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127 nips-2012-Fast Bayesian Inference for Non-Conjugate Gaussian Process Regression

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Author: Emtiyaz Khan, Shakir Mohamed, Kevin P. Murphy

Abstract: We present a new variational inference algorithm for Gaussian process regression with non-conjugate likelihood functions, with application to a wide array of problems including binary and multi-class classiﬁcation, and ordinal regression. Our method constructs a concave lower bound that is optimized using an efﬁcient ﬁxed-point updating algorithm. We show that the new algorithm has highly competitive computational complexity, matching that of alternative approximate inference methods. We also prove that the use of concave variational bounds provides stable and guaranteed convergence – a property not available to other approaches. We show empirically for both binary and multi-class classiﬁcation that our new algorithm converges much faster than existing variational methods, and without any degradation in performance. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Our method constructs a concave lower bound that is optimized using an efﬁcient ﬁxed-point updating algorithm. [sent-3, score-0.388]

2 We also prove that the use of concave variational bounds provides stable and guaranteed convergence – a property not available to other approaches. [sent-5, score-0.548]

3 We show empirically for both binary and multi-class classiﬁcation that our new algorithm converges much faster than existing variational methods, and without any degradation in performance. [sent-6, score-0.309]

4 For real-valued outputs, we can combine the GP prior with a Gaussian likelihood and perform exact posterior inference in closed form. [sent-8, score-0.241]

5 However, in other cases, such as classiﬁcation, the likelihood is no longer conjugate to the GP prior, and exact inference is no longer tractable. [sent-9, score-0.145]

6 In this paper, we use a variational approach, where we compute a lower bound to the log marginal likelihood using Jensen’s inequality. [sent-19, score-0.551]

7 Unfortunately, most of these methods are slow, since they attempt to solve for the posterior covariance matrix, which has size O(N 2 ), where N is the number of data points. [sent-22, score-0.147]

8 In [19], a reparameterization was proposed that only requires computing O(N ) variational parameters. [sent-23, score-0.358]

9 In this paper, we propose a new lower bound that is concave, and derive an efﬁcient iterative algorithm for its maximization. [sent-25, score-0.147]

10 Table 1: Gaussian process regression (top left) and its graphical model (right), along with the example likelihoods for outputs (bottom left). [sent-28, score-0.163]

11 2 Gaussian Process Regression Gaussian process (GP) regression is a powerful method for non-parametric regression that has gained a great deal of attention as a ﬂexible and accurate modeling approach. [sent-33, score-0.153]

12 Consider N data points with the n’th observation denoted by yn , with corresponding features xn . [sent-34, score-0.13]

13 A Gaussian process model uses a non-linear latent function z(x) to obtain the distribution of the observation y using an appropriate likelihood [15, 18]. [sent-35, score-0.149]

14 A Gaussian process [20] speciﬁes a distribution over z(x), and is a stochastic process that is characterized by a mean function µ(x) and a covariance function Σ(x, x ), which are speciﬁed using a kernel function that depends on the observed features x. [sent-38, score-0.16]

15 We consider frequently encountered data such as binary, ordinal, categorical and count observations, and describe their likelihoods in Table 1. [sent-55, score-0.194]

16 For the case of categorical observations, the latent function z is a vector whose k’th element is the latent function for k’th category. [sent-56, score-0.184]

17 In all cases, the likelihoods we consider are not conjugate to the Gaussian prior distribution and as a result, the posterior distribution is intractable. [sent-59, score-0.163]

18 Similarly, the integrations required in computing the predictive distribution and the marginal likelihood are intractable. [sent-60, score-0.121]

19 To deal with this intractability we make use of variational methods. [sent-61, score-0.213]

20 3 Variational Lower Bound to the Log Marginal Likelihood Inference and model selection are always problematic in any Gaussian process regression using nonconjugate likelihoods due to the fact that the marginal likelihood contains an intractable integral. [sent-62, score-0.284]

21 In this section, we derive a tractable variational lower bound to the marginal likelihood. [sent-63, score-0.41]

22 We show 2 that the lower bound takes a well known form and can be maximized using concave optimization. [sent-64, score-0.388]

23 Throughout the section, we assume scalar zn , with extension to the vector case being straightforward. [sent-65, score-0.164]

24 We begin with the intractable log marginal likelihood L(θ) in Eq. [sent-66, score-0.191]

25 We use a Gaussian posterior with mean m and covariance V. [sent-68, score-0.147]

26 The full set of variational parameters is thus γ = {m, V}. [sent-69, score-0.213]

27 As log is a concave function, we obtain a lower bound LJ (θ, γ) using Jensen’s inequality, given in Eq. [sent-70, score-0.458]

28 The ﬁrst integral is simply the Kullback−Leibler (KL) divergence from the variational Gaussian posterior q(z|m, V) to the GP prior p(z|µ, Σ) as shown in Eq. [sent-72, score-0.309]

29 The second integral can be expressed in terms of the expectation with respect to the marginal q(zn |mn , Vnn ) as shown in the second term of Eq. [sent-75, score-0.129]

30 Here mn is the n’th element of m and Vnn is the n’th diagonal element of V, the two variables collectively denoted by γn . [sent-77, score-0.222]

31 The lower bound LJ is still intractable since the expectation of log p(yn |zn ) is not available in closed form for the distributions listed in Table 1. [sent-78, score-0.258]

32 To derive a tractable lower bound, we make use of local variational bounds (LVB) fb , deﬁned such that E[log p(yn |zn )] ≥ fb (yn , mn , Vnn ), giving us Eq. [sent-79, score-0.973]

33 (6) n=1 We discuss the choice of LVBs in the next section, but ﬁrst discuss the well-known form that the lower bound of Eq. [sent-82, score-0.147]

34 Similarly, the function with respect to V is similar to the graphical lasso [8] or covariance selection problem [7], but is different in that the argument is a covariance matrix instead of a precision matrix [8]. [sent-85, score-0.202]

35 These two objective functions are coupled through the non-linear term fb (·). [sent-86, score-0.267]

36 In our case, this term arises from the likelihood, and is smooth and concave as we discuss in next section. [sent-88, score-0.241]

37 It is straightforward to show that the variational lower bound is strictly concave with respect to γ if fb is jointly concave with respect to mn and Vnn . [sent-89, score-1.336]

38 Strict concavity of terms other than fb is well-known since both the least squares and covariance selection problems are concave. [sent-90, score-0.457]

39 Similar concavity results have been discussed by Braun and McAuliffe [5] for the discrete choice model, and more recently by Challis and Barber [6] for the Bayesian linear model, who consider concavity with respect to the Cholesky factor of V. [sent-91, score-0.254]

40 We consider concavity with respect to V instead of its Cholesky factor, which allows us to exploit the special structure of V, as explained in Section 5. [sent-92, score-0.146]

41 4 Concave Local Variational Bounds In this section, we describe concave LVBs for various likelihoods. [sent-93, score-0.241]

42 We describe the LVBs for the likelihoods given in Table 1 with z being a scalar for count, binary, and ordinal data, but a vector of length K for categorical data, K being the number of classes. [sent-95, score-0.257]

43 For the Poison distribution, the expectation is available in closed form and we do not need any bounding: E[log p(y|η)] = ym − exp(m + v/2) − log y! [sent-97, score-0.141]

44 This function is jointly concave with respect to m and v since the exponential is a convex function. [sent-99, score-0.309]

45 3 For binary data, we use the piecewise linear/quadratic bounds proposed by [16], which is a bound on the logistic-log-partition (LLP) function log(1 + exp(x)) and can be used to obtain a bound over the sigmoid function σ(x). [sent-100, score-0.401]

46 The ﬁnal bound can be expressed as sum of R pieces: E(log p(y|η)) = R fb (y, m, v) = ym − r=1 fbr (m, v) where fbr is the expectation of r’th quadratic piece. [sent-101, score-0.593]

47 The function fbr is jointly concave with respect to m, v and their gradients are available in closed-form. [sent-102, score-0.424]

48 An important property of the piecewise bound is that its maximum error is bounded and can be driven to zero by increasing the number of pieces. [sent-103, score-0.17]

49 For this reason, this bound always performs better than other existing bounds, such as Jaakola’s bound [12], given that the number of pieces is chosen appropriately. [sent-106, score-0.247]

50 Finally, the cumulative logit likeilhood for ordinal observations depends on σ(x) and its expectation can be bounded using piecewise bounds in a similar way. [sent-107, score-0.387]

51 For the multinomial logit distribution, we can use the bounds proposed by [3] and [4], both leading to concave LVBs. [sent-108, score-0.487]

52 The ﬁrst bound takes the form fb (y, m, V) = yT m − lse(m + v/2) with y represented using a 1-of-K encoding. [sent-109, score-0.358]

53 This function is jointly concave with respect to m and v, which can be shown by noting the fact that the log-sum-exp function is convex. [sent-110, score-0.309]

54 The second bound is the product of sigmoids bound proposed by [4] which bounds the likelihood with product of sigmoids (see Eq. [sent-111, score-0.398]

55 We can also use piecewise linear/quadratic bound to bound each sigmoid. [sent-113, score-0.261]

56 Alternatively, we can use the recently proposed stick-breaking likelihood of [14] which uses piecewise bounds as well. [sent-114, score-0.199]

57 Finally, note that the original log-likelihood may not be concave itself, but if it is such that LJ has a unique solution, then designing a concave variational lower bound will allow us to use concave optimization to efﬁciently maximize the lower bound. [sent-115, score-1.139]

58 5 Existing Algorithms for Variational Inference In this section, we assume that for each output yn there is a corresponding scalar latent function zn . [sent-116, score-0.333]

59 In variational inference, we ﬁnd the approximate Gaussian posterior distribution with mean m and covariance V that maximizes Eq. [sent-118, score-0.36]

60 The simplest approach is to use gradient-based methods for optimization, but this can be problematic since the number of variational parameters is quadratic in N due to the covariance matrix V. [sent-120, score-0.295]

61 The authors of [19] speculate that this may perhaps be the reason behind limited use of Gaussian variational approximations. [sent-121, score-0.213]

62 v m 7 and 8 and equate to zero, using gn := ∂fb (yn , mn , vn )/∂mn and gn := ∂fb (yn , mn , vn )/∂vn . [sent-124, score-0.398]

63 Also, gm and gv are the vectors of these gradients, and diag(gv ) is the matrix with gv as its diagonal. [sent-125, score-0.273]

64 This property can be exploited to reduce the number of variational parameters. [sent-127, score-0.213]

65 Opper and Archambeau [19] (and [18]) propose a reparameterization to reduce the number of parameters to O(N ). [sent-128, score-0.145]

66 At the maximum (but not everywhere), α and λ will be equal to gm and gv respectively. [sent-130, score-0.162]

67 Therefore, instead of solving the ﬁxed-point equations to obtain m and V, we can reparameterize the lower bound with respect to α and λ. [sent-131, score-0.185]

68 6 and after simpliﬁcation using the matrix inversion and determinant lemmas, we get the following new objective function (for a detailed derivation, see [18]), N 1 2 − log(|Bλ ||diag(λ)|) + Tr(B−1 Σ) − αT Σα + λ fb (yn , mn , Vnn ), n=1 4 (11) with Bλ = diag(λ)−1 + Σ. [sent-134, score-0.463]

69 The new lower bound involves vectors of size N , reducing the number of variational parameters to O(N ). [sent-137, score-0.36]

70 The problem with this reparameterization is that the new lower bound is no longer concave, even though it has a unique maximum. [sent-138, score-0.292]

71 We substitute the reparameterization V = (Σ−1 + λ)−1 to get a new function f (λ) = 1 [− log(1 + Σλ) − (1 + Σλ)−1 ]/2. [sent-142, score-0.233]

72 Clearly, this derivative is negative for λ < 1/Σ and non-negative otherwise, making the function neither concave nor convex. [sent-144, score-0.241]

73 With the reparameterization, we loose concavity and therefore the algorithm may have slow convergence. [sent-146, score-0.152]

74 6 Fast Convergent Variational Inference using Coordinate Ascent We now derive an algorithm that reduces the number of variational parameters to 2N while maintaining concavity. [sent-148, score-0.213]

75 14 at the solution, where g22 is the gradient of fb with respect to v22 . [sent-167, score-0.305]

76 v 0 = k22 − Ω22 + 2g22 (14) v 0 = k22 + 1/v22 − Ω22 + 2g22 (15) f (v) = log(v) − (Ω22 − k22 )v + 2fb (y2 , m22 , v) (16) The function f (v) is a strictly concave function and can be optimized by iterating the following v update: v22 ← 1/(Ω22 − k22 − 2g22 ). [sent-174, score-0.271]

77 and denote the old old old values by k22 and v22 . [sent-184, score-0.687]

78 old old K−1 k12 = −(k22 − k22 )vold = −vold /v22 12 12 11 K−1 = Vold − 11 11 K−1 k12 kT K−1 12 11 11 old k22 − k22 old = Vold − vold (vold )T /v22 11 12 12 (17) (18) Note that both K−1 and k12 do not change after the ﬁxed point iteration. [sent-194, score-1.135]

79 This is indeed the case for us since each ﬁxed point iteration solves a concave problem of the form given by Eq. [sent-210, score-0.241]

80 We evaluate the performance of our fast variational inference algorithm against existing inference methods for 1 http://www. [sent-225, score-0.425]

81 For binary classiﬁcation, we use the UCI ionosphere data (with 351 data examples containing 34 features). [sent-254, score-0.134]

82 We consider GP classiﬁcation using the Bernoulli logit likelihood, for which we use the piecewise bound of [16] with 20 pieces. [sent-257, score-0.317]

83 In all cases, due to non-concavity, the optimization of the Opper and Archambeau reparameterization (black curve with squares) convergence slowly, passing through ﬂat regions of the objective and requiring a large number of computations to reach convergence. [sent-268, score-0.19]

84 We show results comparing the lower bound vs the number of ﬂops taken in Figure 1(b), for four hyperparameter settings {log(s), log(σ)}. [sent-278, score-0.18]

85 5) 20K 40K 60K 80K 100K 0 10K Mega−flops Mega−Flops (a) Ionosphere data 20K 30K 40K 50K Mega−flops (b) Forensic glass data Figure 1: Convergence results for (a) the binary classiﬁcation on the ionosphere data set and (b) the multi-class classiﬁcation on the glass dataset. [sent-296, score-0.244]

86 We plot the negative of the lower bound vs the number of ﬂops. [sent-297, score-0.147]

87 where both methods reach the same lower bound value, but the existing approach converging much slower, with our algorithm always converged within 20 iterations. [sent-300, score-0.181]

88 8 Discussion In this paper we have presented a new variational inference algorithm for non-conjugate GP regression. [sent-301, score-0.287]

89 We derived a concave variational lower bound to the log marginal likelihood, and used concavity to develop an efﬁcient optimization algorithm. [sent-302, score-0.829]

90 Our algorithm uses an efﬁcient approach where we update the marginals of the posterior and then do a rank one update of the covariance matrix. [sent-306, score-0.273]

91 Both EP and our algorithm update posterior marginals, followed by a rank-one update of the covariance. [sent-312, score-0.191]

92 The derivation of our algorithm is based on the observation that the posterior covariance has a special structure, and does not directly use the concavity of the lower bound. [sent-315, score-0.367]

93 An alternate derivation based on the Fenchel duality exists and shows that the ﬁxed-point iterations compute dual variables which are related to the gradients of fb . [sent-316, score-0.391]

94 Efﬁcient bounds for the softmax and applications to approximate inference in hybrid models. [sent-341, score-0.123]

95 Concave Gaussian variational approximations for inference in largescale Bayesian linear models. [sent-351, score-0.287]

96 Data augmentation and MCMC for binary and multiu u nomial logit models. [sent-366, score-0.241]

97 Variational Bayesian multinomial probit regression with Gaussian process priors. [sent-371, score-0.146]

98 A variational approach to Bayesian logistic regression problems and their extensions. [sent-381, score-0.27]

99 A stick-breaking likelihood for categorical data analysis with latent Gaussian models. [sent-394, score-0.187]

100 Data augmentation, frequentist estimation, and the Bayesian analysis of multinomial logit models. [sent-446, score-0.197]

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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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