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233 nips-2012-Multiresolution Gaussian Processes

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Author: David B. Dunson, Emily B. Fox

Abstract: We propose a multiresolution Gaussian process to capture long-range, nonMarkovian dependencies while allowing for abrupt changes and non-stationarity. The multiresolution GP hierarchically couples a collection of smooth GPs, each deﬁned over an element of a random nested partition. Long-range dependencies are captured by the top-level GP while the partition points deﬁne the abrupt changes. Due to the inherent conjugacy of the GPs, one can analytically marginalize the GPs and compute the marginal likelihood of the observations given the partition tree. This property allows for efﬁcient inference of the partition itself, for which we employ graph-theoretic techniques. We apply the multiresolution GP to the analysis of magnetoencephalography (MEG) recordings of brain activity.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We propose a multiresolution Gaussian process to capture long-range, nonMarkovian dependencies while allowing for abrupt changes and non-stationarity. [sent-7, score-0.297]

2 The multiresolution GP hierarchically couples a collection of smooth GPs, each deﬁned over an element of a random nested partition. [sent-8, score-0.17]

3 Long-range dependencies are captured by the top-level GP while the partition points deﬁne the abrupt changes. [sent-9, score-0.342]

4 Due to the inherent conjugacy of the GPs, one can analytically marginalize the GPs and compute the marginal likelihood of the observations given the partition tree. [sent-10, score-0.366]

5 This property allows for efﬁcient inference of the partition itself, for which we employ graph-theoretic techniques. [sent-11, score-0.185]

6 We apply the multiresolution GP to the analysis of magnetoencephalography (MEG) recordings of brain activity. [sent-12, score-0.173]

7 However, GPs typically assume smoothness properties that can blur key elements of the signal if abrupt changes occur. [sent-15, score-0.196]

8 Likewise, a changepoint [21] or partition [8] model between smooth functions fails to capture long range dependencies spanning changepoints. [sent-17, score-0.29]

9 Key to the data analysis is the ability to share information about the overall trajectory between the single trials without forcing unrealistic smoothness assumptions on the single trials themselves. [sent-29, score-0.325]

10 In order to capture both long-range dependencies and potential discontinuities, we propose a multiresolution GP (mGP) that hierarchically couples a collection of smooth GPs, each deﬁned over an element of a nested partition set. [sent-30, score-0.387]

11 The top-level GP captures a smooth global trajectory, while the partition points deﬁne abrupt changes in correlation induced by the lower-level GPs. [sent-31, score-0.48]

12 Due to the inherent conjugacy of the GPs, conditioned on the partition points the resulting function at the bottom level is marginally GP-distributed with a partition-dependent (and thus non-stationary) covariance function. [sent-32, score-0.364]

13 The correlation between any two observations yi and yj generated by the mGP at locations xi and xj is a function of the distance ||xi − xj || and which partition sets contain both xi and xj . [sent-33, score-0.557]

14 In a standard regression setting, the marginal GP structure of the mGP allows us to compute the marginal likelihood of the data conditioned on the partition, enabling efﬁcient inference of the partition itself. [sent-34, score-0.323]

15 We integrate over the hierarchy of GPs and only sample the partition points. [sent-35, score-0.215]

16 Recursing down the tree, each Hierarchical segmentation produced by recursive minimization of partition has a GP with mean given by its parent function restricted to that set. [sent-39, score-0.287]

17 proposal distribution, we borrow the graph-theoretic idea of normalized cuts [22] often used in image segmentation. [sent-41, score-0.182]

18 Our inferences integrate over the partition tree, allowing blurring of discontinuities and producing functions which can appear smooth when discontinuities are not present in the data. [sent-42, score-0.406]

19 Then, over each partition set Aℓ we independently draw i f ℓ (Aℓ ) ∼ GP(f ℓ−1 (Aℓ ), cℓ ). [sent-84, score-0.185]

20 (2) i i i That is, the mean of the GP is given by the parent function restricted to the current partition set. [sent-85, score-0.287]

21 Due to the conditional independence of these draws, f ℓ can have discontinuities at the partition points. [sent-86, score-0.272]

22 − x′ ||2 ), 2 ℓ 2 Covariance Function We assume a squared exponential kernel cℓ = dℓ exp(−κℓ ||x i i i ∞ encouraging local smoothness over each partition set Aℓ . [sent-92, score-0.227]

23 We focus on dℓ = dℓ with ℓ=1 (d ) < 1 i i for ﬁnite variance regardless of tree depth and additionally encouraging lower levels to vary less from their parent function, providing regularization and robustness to the choice of L. [sent-93, score-0.275]

24 One can think of this formulation as akin to a fractal process: zooming in on any partition, the locally deﬁned function has the same smoothness as that of its parent over the larger partition. [sent-95, score-0.175]

25 (3) ℓ=1 Due to the inherent conjugacy of the GPs, one can analytically marginalize the hierarchy of GPs conditioned on the partition tree A yielding L−1 g | A ∼ GP(0, c∗ ), A cℓ IAℓ . [sent-103, score-0.456]

26 (4) provides an interpretation of the mGP i i as a (marginally) partition-dependent GP, where the partition A deﬁnes the discontinuities in the covariance function c∗ . [sent-106, score-0.326]

27 The covariance function encodes local smoothness of g and discontinuities A at the partition points. [sent-107, score-0.368]

28 A The correlation between any two observations yi and yj at locations xi and xj generated as in Eq. [sent-109, score-0.239]

29 (1) ℓ is a function of how many tree levels contain both xi and xj and the distance ||xi − xj ||. [sent-110, score-0.306]

30 Let ri ℓ index the partition set such that xi ∈ Arℓ and Lij the lowest level for which xi and xj fall into the i ℓ ℓ same set (i. [sent-111, score-0.337]

31 Then, for xi = xj , Lij ℓ ℓ ℓ=0 cri (xi , xj ) corr(yi , yj | A) = k∈{i,j} (σ 2 + 1 L−1 ℓ 2 ℓ ℓ=0 crk (xk , xk )) = Lij ℓ=0 dℓ exp(−κ||xi − xj ||2 /||Aℓ ℓ ||2 ) 2 r 2 i σ2 + L−1 ℓ ℓ=0 d (5) where the second equality follows from assuming the previously described kernels. [sent-114, score-0.203]

32 0 otherwise (6) The level-speciﬁc covariance matrix Kℓ is block-diagonal with structure determined by the levelspeciﬁc partition Aℓ . [sent-122, score-0.239]

33 (8) ℓ=ℓ′ +1 A key advantage of the mGP is the conditional conjugacy of the latent GPs that allows us to compute the likelihood of the data simply conditioned on the hierarchical partition A (see Eq. [sent-130, score-0.388]

34 This fact is fundamental to the efﬁciency of the partition inference procedure described in Sec. [sent-132, score-0.185]

35 To capture the common global trajectory of these trials while still allowing for trial-speciﬁc variability, we model each as a realization from an mGP with a shared parent function f 0 . [sent-135, score-0.343]

36 For simplicity, and due to the motivating MEG application, we additionally assume shared changepoints between the trials, though this assumption can also be relaxed. [sent-137, score-0.172]

37 (e) Hierarchical segmentation produced by recursive minimization of normalized cut objective. [sent-141, score-0.163]

38 , yn }, we model (j) yi (j) (j) = g (j) (xi ) + ǫi , ǫi ∼ N (0, σ 2 ), (9) with g (j) = f L−1,(j) generated from a trial-speciﬁc GP hierarchy f 0 → f 1,(j) → · · · → f L−1,(j) with shared parent f 0 . [sent-145, score-0.167]

39 , y(J) } are generated from a shared parent function f 0 , the marginal likelihood does not decompose over trials. [sent-179, score-0.215]

40 5 Inference of the Hierarchical Partition In the formulation so far, we have assumed that the hierarchical partition A is given. [sent-184, score-0.267]

41 A key question is to infer the partition from the data. [sent-185, score-0.185]

42 In what follows, we assume a balanced binary tree for A. [sent-188, score-0.179]

43 4 Partition Prior We consider a prior solely on the partition points {z1 , . [sent-190, score-0.185]

44 , z2L−1 −1 } rather than taking tree level into account as well. [sent-193, score-0.17]

45 Generatively, one can think of drawing 2L−1 − 1 partition points from F and deterministically forming a balanced binary tree A from these. [sent-196, score-0.364]

46 Also, despite common deployment, taking the partition point at level ℓ as uniformly distributed over the parent set Aℓ−1 yields high mass on A with small Aℓ . [sent-200, score-0.314]

47 Partition Proposal Although stochastic tree search algorithms tend to be inefﬁcient in general, we can harness the well-deﬁned correlation structure associated with a given hierarchical partition to much more efﬁciently search the tree space. [sent-204, score-0.627]

48 One can think of every observed location xi as a node in a graph with edge weights between xi and xj deﬁned by the magnitude of the correlation of yi and yj . [sent-205, score-0.199]

49 Based on this interpretation, the partition points of A correspond to graph cuts that bisect small edge weights, as graphically depicted in Fig. [sent-206, score-0.267]

50 The normalized cut metric balances between the cost of edge weights cut and the connectivity of the cut component, thus avoiding cuts that separate small sets. [sent-211, score-0.491]

51 Instead of deterministically selecting cut points, we employ the cut 1 cut 2 cut 2 normalized cut objective as a proposal distribution. [sent-214, score-0.715]

52 Let the cost matrix W be the absolute value of the empirical correlation matrix computed from trials {y(1) , . [sent-215, score-0.171]

53 At level ℓ, each Aℓ is partitioned via a normalized cut proposal based on the submatrix of W correi sponding to the locations xi ∈ Aℓ . [sent-223, score-0.335]

54 The probability of any partition A under the speciﬁed proposal i distribution is simply computed as the product of the sequence of conditional probabilities of each cut. [sent-224, score-0.245]

55 This procedure generates cut points only at the observed locations xi . [sent-225, score-0.208]

56 More formally, the partition point in X is proposed as uniformly distributed between xi and xi+1 . [sent-226, score-0.224]

57 Markov Chain Monte Carlo An importance sampler draws hierarchical partitions A(m) ∼ q, with the proposal distribution q deﬁned as above, and then weights the samples by p(A(m) )/q(A(m) ) to obtain posterior draws [19]. [sent-228, score-0.196]

58 (16) The tailoring of the proposal distribution q to this application based on normalized cuts dramatically aids in improving the acceptance rate relative to more naive tree proposals. [sent-231, score-0.325]

59 One beneﬁt of the MCMC approach over importance sampling is the ability to include more intricate tree proposals to increase efﬁciency. [sent-233, score-0.167]

60 We choose to interleave both local and global tree proposals. [sent-234, score-0.186]

61 , a partition set Aℓ ) and then propose a i new sequence of cuts for all children of this node. [sent-237, score-0.267]

62 We adapt the proposal distribution for node selection to encourage more global searches at ﬁrst and then shift towards a greater balance between local and global searches as the sampling progresses. [sent-239, score-0.206]

63 Using dynamic programming, the cost associated with the normalized cuts proposal is O(n2 (L − 1)). [sent-242, score-0.182]

64 These methods capture abrupt changes, but do not allow for long-range dependencies spanning changepoints nor a functional data hierarchical structure, both inherent to our multiresolution perspective. [sent-248, score-0.496]

65 [10, 11] consider covariance functions deﬁned on a phylogenetic tree such that the covariance between function-valued traits depends on both their spatial distance and evolutionary time spanned via a common ancestor. [sent-251, score-0.275]

66 Here, the tree deﬁnes the strength and structure of sharing between a collection of functions rather than abrupt changes within the function. [sent-252, score-0.339]

67 The Bayesian rose tree of [3] considers a mixture of GP experts, as in [14, 17], but using Bayesian hierarchical clustering with arbitrary branching structure in place of a Dirichlet process mixture. [sent-253, score-0.249]

68 [25]): the variables deﬁne a Markov process on a typically balanced, binary tree and higher-level nodes capture coarser level information about the process. [sent-256, score-0.218]

69 At a high level, the mGP differs from previous GP-based tree models in that the nodes of our tree represent GPs over a contiguous subset of the input space X constrained in a hierarchical fashion. [sent-258, score-0.398]

70 Thus, the mGP combines ideas of GP-based tree models and GP-based partition models. [sent-259, score-0.328]

71 3, one can formulate an mGP as an additive GP where each GP in the sum decomposes independently over the level-speciﬁc partition of the input space X . [sent-261, score-0.208]

72 1 Synthetic Experiments To assess our ability to infer a hierarchical partition via the proposed MCMC sampler, we generated 100 trials of length 200 from a 5-level mGP with a shared parent function f 0 . [sent-264, score-0.524]

73 3, along with the empirical correlation matrix that is used as the cost matrix for the normalized cuts proposals. [sent-272, score-0.173]

74 Log likelihood under the true partition (cyan) and minimized normalized cut partition of Fig. [sent-290, score-0.579]

75 (e) Predictive log likelihood of 10 heldout sequences for GP, hGP, and mGP with L = 2, 5(true), 7, 10. [sent-293, score-0.19]

76 The ﬁrst 1000 iterations used pure global tree searches; the sampler was then tempered to uniform node proposals. [sent-296, score-0.186]

77 5, which also displays the true hierarchical partition and MAP estimate. [sent-298, score-0.267]

78 To assess sensitivity to the choice of L, we compare the predictive log-likelihood of 10 heldout test sequences under an mGP with 2, 5, 7, and 10 levels. [sent-303, score-0.199]

79 However, overestimating L has minimal inﬂuence on predictive likelihood since lower tree levels capture ﬁner details and have less overall effect. [sent-306, score-0.274]

80 2 MEG Analysis We analyzed magnetoencephalography (MEG) recordings of neuronal activity collected from a helmet with gradiometers distributed over 102 locations around the head. [sent-316, score-0.163]

81 Efﬁcient sharing of information between the single trials is important for tasks such as word classiﬁcation [7]. [sent-321, score-0.194]

82 [7] propose a 2-level hierarchical GP (hGP): a parent GP captures the common global trajectory, as in the mGP, and each trial-speciﬁc GP is centered about the entire parent function1. [sent-324, score-0.329]

83 The mGP instead models the trial-speciﬁc variability with a multi-level tree of GPs deﬁned as deviations from the parent function over local partitions, allowing for abrupt changes relative to the smooth global trajectory. [sent-326, score-0.489]

84 We ran 3 independent MCMC chains for 3000 iterations with both global and local tree searches. [sent-331, score-0.186]

85 We compare the predictive performance of the mGP in terms of MSE of heldout segments relative to a GP and hGP, each with similarly optimized hyperparameters. [sent-336, score-0.199]

86 The predictive mean conditioned on data up to the heldout time is straightforwardly derived from Eq. [sent-337, score-0.227]

87 GP Visual Frontal Parietal Temporal 25 (b) (c) (d) Figure 6: Per-lobe comparison of mGP to (a) GP and (b) hGP: For various values of τ , % decrease in predictive ∗ ∗ MSE of heldout yτ :τ +30 conditioned on y1:τ −1 and 15 training sequences. [sent-348, score-0.227]

88 (c) For a visual cortex sensor and word hammer, plots of test data, empirical mean (MLE), and hGP and mGP predictive mean for entire heldout y∗ . [sent-349, score-0.305]

89 (d) Boxplots of predictive log likelihood of heldout y∗ for the mGP and wavelet-based method of [15]. [sent-350, score-0.245]

90 The results clearly indicate that the mGP consistently better captures the features of the data, and particularly for sensors with large abrupt changes such as in the visual cortex. [sent-351, score-0.245]

91 The heldout trials for a visual cortex sensor are displayed in Fig. [sent-352, score-0.338]

92 The posterior distribution of inferred changepoints at level 1, also broken down by cortical region, are displayed in Fig. [sent-355, score-0.169]

93 Visual L=1 Frontal L=1 igloo house church apartment barn hammer saw screwdriver pliers chisel 0 0. [sent-361, score-0.278]

94 5 1 Time (sec) Temporal L=1 igloo house church apartment barn hammer saw screwdriver pliers chisel 0 0. [sent-363, score-0.278]

95 5 1 Time (sec) Figure 7: Inferred changepoints at level 1 aggregated over sensors within each lobe: visual (top-left), frontal (top-right), parietal (bottom-left), and temporal (bottom-right). [sent-365, score-0.346]

96 The wfmm enables analysis in a multivariate setting, but for a direct comparison we simply apply the wfmm to each word and sensor independently. [sent-369, score-0.194]

97 6(d) shows boxplots of the predictive heldout log likelihood of the test trials under the mGP and wfmm. [sent-371, score-0.393]

98 In particular, the mGP provides a hierarchical functional data analysis framework for modeling (i) strong, locally smooth sharing of information, (ii) global long-range correlations, and (iii) abrupt changes. [sent-375, score-0.375]

99 For hyperparameter inference, we anticipate that joint sampling with the partition would mix poorly, and consider it a topic for future exploration. [sent-380, score-0.21]

100 Another interesting topic is to explore proposals for more general tree structures. [sent-381, score-0.167]

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As is common we assume this to have zero mean, so the multi-task GP is fully speciﬁed by the covariances fτ (x)fτ (x ) = C(τ, x, τ , x ). For this covariance we take the ﬂexible form from [5], fτ (x)fτ (x ) = Dτ τ C(x, x ). Here C(x, x ) determines the covariance between function values at different input points, encoding “spatial” behaviour such as smoothness and the lengthscale(s) over which the functions vary, while the matrix D is a free-form inter-task covariance matrix. One of the attractions of GPs for regression is that, even though they are non-parametric models with (in general) an inﬁnite number of degrees of freedom, predictions can be made in closed form, see e.g. [1]. For a test point x for task τ , one would predict as output the mean of fτ (x) over the (Gaussian) posterior, which is y T K −1 kτ (x). Here K is the n × n Gram matrix with entries 2 K m = Dτ τm C(x , xm ) + στ δ m , while kτ (x) is a vector with the n entries kτ, = Dτ τ C(x , x). The error bar would be taken as the square root of the posterior variance of fτ (x), which is T Vτ (x) = Dτ τ C(x, x) − kτ (x)K −1 kτ (x) (1) The learning curve for task τ is deﬁned as the mean-squared prediction error, averaged over the location of test input x and over all data sets with a speciﬁed number of examples for each task, say n1 for task 1 and so on. As is standard in learning curve analysis we consider a matched scenario where the training outputs y are generated from the same prior and noise model that we use for inference. In this case the mean-squared prediction error ˆτ is the Bayes error, and is given by the average posterior variance [1], i.e. ˆτ = Vτ (x) x . To obtain the learning curve this is averaged over the location of the training inputs x : τ = ˆτ . This average presents the main challenge for learning curve prediction because the training inputs feature in a highly nonlinear way in Vτ (x). Note that the training outputs, on the other hand, do not appear in the posterior variance Vτ (x) and so do not need to be averaged over. We now want to write the Bayes error ˆτ in a form convenient for performing, at least approximately, the averages required for the learning curve. Assume that all training inputs x , and also the test input x, are drawn from the same distribution P (x). One can decompose the input-dependent part of the covariance function into eigenfunctions relative to P (x), according to C(x, x ) = i λi φi (x)φi (x ). The eigenfunctions are deﬁned by the condition C(x, x )φi (x ) x = λi φi (x) and can be chosen to be orthonormal with respect to P (x), φi (x)φj (x) x = δij . The sum over i here is in general inﬁnite (unless the covariance function is degenerate, as e.g. for the dot product kernel C(x, x ) = x · x ). To make the algebra below as simple as possible, we let the eigenvalues λi be arranged in decreasing order and truncate the sum to the ﬁnite range i = 1, . . . , M ; M is then some large effective feature space dimension and can be taken to inﬁnity at the end. 2 In terms of the above eigenfunction decomposition, the Gram matrix has elements K m = Dτ 2 λi φi (x )φi (xm )+στ δ τm m δτ = i ,τ φi (x )λi δij Dτ τ φj (xm )δτ 2 ,τm +στ δ m i,τ,j,τ or in matrix form K = ΨLΨT + Σ where Σ is the diagonal matrix from the noise variances and Ψ = δτ ,iτ ,τ φi (x ), Liτ,jτ = λi δij Dτ τ (2) Here Ψ has its second index ranging over M (number of kernel eigenvalues) times T (number of tasks) values; L is a square matrix of this size. In Kronecker (tensor) product notation, L = D ⊗ Λ if we deﬁne Λ as the diagonal matrix with entries λi δij . The Kronecker product is convenient for the simpliﬁcations below; we will use that for generic square matrices, (A ⊗ B)(A ⊗ B ) = (AA ) ⊗ (BB ), (A ⊗ B)−1 = A−1 ⊗ B −1 , and tr (A ⊗ B) = (tr A)(tr B). In thinking about the mathematical expressions, it is often easier to picture Kronecker products over feature spaces and tasks as block matrices. For example, L can then be viewed as consisting of T × T blocks, each of which is proportional to Λ. To calculate the Bayes error, we need to average the posterior variance Vτ (x) over the test input x. The ﬁrst term in (1) then becomes Dτ τ C(x, x) = Dτ τ tr Λ. In the second one, we need to average kτ, (x)kτ,m = Dτ τ C(x , x)C(x, xm ) x Dτm τ = x Dτ τ λi λj φi (x ) φi (x)φj (x) x φj (xm )Dτm τ ij = Dτ τ Ψl,iτ λi λj δij Ψm,jτ Dτ τ i,τ ,j,τ T In matrix form this is kτ (x)kτ (x) x = Ψ[(Deτ eT D) ⊗ Λ2 ]ΨT = ΨMτ ΨT Here the last τ equality deﬁnes Mτ , and we have denoted by eτ the T -dimensional vector with τ -th component equal to one and all others zero. Multiplying by the inverse Gram matrix K −1 and taking the trace gives the average of the second term in (1); combining with the ﬁrst gives the Bayes error on task τ ˆτ = Vτ (x) x = Dτ τ tr Λ − tr ΨMτ ΨT (ΨLΨT + Σ)−1 Applying the Woodbury identity and re-arranging yields = Dτ τ tr Λ − tr Mτ ΨT Σ−1 Ψ(I + LΨT Σ−1 Ψ)−1 = ˆτ Dτ τ tr Λ − tr Mτ L−1 [I − (I + LΨT Σ−1 Ψ)−1 ] But tr Mτ L−1 = tr {[(Deτ eT D) ⊗ Λ2 ][D ⊗ Λ]−1 } τ = tr {[Deτ eT ] ⊗ Λ} = eT Deτ tr Λ = Dτ τ tr Λ τ τ so the ﬁrst and second terms in the expression for ˆτ cancel and one has = tr Mτ L−1 (I + LΨT Σ−1 Ψ)−1 = tr L−1 Mτ L−1 (L−1 + ΨT Σ−1 Ψ)−1 = tr [D ⊗ Λ]−1 [(Deτ eT D) ⊗ Λ2 ][D ⊗ Λ]−1 (L−1 + ΨT Σ−1 Ψ)−1 τ = ˆτ tr [eτ eT ⊗ I](L−1 + ΨT Σ−1 Ψ)−1 τ The matrix in square brackets in the last line is just a projector Pτ onto task τ ; thought of as a matrix of T × T blocks (each of size M × M ), this has an identity matrix in the (τ, τ ) block while all other blocks are zero. We can therefore write, ﬁnally, for the Bayes error on task τ , ˆτ = tr Pτ (L−1 + ΨT Σ−1 Ψ)−1 (3) Because Σ is diagonal and given the deﬁnition (2) of Ψ, the matrix ΨT Σ−1 Ψ is a sum of contributions from the individual training examples = 1, . . . , n. This will be important for deriving the learning curve approximation below. We note in passing that, because τ Pτ = I, the sum of the Bayes errors on all tasks is τ ˆτ = tr (L−1 +ΨT Σ−1 Ψ)−1 , in close analogy to the corresponding expression for the single-task case [13]. 3 3 Learning curve prediction To obtain the learning curve τ = ˆτ , we now need to carry out the average . . . over the training inputs. To help with this, we can extend an approach for the single-task scenario [13] and deﬁne a response or resolvent matrix G = (L−1 + ΨT Σ−1 Ψ + τ vτ Pτ )−1 with auxiliary parameters vτ that will be set back to zero at the end. One can then ask how G = G and hence τ = tr Pτ G changes with the number nτ of training points for task τ . Adding an example at position x for task −2 τ increases ΨT Σ−1 Ψ by στ φτ φT , where φτ has elements (φτ )iτ = φi (x)δτ τ . Evaluating the τ −1 −2 difference (G + στ φτ φT )−1 − G with the help of the Woodbury identity and approximating it τ with a derivative gives Gφτ φT G ∂G τ =− 2 ∂nτ στ + φT Gφτ τ This needs to be averaged over the new example and all previous ones. If we approximate by averaging numerator and denominator separately we get 1 ∂G ∂G = 2 ∂nτ στ + tr Pτ G ∂vτ (4) Here we have exploited for the average over x that the matrix φτ φT x has (i, τ ), (j, τ )-entry τ φi (x)φj (x) x δτ τ δτ τ = δij δτ τ δτ τ , hence simply φτ φT x = Pτ . We have also used the τ auxiliary parameters to rewrite − GPτ G = ∂ G /∂vτ = ∂G/∂vτ . Finally, multiplying (4) by Pτ and taking the trace gives the set of quasi-linear partial differential equations ∂ τ 1 = 2 ∂nτ στ + τ ∂ τ ∂vτ (5) The remaining task is now to ﬁnd the functions τ (n1 , . . . , nT , v1 , . . . , vT ) by solving these differential equations. We initially attempted to do this by tracking the τ as examples are added one task at a time, but the derivation is laborious already for T = 2 and becomes prohibitive beyond. Far more elegant is to adapt the method of characteristics to the present case. We need to ﬁnd a 2T -dimensional surface in the 3T -dimensional space (n1 , . . . , nT , v1 , . . . , vT , 1 , . . . , T ), which is speciﬁed by the T functions τ (. . .). A small change (δn1 , . . . , δnT , δv1 , . . . , δvT , δ 1 , . . . , δ T ) in all 3T coordinates is tangential to this surface if it obeys the T constraints (one for each τ ) δ τ ∂ τ ∂ τ δnτ + δvτ ∂nτ ∂vτ = τ 2 From (5), one sees that this condition is satisﬁed whenever δ τ = 0 and δnτ = −δvτ (στ + τ ) It follows that all the characteristic curves given by τ (t) = τ,0 = const., vτ (t) = vτ,0 (1 − t), 2 nτ (t) = vτ,0 (στ + τ,0 ) t for t ∈ [0, 1] are tangential to the solution surface for all t, so lie within this surface if the initial point at t = 0 does. Because at t = 0 there are no training examples (nτ (0) = 0), this initial condition is satisﬁed by setting −1 τ,0 = tr Pτ −1 L + vτ ,0 Pτ τ Because t=1 τ (t) is constant along the characteristic curve, we get by equating the values at t = 0 and −1 τ,0 = tr Pτ L −1 + vτ ,0 Pτ = τ ({nτ = vτ 2 ,0 (στ + τ ,0 )}, {vτ = 0}) τ Expressing vτ ,0 in terms of nτ gives then τ = tr Pτ L−1 + τ nτ 2 στ + −1 Pτ (6) τ This is our main result: a closed set of T self-consistency equations for the average Bayes errors 2 τ . Given L as deﬁned by the eigenvalues λi of the covariance function, the noise levels στ and the 4 number of examples nτ for each task, it is straightforward to solve these equations numerically to ﬁnd the average Bayes error τ for each task. The r.h.s. of (6) is easiest to evaluate if we view the matrix inside the brackets as consisting of M × M blocks of size T × T (which is the reverse of the picture we have used so far). The matrix is then block diagonal, with the blocks corresponding to different eigenvalues λi . Explicitly, because L−1 = D −1 ⊗ Λ−1 , one has τ λ−1 D −1 + diag({ i = i 4 2 στ nτ + −1 }) τ (7) ττ Results and discussion We now consider the consequences of the approximate prediction (7) for multi-task learning curves in GP regression. A trivial special case is the one of uncorrelated tasks, where D is diagonal. Here one recovers T separate equations for the individual tasks as expected, which have the same form as for single-task learning [13]. 4.1 Pure transfer learning Consider now the case of pure transfer learning, where one is learning a task of interest (say τ = 1) purely from examples for other tasks. What is the lowest average Bayes error that can be obtained? Somewhat more generally, suppose we have no examples for the ﬁrst T0 tasks, n1 = . . . = nT0 = 0, but a large number of examples for the remaining T1 = T − T0 tasks. Denote E = D −1 and write this in block form as E00 E01 E= T E01 E11 2 Now multiply by λ−1 and add in the lower right block a diagonal matrix N = diag({nτ /(στ + i −1 −1 τ )}τ =T0 +1,...,T ). The matrix inverse in (7) then has top left block λi [E00 + E00 E01 (λi N + −1 −1 T T E11 − E01 E00 E01 )−1 E01 E00 ]. As the number of examples for the last T1 tasks grows, so do all −1 (diagonal) elements of N . In the limit only the term λi E00 survives, and summing over i gives −1 −1 1 = tr Λ(E00 )11 = C(x, x) (E00 )11 . The Bayes error on task 1 cannot become lower than this, placing a limit on the beneﬁts of pure transfer learning. That this prediction of the approximation (7) for such a lower limit is correct can also be checked directly: once the last T1 tasks fτ (x) (τ = T0 + 1, . . . T ) have been learn perfectly, the posterior over the ﬁrst T0 functions is, by standard Gaussian conditioning, a GP with covariance C(x, x )(E00 )−1 . Averaging the posterior variance of −1 f1 (x) then gives the Bayes error on task 1 as 1 = C(x, x) (E00 )11 , as found earlier. This analysis can be extended to the case where there are some examples available also for the ﬁrst T0 tasks. One ﬁnds for the generalization errors on these tasks the prediction (7) with D −1 replaced by E00 . This is again in line with the above form of the GP posterior after perfect learning of the remaining T1 tasks. 4.2 Two tasks We next analyse how well the approxiation (7) does in predicting multi-task learning curves for T = 2 tasks. Here we have the work of Chai [21] as a baseline, and as there we choose D= 1 ρ ρ 1 The diagonal elements are ﬁxed to unity, as in a practical application where one would scale both task functions f1 (x) and f2 (x) to unit variance; the degree of correlation of the tasks is controlled by ρ. We ﬁx π2 = n2 /n and plot learning curves against n. In numerical simulations we ensure integer values of n1 and n2 by setting n2 = nπ2 , n1 = n − n2 ; for evaluation of (7) we use 2 2 directly n2 = nπ2 , n1 = n(1 − π2 ). For simplicity we consider equal noise levels σ1 = σ2 = σ 2 . As regards the covariance function and input distribution, we analyse ﬁrst the scenario studied in [21]: a squared exponential (SE) kernel C(x, x ) = exp[−(x − x )2 /(2l2 )] with lengthscale l, and one-dimensional inputs x with a Gaussian distribution N (0, 1/12). The kernel eigenvalues λi 5 1 1 1 1 ε1 ε1 0.8 1 1 ε1 ε1 0.8 0.001 1 ε1 0.8 0.001 n 10000 ε1 1 0.01 1 n 10000 0.6 0.6 0.4 0.4 0.4 0.2 0.2 n 1000 0.6 0.2 0 0 100 200 n 300 400 0 500 0 100 200 n 300 400 500 0 0 100 200 n 300 400 500 Figure 1: Average Bayes error for task 1 for two-task GP regression with kernel lengthscale l = 0.01, noise level σ 2 = 0.05 and a fraction π2 = 0.75 of examples for task 2. Solid lines: numerical simulations; dashed lines: approximation (7). Task correlation ρ2 = 0, 0.25, 0.5, 0.75, 1 from top to bottom. Left: SE covariance function, Gaussian input distribution. Middle: SE covariance, uniform inputs. Right: OU covariance, uniform inputs. Log-log plots (insets) show tendency of asymptotic uselessness, i.e. bunching of the ρ < 1 curves towards the one for ρ = 0; this effect is strongest for learning of smooth functions (left and middle). are known explicitly from [22] and decay exponentially with i. Figure 1(left) compares numerically simulated learning curves with the predictions for 1 , the average Bayes error on task 1, from (7). Five pairs of curves are shown, for ρ2 = 0, 0.25, 0.5, 0.75, 1. Note that the two extreme values represent single-task limits, where examples from task 2 are either ignored (ρ = 0) or effectively treated as being from task 1 (ρ = 1). Our predictions lie generally below the true learning curves, but qualitatively represent the trends well, in particular the variation with ρ2 . The curves for the different ρ2 values are fairly evenly spaced vertically for small number of examples, n, corresponding to a linear dependence on ρ2 . As n increases, however, the learning curves for ρ < 1 start to bunch together and separate from the one for the fully correlated case (ρ = 1). The approximation (7) correctly captures this behaviour, which is discussed in more detail below. Figure 1(middle) has analogous results for the case of inputs x uniformly distributed on the interval [0, 1]; the λi here decay exponentially with i2 [17]. Quantitative agreement between simulations and predictions is better for this case. The discussion in [17] suggests that this is because the approximation method we have used implicitly neglects spatial variation of the dataset-averaged posterior variance Vτ (x) ; but for a uniform input distribution this variation will be weak except near the ends of the input range [0, 1]. Figure 1(right) displays similar results for an OU kernel C(x, x ) = exp(−|x − x |/l), showing that our predictions also work well when learning rough (nowhere differentiable) functions. 4.3 Asymptotic uselessness The two-task results above suggest that multi-task learning is less useful asymptotically: when the number of training examples n is large, the learning curves seem to bunch towards the curve for ρ = 0, where task 2 examples are ignored, except when the two tasks are fully correlated (ρ = 1). We now study this effect. When the number of examples for all tasks becomes large, the Bayes errors τ will become small 2 and eventually be negligible compared to the noise variances στ in (7). One then has an explicit prediction for each τ , without solving T self-consistency equations. If we write, for T tasks, 2 nτ = nπτ with πτ the fraction of examples for task τ , and set γτ = πτ /στ , then for large n τ = i λ−1 D −1 + nΓ i −1 ττ = −1/2 −1 [λi (Γ1/2 DΓ1/2 )−1 i (Γ + nI]−1 Γ−1/2 )τ τ 1/2 where Γ = diag(γ1 , . . . , γT ). Using an eigendecomposition of the symmetric matrix Γ T T a=1 δa va va , one then shows in a few lines that (8) can be written as τ −1 ≈ γτ 2 a (va,τ ) δa g(nδa ) 6 (8) 1/2 DΓ = (9) 1 1 1 50000 ε 5000 r 0.1 ε 0.5 n=500 10 100 1000 n 0.1 0 0 0.2 0.4 ρ 2 0.6 0.8 1 1 10 100 1000 n Figure 2: Left: Bayes error (parameters as in Fig. 1(left), with n = 500) vs ρ2 . To focus on the error reduction with ρ, r = [ 1 (ρ) − 1 (1)]/[ 1 (0) − 1 (1)] is shown. Circles: simulations; solid line: predictions from (7). Other lines: predictions for larger n, showing the approach to asymptotic uselessness in multi-task learning of smooth functions. Inset: Analogous results for rough functions (parameters as in Fig. 1(right)). Right: Learning curve for many-task learning (T = 200, parameters otherwise as in Fig. 1(left) except ρ2 = 0.8). Notice the bend around 1 = 1 − ρ = 0.106. Solid line: simulations (steps arise because we chose to allocate examples to tasks in order τ = 1, . . . , T rather than randomly); dashed line: predictions from (7). Inset: Predictions for T = 1000, with asymptotic forms = 1 − ρ + ρ˜ and = (1 − ρ)¯ for the two learning stages shown as solid lines. −1 where g(h) = tr (Λ−1 + h)−1 = + h)−1 and va,τ is the τ -th component of the a-th i (λi eigenvector va . This is the general asymptotic form of our prediction for the average Bayes error for task τ . To get a more explicit result, consider the case where sample functions from the GP prior have (mean-square) derivatives up to order r. The kernel eigenvalues λi then decay as1 i−(2r+2) for large i, and using arguments from [17] one deduces that g(h) ∼ h−α for large h, with α = (2r +1)/(2r + 2). In (9) we can then write, for large n, g(nδa ) ≈ (δa /γτ )−α g(nγτ ) and hence τ ≈ g(nγτ ){ 2 1−α } a (va,τ ) (δa /γτ ) (10) 2 When there is only a single task, δ1 = γ1 and this expression reduces to 1 = g(nγ1 ) = g(n1 /σ1 ). 2 Thus g(nγτ ) = g(nτ /στ ) is the error we would get by ignoring all examples from tasks other than τ , and the term in {. . .} in (10) gives the “multi-task gain”, i.e. the factor by which the error is reduced because of examples from other tasks. (The absolute error reduction always vanishes trivially for n → ∞, along with the errors themselves.) One observation can be made directly. Learning of very smooth functions, as deﬁned e.g. by the SE kernel, corresponds to r → ∞ and hence α → 1, so the multi-task gain tends to unity: multi-task learning is asymptotically useless. The only exception occurs when some of the tasks are fully correlated, because one or more of the eigenvalues δa of Γ1/2 DΓ1/2 will then be zero. Fig. 2(left) shows this effect in action, plotting Bayes error against ρ2 for the two-task setting of Fig. 1(left) with n = 500. Our predictions capture the nonlinear dependence on ρ2 quite well, though the effect is somewhat weaker in the simulations. For larger n the predictions approach a curve that is constant for ρ < 1, signifying negligible improvement from multi-task learning except at ρ = 1. It is worth contrasting this with the lower bound from [21], which is linear in ρ2 . While this provides a very good approximation to the learning curves for moderate n [21], our results here show that asymptotically this bound can become very loose. When predicting rough functions, there is some asymptotic improvement to be had from multi-task learning, though again the multi-task gain is nonlinear in ρ2 : see Fig. 2(left, inset) for the OU case, which has r = 1). A simple expression for the gain can be obtained in the limit of many tasks, to which we turn next. 1 See the discussion of Sacks-Ylvisaker conditions in e.g. [1]; we consider one-dimensional inputs here though the discussion can be generalized. 7 4.4 Many tasks We assume as for the two-task case that all inter-task correlations, Dτ,τ with τ = τ , are equal to ρ, while Dτ,τ = 1. This setup was used e.g. in [23], and can be interpreted as each task having a √ component proportional to ρ of a shared latent function, with an independent task-speciﬁc signal in addition. We assume for simplicity that we have the same number nτ = n/T of examples for 2 each task, and that all noise levels are the same, στ = σ 2 . Then also all Bayes errors τ = will be the same. Carrying out the matrix inverses in (7) explicitly, one can then write this equation as = gT (n/(σ 2 + ), ρ) (11) where gT (h, ρ) is related to the single-task function g(h) from above by gT (h, ρ) = 1−ρ T −1 (1 − ρ)g(h(1 − ρ)/T ) + ρ + T T g(h[ρ + (1 − ρ)/T ]) (12) Now consider the limit T → ∞ of many tasks. If n and hence h = n/(σ 2 + ) is kept ﬁxed, gT (h, ρ) → (1 − ρ) + ρg(hρ); here we have taken g(0) = 1 which corresponds to tr Λ = C(x, x) x = 1 as in the examples above. One can then deduce from (11) that the Bayes error for any task will have the form = (1 − ρ) + ρ˜, where ˜ decays from one to zero with increasing n as for a single task, but with an effective noise level σ 2 = (1 − ρ + σ 2 )/ρ. Remarkably, then, ˜ even though here n/T → 0 so that for most tasks no examples have been seen, the Bayes error for each task decreases by “collective learning” to a plateau of height 1 − ρ. The remaining decay of to zero happens only once n becomes of order T . Here one can show, by taking T → ∞ at ﬁxed h/T in (12) and inserting into (11), that = (1 − ρ)¯ where ¯ again decays as for a single task but with an effective number of examples n = n/T and effective noise level σ 2 /(1 − ρ). This ﬁnal stage of ¯ ¯ learning therefore happens only when each task has seen a considerable number of exampes n/T . Fig. 2(right) validates these predictions against simulations, for a number of tasks (T = 200) that is in the same ballpark as in the many-tasks application example of [24]. The inset for T = 1000 shows clearly how the two learning curve stages separate as T becomes larger. Finally we can come back to the multi-task gain in the asymptotic stage of learning. For GP priors with sample functions with derivatives up to order r as before, the function ¯ from above will decay as (¯ /¯ 2 )−α ; since = (1 − ρ)¯ and σ 2 = σ 2 /(1 − ρ), the Bayes error is then proportional n σ ¯ to (1 − ρ)1−α . This multi-task gain again approaches unity for ρ < 1 for smooth functions (α = (2r + 1)/(2r + 2) → 1). Interestingly, for rough functions (α < 1), the multi-task gain decreases for small ρ2 as 1 − (1 − α) ρ2 and so always lies below a linear dependence on ρ2 initially. This shows that a linear-in-ρ2 lower error bound cannot generally apply to T > 2 tasks, and indeed one can verify that the derivation in [21] does not extend to this case. 5 Conclusion We have derived an approximate prediction (7) for learning curves in multi-task GP regression, valid for arbitrary inter-task correlation matrices D. This can be evaluated explicitly knowing only the kernel eigenvalues, without sampling or recourse to single-task learning curves. The approximation shows that pure transfer learning has a simple lower error bound, and provides a good qualitative account of numerically simulated learning curves. Because it can be used to study the asymptotic behaviour for large training sets, it allowed us to show that multi-task learning can become asymptotically useless: when learning smooth functions it reduces the asymptotic Bayes error only if tasks are fully correlated. For the limit of many tasks we found that, remarkably, some initial “collective learning” is possible even when most tasks have not seen examples. A much slower second learning stage then requires many examples per task. The asymptotic regime of this also showed explicitly that a lower error bound that is linear in ρ2 , the square of the inter-task correlation, is applicable only to the two-task setting T = 2. In future work it would be interesting to use our general result to investigate in more detail the consequences of speciﬁc choices for the inter-task correlations D, e.g. to represent a lower-dimensional latent factor structure. 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