparse Gaussian Processes Distributed Gaussian Processes Distributed Gaussian Processes Marc 15th September

12 GP Factor Graph Inducing function values Training function values f i f(u1) f(um ) f(x1) f(xn ) f 1 f L Training data Test data Probabilistic graphical model (factor graph) of a GP All function values are jointly Gaussian distributed (e.g., training and test function values) Distributed Gaussian Processes Marc 15th September

13 GP Factor Graph Inducing function values Training function values f i f(u1) f(um ) f(x1) f(xn ) f 1 f L Training data Test data Probabilistic graphical model (factor graph) of a GP All function values are jointly Gaussian distributed (e.g., training and test function values) GP prior «ff «ff 0 K f f K f pp f, f q N, 0 K f K Distributed Gaussian Processes Marc 15th September

14 Inducing Variables Inducing function values f(u1) f(um ) Training function values f i Hypothetical function values u j f(x1) f(xn ) f 1 f L Training data Test data Introduce inducing function values f u Hypothetical function values Distributed Gaussian Processes Marc 15th September

15 Inducing Variables Inducing function values f(u1) f(um ) Training function values f i Hypothetical function values u j f(x1) f(xn ) f 1 f L Training data Test data Introduce inducing function values f u 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 values Distributed Gaussian Processes Marc 15th September

16 Central Approximation Scheme Inducing function values f(u1) f(um ) f(x1) f(xn ) f 1 f L Training data Test data Approximation: Training and test set are conditionally independent given the inducing function values: f K f f u Distributed Gaussian Processes Marc 15th September

17 Central Approximation Scheme Inducing function values f(u1) f(um ) f(x1) f(xn ) f 1 f L Training data Test data Approximation: Training and test set are conditionally independent given the inducing function values: f K f f u Then, the effective GP prior is ż «ff» fi qp f, f q pp f f u qpp f f u qpp f u qd f u N 0, K f f Q f fl, 0 Q f K Q f : K fu K 1 f u f u K fu f Nyström approximation Distributed Gaussian Processes Marc 15th September

18 FI(T)C Sparse Approximation Inducing function values f(u1) f(um ) f(x1) f(xn ) f 1 f L Assume that training (and test sets) are fully independent given the inducing variables (Snelson & Ghahramani, 2006) Training data Test data Distributed Gaussian Processes Marc 15th September

19 FI(T)C Sparse Approximation Inducing function values f(u1) f(um ) f(x1) f(xn ) f 1 f L Assume that training (and test sets) are fully independent given the inducing variables (Snelson & Ghahramani, 2006) Training data Test data Effective GP prior with this approximation «ff» fi qp f, f q N 0, Q f f diagpq f f K f f q Q f fl 0 Q f K Q diagpq K q can be used instead of K FIC Distributed Gaussian Processes Marc 15th September

20 FI(T)C Sparse Approximation Inducing function values f(u1) f(um ) f(x1) f(xn ) f 1 f L Assume that training (and test sets) are fully independent given the inducing variables (Snelson & Ghahramani, 2006) Training data Test data Effective GP prior with this approximation «ff» fi qp f, f q N 0, Q f f diagpq f f K f f q Q f fl 0 Q f K Q diagpq K q can be used instead of K FIC Training: OpNM 2 q, Prediction: OpM 2 q Distributed Gaussian Processes Marc 15th September

21 Inducing Inputs FI(T)C sparse approximation exploits inducing function values f pu i q, where u i are the corresponding inputs Distributed Gaussian Processes Marc 15th September

22 Inducing Inputs FI(T)C sparse approximation exploits inducing function values f pu i q, where u i are the corresponding inputs These inputs are unknown a priori Find optimal ones Distributed Gaussian Processes Marc 15th September

23 Inducing Inputs FI(T)C sparse approximation exploits inducing function values f pu i q, where u i are the corresponding inputs These inputs are unknown a priori Find optimal ones Find them by maximizing the FI(T)C marginal likelihood with respect to the inducing inputs (and the standard hyper-parameters): u 1:M P arg max q FITC py X, u 1:M, θq u 1:M Distributed Gaussian Processes Marc 15th September

24 Inducing Inputs FI(T)C sparse approximation exploits inducing function values f pu i q, where u i are the corresponding inputs These inputs are unknown a priori Find optimal ones Find them by maximizing the FI(T)C marginal likelihood with respect to the inducing inputs (and the standard hyper-parameters): u 1:M P arg max q FITC py X, u 1:M, θq u 1:M Intuitively: The marginal likelihood is not only parameterized by the hyper-parameters θ, but also by the inducing inputs u 1:M. Distributed Gaussian Processes Marc 15th September

25 Inducing Inputs FI(T)C sparse approximation exploits inducing function values f pu i q, where u i are the corresponding inputs These inputs are unknown a priori Find optimal ones Find them by maximizing the FI(T)C marginal likelihood with respect to the inducing inputs (and the standard hyper-parameters): u 1:M P arg max q FITC py X, u 1:M, θq u 1:M Intuitively: The marginal likelihood is not only parameterized by the hyper-parameters θ, but also by the inducing inputs u 1:M. End up with a high-dimensional non-convex optimization problem with MD additional parameters Distributed Gaussian Processes Marc 15th September

26 FITC Example Pink: Original data Red crosses: Initialization of inducing inputs Blue crosses: Location of inducing inputs after optimization Figure from Ed Snelson Efficient compression of the original data set Distributed Gaussian Processes Marc 15th September

27 Summary Sparse Gaussian Processes Sparse approximations typically approximate a GP with N data points by a model with M! N data points Distributed Gaussian Processes Marc 15th September

28 Summary Sparse Gaussian Processes Sparse approximations typically approximate a GP with N data points by a model with M! N data points Selection of these M data points can be tricky and may involve non-trivial computations (e.g., optimizing inducing inputs) Distributed Gaussian Processes Marc 15th September

29 Summary Sparse Gaussian Processes Sparse approximations typically approximate a GP with N data points by a model with M! N data points Selection of these M data points can be tricky and may involve non-trivial computations (e.g., optimizing inducing inputs) Simple (random) subset selection is fast and generally robust (Chalupka et al., 2013) Distributed Gaussian Processes Marc 15th September

30 Summary Sparse Gaussian Processes Sparse approximations typically approximate a GP with N data points by a model with M! N data points Selection of these M data points can be tricky and may involve non-trivial computations (e.g., optimizing inducing inputs) Simple (random) subset selection is fast and generally robust (Chalupka et al., 2013) Computational complexity: OpM 3 q or OpNM 2 q for training Distributed Gaussian Processes Marc 15th September

31 Summary Sparse Gaussian Processes Sparse approximations typically approximate a GP with N data points by a model with M! N data points Selection of these M data points can be tricky and may involve non-trivial computations (e.g., optimizing inducing inputs) Simple (random) subset selection is fast and generally robust (Chalupka et al., 2013) Computational complexity: OpM 3 q or OpNM 2 q for training Practical limit M ď Often: M P Op10 2 q in the case of inducing variables Distributed Gaussian Processes Marc 15th September

32 Summary Sparse Gaussian Processes Sparse approximations typically approximate a GP with N data points by a model with M! N data points Selection of these M data points can be tricky and may involve non-trivial computations (e.g., optimizing inducing inputs) Simple (random) subset selection is fast and generally robust (Chalupka et al., 2013) Computational complexity: OpM 3 q or OpNM 2 q for training Practical limit M ď Often: M P Op10 2 q in the case of inducing variables If we set M N{100, i.e., each inducing function value summarizes 100 real function values, our practical limit is N P Op10 6 q Distributed Gaussian Processes Marc 15th September

33 Table of Contents Gaussian Processes Sparse Gaussian Processes Distributed Gaussian Processes Distributed Gaussian Processes Marc 15th September

34 Distributed Gaussian Processes Joint work with Jun Wei Ng Distributed Gaussian Processes Marc 15th September

35 An Orthogonal Approximation: Distributed GPs Data set Standard GP Kernel matrix 1 O(N 3 ) GP GP GP GP Distributed GP 1 = O(MP 3 ) Distributed Gaussian Processes Marc 15th September

36 An Orthogonal Approximation: Distributed GPs Data set Standard GP Kernel matrix 1 O(N 3 ) GP GP GP GP Distributed GP 1 = O(MP 3 ) Randomly split the full data set into M chunks Distributed Gaussian Processes Marc 15th September

37 An Orthogonal Approximation: Distributed GPs Data set Standard GP Kernel matrix 1 O(N 3 ) GP GP GP GP Distributed GP 1 = O(MP 3 ) Randomly split the full data set into M chunks Place M independent GP models (experts) on these small chunks Distributed Gaussian Processes Marc 15th September

38 An Orthogonal Approximation: Distributed GPs Data set Standard GP Kernel matrix 1 O(N 3 ) GP GP GP GP Distributed GP 1 = 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 Distributed Gaussian Processes Marc 15th September

39 An Orthogonal Approximation: Distributed GPs Data set Standard GP Kernel matrix 1 O(N 3 ) GP GP GP GP Distributed GP 1 = 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 Distributed Gaussian Processes Marc 15th September

40 An Orthogonal Approximation: Distributed GPs Data set Standard GP Kernel matrix 1 O(N 3 ) GP GP GP GP Distributed GP 1 = 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 15th September

41 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 «ÿ M log p k 1 kpy pkq X pkq, θq Distributed Gaussian Processes Marc 15th September

42 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 «ÿ M log p k 1 kpy pkq X pkq, θq Distributed optimization and training straightforward Distributed Gaussian Processes Marc 15th September

43 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 «ÿ M log p k 1 kpy pkq X pkq, θq Distributed optimization and training straightforward Computational complexity: OpMP 3 q [instead of OpN 3 q] But distributed over many machines Memory footprint: OpMP 2 ` NDq [instead of OpN 2 ` NDq] Distributed Gaussian Processes Marc 15th September

44 Empirical Training Time Computation time of NLML and its gradients (s) Computation time (DGP) 10 2 Computation time (Full GP) Computation time (FITC) Number of GP experts (DGP) Training data set size Number of GP experts (DGP) NLML is proportional to training time Distributed Gaussian Processes Marc 15th September

45 Empirical Training Time Computation time of NLML and its gradients (s) Computation time (DGP) 10 2 Computation time (Full GP) Computation time (FITC) Number of GP experts (DGP) Training data set size 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 15th September

46 Practical Training Times Training* with N 10 6, D 1 on a laptop: «10 30 min Training* with N 5 ˆ 10 6, D 8 on a workstation: «4 hours Distributed Gaussian Processes Marc 15th September

47 Practical Training Times Training* with N 10 6, D 1 on a laptop: «10 30 min Training* with N 5 ˆ 10 6, D 8 on a workstation: «4 hours *: Maximize the marginal likelihood, stop when converged** Distributed Gaussian Processes Marc 15th September

48 Practical Training Times Training* with N 10 6, D 1 on a laptop: «10 30 min Training* with N 5 ˆ 10 6, D 8 on a workstation: «4 hours *: Maximize the marginal likelihood, stop when converged** **: Convergence often after line searches*** Distributed Gaussian Processes Marc 15th September

49 Practical Training Times Training* with N 10 6, D 1 on a laptop: «10 30 min Training* with N 5 ˆ 10 6, D 8 on a workstation: «4 hours *: Maximize the marginal likelihood, stop when converged** **: Convergence often after line searches*** ***: Line search «2 3 evaluations of marginal likelihood and its gradient (usually OpN 3 q) Distributed Gaussian Processes Marc 15th September

50 Predictions with the Distributed GP µ, σ Prediction of each GP expert is Gaussian N `µ i, σ 2 i How to combine them to an overall prediction N `µ, σ 2? Distributed Gaussian Processes Marc 15th September

51 Predictions with the Distributed GP µ, σ Prediction of each GP expert is Gaussian N `µ i, σi 2 How to combine them to an overall prediction N `µ, σ 2? 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 15th September

52 Objectives µ, σ µ, σ µ 1, σ 1 µ 2, σ 2 µ 11, σ 11 µ 12, σ 12 µ 13, σ 13 µ 21, σ 21 µ 22, σ 22 µ 23, σ 23 Figure: Two computational graphs Scale to large data sets Distributed Gaussian Processes Marc 15th September

53 Objectives µ, σ µ, σ µ 1, σ 1 µ 2, σ 2 µ 11, σ 11 µ 12, σ 12 µ 13, σ 13 µ 21, σ 21 µ 22, σ 22 µ 23, σ 23 Figure: Two computational graphs Scale to large data sets Good approximation of full GP ( ground truth ) Distributed Gaussian Processes Marc 15th September

54 Objectives µ, σ µ, σ µ 1, σ 1 µ 2, σ 2 µ 11, σ 11 µ 12, σ 12 µ 13, σ 13 µ 21, σ 21 µ 22, σ 22 µ 23, σ 23 Figure: Two computational graphs Scale to large data sets Good approximation of full GP ( ground truth ) Predictions independent of computational graph Runs on heterogeneous computing infrastructures (laptop, cluster,...) Distributed Gaussian Processes Marc 15th September

55 Objectives µ, σ µ, σ µ 1, σ 1 µ 2, σ 2 µ 11, σ 11 µ 12, σ 12 µ 13, σ 13 µ 21, σ 21 µ 22, σ 22 µ 23, σ 23 Figure: Two computational graphs Scale to large data sets Good approximation of full GP ( ground truth ) Predictions independent of computational graph Runs on heterogeneous computing infrastructures (laptop, cluster,...) Reasonable predictive variances Distributed Gaussian Processes Marc 15th September

56 Running Example Investigate various product-of-experts models Same training procedure, but different mechanisms for predictions Distributed Gaussian Processes Marc 15th September

57 Product of GP Experts Prediction model (independent predictors): Mź pp f x, Dq9 p k p f x, D pkq q, k 1 p k p f x, D pkq q N ` f µ k px q, σ 2 k px q Distributed Gaussian Processes Marc 15th September

58 Product of GP Experts Prediction model (independent predictors): Mź pp f x, Dq9 p k p f x, D pkq q, k 1 p k p f x, D pkq q N ` f µ k px q, σ 2 k px q Predictive precision (inverse variance) and mean: pσ poe q 2 ÿ σ 2 px q k µ poe k pσ poe q ÿ 2 σ 2 px qµ k k px q k Distributed Gaussian Processes Marc 15th September

59 Computational Graph Prediction: Mź pp f x, Dq9 p k p f x, D pkq q k 1 Distributed Gaussian Processes Marc 15th September

60 Computational Graph PoE PoE GP Experts Prediction: Mź pp f x, Dq9 p k p f x, D pkq q Multiplication is associative: a b c d pa bq pc dq Mź Lź źl k ÿ p k p f D pkq q p ki p f D pkiq q, L k M k 1 k 1 k 1i 1 k Independent of computational graph Distributed Gaussian Processes Marc 15th September

61 Product of GP Experts Unreasonable variances for M ą 1: Distributed Gaussian Processes Marc 15th September

62 Product of GP Experts Unreasonable variances for M ą 1: pσ poe q 2 ÿ σ 2 px q k The more experts the more certain the prediction, even if every expert itself is very uncertain Cannot fall back to the prior Distributed Gaussian Processes Marc 15th September k

63 Generalized Product of GP Experts Weight the responsiblity of each expert in PoE with β k Distributed Gaussian Processes Marc 15th September

64 Generalized Product of GP Experts Weight the responsiblity of each expert in PoE with β k Prediction model (independent predictors): pp f x, Dq9 Mź k 1 p β k k p f x, D pkq q p k p f x, D pkq q N ` f µ k px q, σ 2 k px q Distributed Gaussian Processes Marc 15th September

65 Generalized Product of GP Experts Weight the responsiblity of each expert in PoE with β k Prediction model (independent predictors): pp f x, Dq9 Mź k 1 p β k k p f x, D pkq q p k p f x, D pkq q N ` f µ k px q, σ 2 k px q Predictive precision and mean: pσ gpoe q 2 ÿ βkσ 2 px q k µ gpoe pσ gpoe k q ÿ 2 βkσ 2 px q µ k k px q k Distributed Gaussian Processes Marc 15th September

66 Generalized Product of GP Experts Weight the responsiblity of each expert in PoE with β k Prediction model (independent predictors): pp f x, Dq9 Mź k 1 p β k k p f x, D pkq q p k p f x, D pkq q N ` f µ k px q, σ 2 k px q Predictive precision and mean: pσ gpoe q 2 ÿ βkσ 2 px q k µ gpoe pσ gpoe k q ÿ 2 βkσ 2 px q µ k k px q With ř k β k 1, the model can fall back to the prior Log-opinion pool model (Heskes, 1998) Distributed Gaussian Processes Marc 15th September k

67 Computational Graph PoE PoE gpoe GP Experts Prediction: pp f x, Dq9 Mź k 1 p β k k p f x, D pkq q Lź źl k k 1 i 1 p β k i k i p f D pk iq q, ÿ β ki 1 k,i Distributed Gaussian Processes Marc 15th September

68 Computational Graph PoE PoE gpoe GP Experts Prediction: pp f x, Dq9 Mź k 1 p β k k p f x, D pkq q Lź źl k k 1 i 1 p β k i k i p f D pk iq q, ÿ β ki 1 k,i Independent of computational graph if ř k,i β k i 1 Distributed Gaussian Processes Marc 15th September

69 Computational Graph PoE PoE gpoe GP Experts Prediction: pp f x, Dq9 Mź k 1 p β k k p f x, D pkq q Lź źl k k 1 i 1 p β k i k i p f D pk iq q, ÿ β ki 1 k,i Independent of computational graph if ř k,i β k i 1 A priori setting of β ki required β ki 1{M a priori ( ) Distributed Gaussian Processes Marc 15th September

70 Generalized Product of GP Experts Same mean as PoE Model no longer overconfident and falls back to prior Very conservative variances Distributed Gaussian Processes Marc 15th September

71 Bayesian Committee Machine Apply Bayes theorem when combining predictions (and not only for computing predictions) Distributed Gaussian Processes Marc 15th September

72 Bayesian Committee Machine Apply Bayes theorem when combining predictions (and not only for computing predictions) Prediction model (D pjq K D pkq f ): ś M k 1 pp f x, Dq9 p kp f x, D pkq q p M 1 p f q Distributed Gaussian Processes Marc 15th September

73 Bayesian Committee Machine Apply Bayes theorem when combining predictions (and not only for computing predictions) Prediction model (D pjq K D pkq f ): Predictive precision and mean: ś M k 1 pp f x, Dq9 p kp f x, D pkq q p M 1 p f q pσ bcm q 2 ÿ M µ bcm σ 2 k 1 k pσ bcm q ÿ 2 M px q pm 1qσ 2 σ 2 k 1 k px qµ k px q Distributed Gaussian Processes Marc 15th September

74 Bayesian Committee Machine Apply Bayes theorem when combining predictions (and not only for computing predictions) Prediction model (D pjq K D pkq f ): Predictive precision and mean: ś M k 1 pp f x, Dq9 p kp f x, D pkq q p M 1 p f q pσ bcm q 2 ÿ M µ bcm σ 2 k 1 k pσ bcm q ÿ 2 M px q pm 1qσ 2 σ 2 k 1 k px qµ k px q Product of GP experts, divided by M 1 times the prior Distributed Gaussian Processes Marc 15th September

75 Bayesian Committee Machine Apply Bayes theorem when combining predictions (and not only for computing predictions) Prediction model (D pjq K D pkq f ): Predictive precision and mean: ś M k 1 pp f x, Dq9 p kp f x, D pkq q p M 1 p f q pσ bcm q 2 ÿ M µ bcm σ 2 k 1 k pσ bcm q ÿ 2 M px q pm 1qσ 2 σ 2 k 1 k px qµ k px q Product of GP experts, divided by M 1 times the prior Guaranteed to fall back to the prior outside data regime Distributed Gaussian Processes Marc 15th September

76 Computational Graph Prediction: ś M k 1 pp f x, Dq9 p kp f x, D pkq q p M 1 p f q Distributed Gaussian Processes Marc 15th September

77 Computational Graph Prediction: ś M k 1 pp f x, Dq9 p kp f x, D pkq q p M 1 p f q ś M k 1 p kp f D pkq q p M 1 p f q ś L ś Lk k 1 i 1 p k i p f D pkiq q p M 1 p f q Distributed Gaussian Processes Marc 15th September

78 Computational Graph Prior PoE PoE GP Experts Prediction: ś M k 1 pp f x, Dq9 p kp f x, D pkq q p M 1 p f q ś M k 1 p kp f D pkq q p M 1 p f q ś L ś Lk k 1 i 1 p k i p f D pkiq q p M 1 p f q Independent of computational graph Distributed Gaussian Processes Marc 15th September

79 Bayesian Committee Machine Variance estimates are about right When leaving the data regime, the BCM can produce junk Robustify Distributed Gaussian Processes Marc 15th September

80 Robust Bayesian Committee Machine Merge gpoe (weighting of experts) with the BCM (Bayes theorem when combining predictions) Distributed Gaussian Processes Marc 15th September

81 Robust Bayesian Committee Machine Merge gpoe (weighting of experts) with the BCM (Bayes theorem when combining predictions) Prediction model (conditional independence D pjq K D pkq f ): pp f x, Dq9 ś M k 1 p β k k p f x, D pkq q p ř k βk 1 p f q Distributed Gaussian Processes Marc 15th September

82 Robust Bayesian Committee Machine Merge gpoe (weighting of experts) with the BCM (Bayes theorem when combining predictions) Prediction model (conditional independence D pjq K D pkq f ): pp f x, Dq9 Predictive precision and mean: pσ rbcm q 2 ÿ M µ rbcm pσ rbcm βkσ 2 k 1 k ś M k 1 p β k k p f x, D pkq q p ř k βk 1 p f q px q `p1 řm q ÿ 2 βkσ 2 px q µ k k px q k k 1 βkqσ 2, Distributed Gaussian Processes Marc 15th September

83 Computational Graph Prediction: pp f x, Dq9 ś M k 1 p β k k p f x, D pkq q p ř k βk 1 p f q ś L ś Lk k 1 i 1 p β k i p ř k i k i β ki 1 p f q p f D pk iq q Distributed Gaussian Processes Marc 15th September

84 Computational Graph Prior PoE PoE gpoe GP Experts Prediction: pp f x, Dq9 ś M k 1 p β k k p f x, D pkq q p ř k βk 1 p f q ś L ś Lk k 1 i 1 p β k i p ř k i k i β ki 1 p f q p f D pk iq q Independent of computational graph, even with arbitrary β ki Distributed Gaussian Processes Marc 15th September

85 Robust Bayesian Committee Machine Does not break down in case of weak experts Robustified Robust version of BCM Reasonable predictions Independent of computational graph (for all choices of β k ) Distributed Gaussian Processes Marc 15th September

86 Empirical Approximation Error RMSE #Points/Expert rbcm BCM gpoe PoE SOD GP Gradient time in sec 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 15th September

87 Empirical Approximation Error (2) NLPD #Points/Expert rbcm BCM gpoe PoE SOD GP Gradient time in sec 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 15th September

88 Large Data Sets: Airline Data (700K) RMSE Airline Delay, 700K DGP, random data assignment 34 Sparse GP rbcm BCM gpoe PoE Dist VGP SVI GP rbcm performs best (r)bcm and (g)poe with 4096 GP experts Gradient time: 13 seconds (12 cores) Inducing inputs: Dist-VGP (Gal et al., 2014), SVI-GP (Hensman et al., 2013) (g)poe and BCM perform not worse than sparse GPs Distributed Gaussian Processes Marc 15th September

89 Large Data Sets: Airline Data (700K) RMSE Airline Delay, 700K DGP, random data assignment 34 DGP, KD- tree data assignment Sparse GP rbcm rbcm KD BCM BCM KD gpoe gpoe KD PoE PoE KDDist VGP SVI GP rbcm performs best (r)bcm and (g)poe with 4096 GP experts Gradient time: 13 seconds (12 cores) Inducing inputs: Dist-VGP (Gal et al., 2014), SVI-GP (Hensman et al., 2013) (g)poe and BCM perform not worse than sparse GPs KD-tree data assignment clearly helps (r)bcm Distributed Gaussian Processes Marc 15th September

90 Summary: Distributed Gaussian Processes µ, σ µ 1, σ 1 µ 2, σ 2 µ 11, σ 11 µ 12, σ 12 µ 13, σ 13 µ 21, σ 21 µ 22, σ 22 µ 23, σ 23 Scale Gaussian processes to large data (beyond 10 6 ) Model conceptually straightforward and easy to train Key: Distributed computation Currently tested with N P Op10 7 q Scales to arbitrarily large data sets (with enough computing power) m.deisenroth@imperial.ac.uk Distributed Gaussian Processes Marc 15th September

91 References [1] Y. Cao and D. J. Fleet. Generalized Product of Experts for Automatic and Principled Fusion of Gaussian Process Predictions. October [2] K. Chalupka, C. K. I. Williams, and I. Murray. A Framework for Evaluating Approximate Methods for Gaussian Process Regression. Journal of Machine Learning Research, 14: , February [3] M. P. Deisenroth and J. W. Ng. Distributed Gaussian Processes. In Proceedings of the International Conference on Machine Learning, [4] M. P. Deisenroth, C. E. Rasmussen, and D. Fox. Learning to Control a Low-Cost Manipulator using Data-Efficient Reinforcement Learning. In Proceedings of Robotics: Science and Systems, Los Angeles, CA, USA, June [5] Y. Gal, M. van der Wilk, and C. E. Rasmussen. Distributed Variational Inference in Sparse Gaussian Process Regression and Latent Variable Models. In Advances in Neural Information Processing Systems [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, [7] T. Heskes. Selecting Weighting Factors in Logarithmic Opinion Pools. In Advances in Neural Information Processing Systems, pages Morgan Kaufman, [8] N. Lawrence. Probabilistic Non-linear Principal Component Analysis with Gaussian Process Latent Variable Models. Journal of Machine Learning Research, 6: , November [9] J. Ng and M. P. Deisenroth. Hierarchical Mixture-of-Experts Model for Large-Scale Gaussian Process Regression. December [10] J. Quiñonero-Candela and C. E. Rasmussen. A Unifying View of Sparse Approximate Gaussian Process Regression. Journal of Machine Learning Research, 6(2): , [11] E. Snelson and Z. Ghahramani. Sparse Gaussian Processes using Pseudo-inputs. In Y. Weiss, B. Schölkopf, and J. C. Platt, editors, Advances in Neural Information Processing Systems 18, pages The MIT Press, Cambridge, MA, USA, [12] M. K. Titsias. Variational Learning of Inducing Variables in Sparse Gaussian Processes. In Proceedings of the International Conference on Artificial Intelligence and Statistics, [13] V. Tresp. A Bayesian Committee Machine. Neural Computation, 12(11): , Distributed Gaussian Processes Marc 15th September