nips nips2007 nips2007-195 knowledge-graph by maker-knowledge-mining

195 nips-2007-The Generalized FITC Approximation

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Author: Andrew Naish-guzman, Sean Holden

Abstract: We present an efﬁcient generalization of the sparse pseudo-input Gaussian process (SPGP) model developed by Snelson and Ghahramani [1], applying it to binary classiﬁcation problems. By taking advantage of the SPGP prior covariance structure, we derive a numerically stable algorithm with O(N M 2 ) training complexity—asymptotically the same as related sparse methods such as the informative vector machine [2], but which more faithfully represents the posterior. We present experimental results for several benchmark problems showing that in many cases this allows an exceptional degree of sparsity without compromising accuracy. Following [1], we locate pseudo-inputs by gradient ascent on the marginal likelihood, but exhibit occasions when this is likely to fail, for which we suggest alternative solutions.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract We present an efﬁcient generalization of the sparse pseudo-input Gaussian process (SPGP) model developed by Snelson and Ghahramani [1], applying it to binary classiﬁcation problems. [sent-5, score-0.147]

2 By taking advantage of the SPGP prior covariance structure, we derive a numerically stable algorithm with O(N M 2 ) training complexity—asymptotically the same as related sparse methods such as the informative vector machine [2], but which more faithfully represents the posterior. [sent-6, score-0.338]

3 We present experimental results for several benchmark problems showing that in many cases this allows an exceptional degree of sparsity without compromising accuracy. [sent-7, score-0.063]

4 Following [1], we locate pseudo-inputs by gradient ascent on the marginal likelihood, but exhibit occasions when this is likely to fail, for which we suggest alternative solutions. [sent-8, score-0.178]

5 However, there is a computational price to pay for this robust framework: the time for training scales as N 3 for N data points, and the cost of prediction is O(N 2 ) per test case. [sent-11, score-0.035]

6 Qui˜ onero-Candela and Rasmussen [8] demonstrated how many of these n schemes are related through different approximations to the joint prior over training and test points. [sent-13, score-0.082]

7 In this paper we consider the “fully independent training conditional” or FITC approximation, which appeared originally in Snelson and Ghahramani [1] as the sparse pseudo-input GP (SPGP). [sent-14, score-0.156]

8 Restricted to a Gaussian noise model, the FITC approximation is entirely tractable; however, for many problems, the Gaussian assumption is inappropriate. [sent-15, score-0.077]

9 In this paper, we describe an extension for non-Gaussian likelihoods, considering as an example probit noise for binary classiﬁcation. [sent-16, score-0.079]

10 This is not only a common problem, but our results bear out the intuition that sparse methods are wellsuited: many data sets enjoy the property that class label does not ﬂuctuate rapidly in the input space, often allowing large regions to be summarized with very few inducing inputs. [sent-17, score-0.256]

11 The informative vector machine (IVM) of Lawrence et al. [sent-19, score-0.044]

12 [2] is another sparse GP method that has been extended to non-Gaussian noise models. [sent-20, score-0.152]

13 It is a subset of data method in which the active set 1 is grown incrementally from the training data using a fast information gain heuristic to ﬁnd at each stage the optimal inclusion. [sent-21, score-0.159]

14 When a threshold number of points have been added, the algorithm terminates: only data accumulated into the active set are relevant for prediction; remaining points inﬂuence the model only in the weak sense of guiding previous steps of the algorithm. [sent-22, score-0.124]

15 Our method is an improvement in three regards: ﬁrstly, the FITC approximation makes use of all the data, yielding for the same active set a closer approximation to the posterior distribution. [sent-23, score-0.296]

16 Secondly, unlike the standard IVM approach, we ﬁt a stable posterior at each iteration, providing more accurate marginal likelihood estimates, and derivatives thereof, to allow more reliable model selection. [sent-24, score-0.232]

17 Finally, we argue with experimental justiﬁcation that the ability to locate inducing inputs independently of the training data, as compared with the greedy approach that drives the IVM, can be a great advantage in ﬁnding the sparsest solutions. [sent-25, score-0.332]

18 We apply Bayes’ rule to obtain the posterior distribution over the f , given the observed X and y, which with the assumption of i. [sent-29, score-0.08]

19 Predictions at X⋆ are made by marginalizing over f in the (Gaussian) joint p(f , f⋆ |X, y, X⋆ ). [sent-33, score-0.029]

20 In order to derive the FITC approximation, we follow [8] and introduce a set of M inducing inputs ¯ ¯ X = {¯ 1 , x2 , . [sent-35, score-0.205]

21 In the ﬁnal expression we make the critical approximation by imposing a conditional independence assumption on the joint prior over training and test cases: communication between them must pass through the bottleneck of the inducing inputs. [sent-40, score-0.263]

22 The FITC approximation follows by letting q(f |u, X) = N f ; Kfu K−1 u , diag (Kﬀ − Qﬀ ) , uu (1) K⋆u K−1 u , uu q(f⋆ |u, X⋆ ) = N f⋆ ; diag (K⋆⋆ − Q⋆⋆ ) , (2) . [sent-41, score-0.256]

23 Of interest for predictions is the posterior distribution over the inducuu ing inputs; this is most efﬁciently obtained via Bayes’ rule after inferring the distribution over f . [sent-43, score-0.106]

24 1 Using (1) and marginalizing over the exact prior on u we obtain the approximate prior on f q(f |X) = N f ; Kfu K−1 u , diag (Kﬀ − Qﬀ ) N (u ; 0 , Kuu ) du uu = N (f ; 0 , Qﬀ + diag (Kﬀ − Qﬀ )) . [sent-44, score-0.306]

25 (3) In the original paper, Snelson and Ghahramani placed the pseudo-inputs randomly and learned their locations by non-linear optimization of the marginal likelihood. [sent-45, score-0.05]

26 We have adopted the idea in this paper, but as emphasized in [8], the FITC approximation is applicable regardless of how the inducing 1 We could also infer the posterior over u directly, rather than marginalizing over the inducing inputs as here. [sent-46, score-0.495]

27 Running EP in this setting, each site maintains a belief about the full M ×M covariance, and we obtain a slower O(N M 3 ) algorithm. [sent-47, score-0.184]

28 Furthermore, calculations to evaluate the derivatives of the log marginal likelihood with ¯ respect to inducing inputs xm are signiﬁcantly complicated by their presence in both prior and likelihood. [sent-48, score-0.407]

29 2 inputs are obtained, and other schemes for their initialization could equally well be married with our algorithm. [sent-49, score-0.098]

30 In the case of classiﬁcation, a sigmoidal function assigns class labels yn ∈ {±1} with a probability that increases monotonically with the latent fn . [sent-50, score-0.236]

31 p(yn |fn , β) = σ(yn (fn + β)) = yn (fn +β) N (z ; 0 , 1) dz. [sent-52, score-0.057]

32 (4) −∞ The posterior distribution p(f |X, y) is only tractable for Gaussian likelihoods, hence we must resort to a further approximation, either by generating Monte Carlo samples from it or ﬁtting deterministically a Gaussian approximation. [sent-53, score-0.08]

33 Suppose we have an intractable distribution over f whose unnormalized form factorizes into a product of terms, such as a dense Gaussian prior t0 (f ) and a series of independent likelihoods ˜ {tn (yn |fn )}N . [sent-56, score-0.078]

34 EP constructs the approximate posterior as a product of scaled site functions tn . [sent-57, score-0.322]

35 n=1 For computational tractability, these sites are usually chosen from an exponential family with natural parameters θ, since in this case their product retains the same functional form as its components. [sent-58, score-0.063]

36 If the prior is of 2 this form, its site function is exact: N p(f |y) = N 1 ˜ t0 (f ) tn (yn |fn ) ≈ q(f ; θ) = t0 (f ) zn tn (fn ; θn ), Z n=1 n=1 (5) where Z is the marginal likelihood and zn are the scale parameters. [sent-60, score-0.55]

37 A simpler optimization minθn KL q n (fn ; θ \n ) q(fn ; θ) then ﬁts only the parameters θn : this is equivalent to moment matching between the two distributions, with scale zn chosen to match the zeroth-order moments. [sent-63, score-0.092]

38 After each site update, the moments at the remaining sites are liable to change, and several iterations may be required before convergence. [sent-64, score-0.282]

39 In the discussion below we omit the moment calculations for the probit model, since they correspond to those of traditional GP classiﬁcation (for more details, consult [9]). [sent-65, score-0.109]

40 Of greater interest is how the mean and covariance structure of the approximate posterior is preserved. [sent-66, score-0.143]

41 Examining the form of the prior (3), we see the covariance consists of a diagonal component D0 and a rank-M term P0 M0 PT , 0 where P0 = Kfu and M0 = K−1 (zero subscripts refer to these initial values; the matrices are uu updated during the course of the EP iterations). [sent-67, score-0.185]

42 EP requires efﬁcient operations for marginalization to obtain p(fn ), and for updating the posterior distribution after reﬁning a site, as well as for refreshing the posterior to avoid loss of numerical precision. [sent-72, score-0.16]

43 Decomposing M = RT R into its Cholesky factor,2 we represent the posterior covariance A and mean h by A = D + PRT RPT , h = ν + Pγ, 2 Care must be taken that the factors share the correct orientation. [sent-73, score-0.143]

44 When our environment offers only upper ` ´ Cholesky factors RT R, the initialization of R0 = chol K−1 can be achieved without computing the explicit uu inverse via the following matrix rotations: “ ` ´T ” R0 := rot180 chol rot180 (Kuu ) \ I . [sent-74, score-0.275]

45 Writing pT = P(n,·) and dn = Dnn , n obtaining marginals in O(M 2 ). [sent-76, score-0.107]

46 hn = νn + pT γ, n Ann = dn + Rpn Now consider a change in the precision at site n by πn . [sent-77, score-0.356]

47 The new covariance Anew is obtained by inverting the sum of the old precision matrix and the change in precision. [sent-79, score-0.094]

48 It is necessary to refresh the covariance and mean every complete EP cycle to avoid loss of precision. [sent-82, score-0.108]

49 −1 Dnew = (I + D0 Π) D0 −1 (O(N )), Pnew = (I + D0 Π) −1 Rnew = rot180 chol rot180 I + R0 PT Π (I + D0 Π) 0 P0 RT 0 (O(N M )), P0 T R0 O(N M 2 ) , where Rnew is obtained being careful to ensure the orientations of the factorizations are not mixed. [sent-83, score-0.086]

50 Reviewing the algorithm above, we see that EP costs are dominated by the O(M 2 ) Cholesky downdate at each site inclusion. [sent-85, score-0.248]

51 Initially, Bayes’ theorem is used to ﬁnd the posterior distribution over u from the inferred posterior over f : p(u|f ) ∝ p(f |u)p(u) = N (u | R−1 c, R−1 CR−T ), 0 0 0 where c = CR0 PT D−1 f 0 0 and C−1 = I + R0 PT D−1 P0 RT . [sent-89, score-0.16]

52 Note that on rare occasions, loss of precision can cause a downdate to result in a non-positive deﬁnite covariance matrix. [sent-91, score-0.158]

53 If this occurs, we should abort the update and refresh the posterior from scratch. [sent-92, score-0.125]

54 In any case, to improve conditioning, it is recommended to add a small multiple of the identity to the prior M0 . [sent-93, score-0.047]

55 3 4 Let our posterior approximation be q(f |y) = N (f ; h , A). [sent-94, score-0.126]

56 4 Model selection EP provides an estimate of the log evidence by matching the 0th-order moments zn at each inclusion. [sent-99, score-0.097]

57 Of interest for model selection are derivatives of the ¯ marginal likelihood with respect to hyperparameters {ξ, X, β}, respectively the kernel parameters, pseudo-input locations, and noise model parameters. [sent-101, score-0.212]

58 Finally, we need derivatives ∇ξ θ prior and ∇X θ prior . [sent-103, score-0.139]

59 The long-winded details are omitted, but by careful consideration ¯ of the covariance structure, it is again possible to limit the complexity to O(N M 2 ). [sent-104, score-0.063]

60 Since we run EP until convergence, our estimates for the marginal likelihood and its derivatives are accurate, allowing us reliablty to ﬁt a model that maximizes the evidence. [sent-105, score-0.124]

61 This is in contrast to the IVM, in which sites excluded from the active set have parameters clamped to zero, and where those included are not iterated to convergence, such that the necessary ﬁxed point conditions do not hold. [sent-106, score-0.187]

62 A particular problem, suffered also by the similar algorithm in [13], is that derivative calculations must be interleaved with site inclusions, and the latter operation tends to disrupt gradient information gained from the previous step. [sent-107, score-0.304]

63 5 The dimensionality of these classiﬁcation problems ranges from two to sixty, and a the size of the training sets is of the order of 400 to 1000. [sent-110, score-0.035]

64 For crabs and the R¨ tsch sets, we average over ten folds of the data; for the synth problem, Ripley has a already divided the data into training and test partitions. [sent-112, score-0.147]

65 Comparisons are made with the full GP classiﬁer, and the SVM, a widely-used discriminative model which in practice is found to yield relatively sparse solutions; we consider also the IVM, a popular framework for building sparse 4 5 Available from http://www. [sent-113, score-0.242]

66 5 Table 1: Test errors and predictive accuracy (smaller is better) for the GP classiﬁer, the support vector machine, the informative vector machine, and the sparse pseudo-input GP classiﬁer. [sent-123, score-0.165]

67 Results on the R¨ tsch data a for the semi-parametric radial basis function network are omitted for lack of space, but available at the site given in footnote 5. [sent-232, score-0.21]

68 In comparison with that model, SPGP tends to give sparser and more accurate results (with the beneﬁt of a sound Bayesian framework). [sent-233, score-0.031]

69 Identical tests were run for a range of active set sizes on the IVM and SPGP classiﬁer, and we have attempted to present the large body of results in its most comprehensible form: we list only the sparsest competitive solution obtained. [sent-234, score-0.156]

70 This means that using M smaller than shown tends to cause a deterioriation in performance, but not that there is no advantage in increasing the value. [sent-235, score-0.031]

71 After all, as M → N we expect error rates to match those of the full model (at least for the IVM, which uses a subset of the training data). [sent-236, score-0.035]

72 6 However, we believe that in exploring the behaviour of a sparse model, the essential question is: what is the greatest sparsity we can achieve without compromising performance? [sent-237, score-0.184]

73 Small values of M for the FITC approximation were found to give remarkably low error rates, and incremented singly would often give an improved approximation. [sent-239, score-0.046]

74 In contrast, the IVM predictions were no better than random guesses for even moderate M —it usually failed if the active set was smaller than a threshold around N/3, where it was simply discarding too much information—and greater step sizes were required for noticeable improvements in performance. [sent-240, score-0.15]

75 More challenging is the task of discriminating 4s from non-4s in the USPS digit database: the data are 256-dimensional, and there are 7291 training and 2007 test points. [sent-242, score-0.035]

76 421 nats—this can be regarded as a failure to generalize, since it corresponds to labelling all test inputs as “not 4”—but given an active set of 400 it reaches error rates of 1. [sent-251, score-0.194]

77 6 Note that the evidence is a poor metric for choosing M since it tends to increase monotonically as the explicative power of the full GP is restored. [sent-254, score-0.031]

78 6 6 Discussion A sparse approximation closely related to FITC is the “deterministic training conditional” (DTC), whose covariance consists solely of the low-rank term LMLT ; it has appeared elsewhere under the name projected latent variables [13]. [sent-255, score-0.292]

79 In generative terms, DTC ﬁrst obtains a posterior process by conditioning on the inducing inputs; observations y are then drawn as noisy samples of the mean of this process. [sent-256, score-0.241]

80 FITC is similar, but the draws are noisy samples from the posterior process itself—hence, while the noise component for DTC is a constant corruption σ 2 , for FITC it grows away from the inducing inputs to Knn +σ 2 . [sent-257, score-0.342]

81 This difﬁculty would not be signiﬁcant if alternative heuristics for building the active set greedily were effective. [sent-262, score-0.124]

82 We hypothesize however that the most informative vectors in the greedy sense of the IVM tend to be those which lie close to the decision boundary. [sent-263, score-0.075]

83 We illustrate with a simple example that, provided the optimization is feasible, very sparse solutions may more easily be found if the inducing inputs can be positioned independently of the data. [sent-266, score-0.326]

84 This allows the size of the active set to grow with the complexity of the problem, rather than with N , the number of training points. [sent-267, score-0.159]

85 5) with a small overlap, giving an optimal error rate of around 13% and in loose terms a complexity which requires an active set of size four. [sent-270, score-0.124]

86 By increasing the size of the training set N in increments from 40 to 400, we obtained the learning curves of ﬁgure 1 for the IVM and FITC models: plotted against N is the size of active set required for the error rate to fall below 15%. [sent-271, score-0.159]

87 We therefore show learning curves for the FITC approximation run using the IVM active set and, generously, optimal kernel parameters. [sent-274, score-0.199]

88 However, a sensible compromise where optimization of all pseudo-inputs is computationally infeasible is to run the IVM to obtain an initial active set, but then switch to the FITC approximation and optimize only kernel parameters, or just a small selection of the pseudo-inputs. [sent-276, score-0.199]

89 7 Conclusions We have presented an efﬁcient and numerically stable way of implementing the sparse FITC model in Gaussian processes. [sent-279, score-0.149]

90 By way of example we considered binary classiﬁcation in which extra data points are introduced to form a continuously adaptable active set. [sent-280, score-0.161]

91 We have demonstrated that the locations of these pseudo-inputs can be ﬁt synchronously with parameters of the kernel, and that 7 We have not compared our model with the RVM since that approximation suffers from nonsensical variance estimates away from the data. [sent-281, score-0.046]

92 Rasmussen and Qui˜ onero-Candela [16] show how it can be “healed” through n augmentation, but the resulting model is no longer sparse in the sense of providing O(M 2 ) predictions. [sent-282, score-0.121]

93 7 150 FITC IVM IVM/FITC Size of active set M 2 100 0 50 -2 8 4 0 40 200 Size of training set N 360 -2 0 2 Figure 1: Left: learning curves for the toy problem described in the text. [sent-283, score-0.159]

94 Right: contours of posterior probability for FITC in ten CG iterations from a random initialization of pseudo-inputs (black dots). [sent-284, score-0.108]

95 In this case, we suggest resorting to a greedy approach, using a fast heuristic like the IVM to build the active set, but adopting the FITC approximation thereafter. [sent-287, score-0.201]

96 Fast sparse Gaussian process methods: the informative vector machine. [sent-294, score-0.191]

97 Bayesian Gaussian process models: PAC-Bayesian generalisation error bounds and sparse approximations. [sent-311, score-0.147]

98 A unifying view of sparse approximate Gausn sian process regression. [sent-314, score-0.147]

99 Fast forward selection to speed up sparse Gaussian process regression. [sent-335, score-0.147]

100 Variable noise and dimensionality reduction for sparse Gaussian processes. [sent-350, score-0.152]

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Maybe more surprisingly, certain algorithms perform well regardless of the assumed rate for the statistical estimation error. 2 2.1 Approximate Optimization Setup Following [6, 2], we consider a space of input-output pairs (x, y) ∈ X × Y endowed with a probability distribution P (x, y). The conditional distribution P (y|x) represents the unknown relationship between inputs and outputs. The discrepancy between the predicted output y and the real output ˆ y is measured with a loss function ℓ(ˆ, y). Our benchmark is the function f ∗ that minimizes the y expected risk E(f ) = that is, ℓ(f (x), y) dP (x, y) = E [ℓ(f (x), y)], f ∗ (x) = arg min E [ ℓ(ˆ, y)| x]. y y ˆ Although the distribution P (x, y) is unknown, we are given a sample S of n independently drawn training examples (xi , yi ), i = 1 . . . n. We deﬁne the empirical risk En (f ) = 1 n n ℓ(f (xi ), yi ) = En [ℓ(f (x), y)]. i=1 Our ﬁrst learning principle consists in choosing a family F of candidate prediction functions and ﬁnding the function fn = arg minf ∈F En (f ) that minimizes the empirical risk. Well known combinatorial results (e.g., [2]) support this approach provided that the chosen family F is sufﬁciently restrictive. Since the optimal function f ∗ is unlikely to belong to the family F, we also deﬁne ∗ ∗ fF = arg minf ∈F E(f ). For simplicity, we assume that f ∗ , fF and fn are well deﬁned and unique. We can then decompose the excess error as ∗ ∗ E [E(fn ) − E(f ∗ )] = E [E(fF ) − E(f ∗ )] + E [E(fn ) − E(fF )] = Eapp + Eest , (1) where the expectation is taken with respect to the random choice of training set. The approximation error Eapp measures how closely functions in F can approximate the optimal solution f ∗ . The estimation error Eest measures the effect of minimizing the empirical risk En (f ) instead of the expected risk E(f ). The estimation error is determined by the number of training examples and by the capacity of the family of functions [2]. Large families1 of functions have smaller approximation errors but lead to higher estimation errors. This tradeoff has been extensively discussed in the literature [2, 3] and lead to excess error that scale between the inverse and the inverse square root of the number of examples [7, 8]. 2.2 Optimization Error Finding fn by minimizing the empirical risk En (f ) is often a computationally expensive operation. Since the empirical risk En (f ) is already an approximation of the expected risk E(f ), it should not be necessary to carry out this minimization with great accuracy. For instance, we could stop an iterative optimization algorithm long before its convergence. ˜ Let us assume that our minimization algorithm returns an approximate solution fn such that ˜ En (fn ) < En (fn ) + ρ ˜ where ρ ≥ 0 is a predeﬁned tolerance. An additional term Eopt = E E(fn ) − E(fn ) then appears ˜ ) − E(f ∗ ) : in the decomposition of the excess error E = E E(fn E ∗ ∗ ˜ = E [E(fF ) − E(f ∗ )] + E [E(fn ) − E(fF )] + E E(fn ) − E(fn ) = Eapp + Eest + Eopt . (2) We call this additional term optimization error. It reﬂects the impact of the approximate optimization on the generalization performance. Its magnitude is comparable to ρ (see section 3.1.) 1 We often consider nested families of functions of the form Fc = {f ∈ H, Ω(f ) ≤ c}. Then, for each value of c, function fn is obtained by minimizing the regularized empirical risk En (f ) + λΩ(f ) for a suitable choice of the Lagrange coefﬁcient λ. We can then control the estimation-approximation tradeoff by choosing λ instead of c. 2.3 The Approximation–Estimation–Optimization Tradeoff This decomposition leads to a more complicated compromise. It involves three variables and two constraints. The constraints are the maximal number of available training example and the maximal computation time. The variables are the size of the family of functions F, the optimization accuracy ρ, and the number of examples n. This is formalized by the following optimization problem. min E = Eapp + Eest + Eopt F ,ρ,n n ≤ nmax T (F, ρ, n) ≤ Tmax subject to (3) The number n of training examples is a variable because we could choose to use only a subset of the available training examples in order to complete the optimization within the alloted time. This happens often in practice. Table 1 summarizes the typical evolution of the quantities of interest with the three variables F, n, and ρ increase. Table 1: Typical variations when F, n, and ρ increase. F Eapp Eest Eopt T (approximation error) (estimation error) (optimization error) (computation time) n ρ ց ր ··· ր ց ··· ր ր ց The solution of the optimization program (3) depends critically of which budget constraint is active: constraint n < nmax on the number of examples, or constraint T < Tmax on the training time. • We speak of small-scale learning problem when (3) is constrained by the maximal number of examples nmax . Since the computing time is not limited, we can reduce the optimization error Eopt to insigniﬁcant levels by choosing ρ arbitrarily small. The excess error is then dominated by the approximation and estimation errors, Eapp and Eest . Taking n = nmax , we recover the approximation-estimation tradeoff that is the object of abundant literature. • We speak of large-scale learning problem when (3) is constrained by the maximal computing time Tmax . Approximate optimization, that is choosing ρ > 0, possibly can achieve better generalization because more training examples can be processed during the allowed time. The speciﬁcs depend on the computational properties of the chosen optimization algorithm through the expression of the computing time T (F, ρ, n). 3 The Asymptotics of Large-scale Learning In the previous section, we have extended the classical approximation-estimation tradeoff by taking into account the optimization error. We have given an objective criterion to distiguish small-scale and large-scale learning problems. In the small-scale case, we recover the classical tradeoff between approximation and estimation. The large-scale case is substantially different because it involves the computational complexity of the learning algorithm. In order to clarify the large-scale learning tradeoff with sufﬁcient generality, this section makes several simpliﬁcations: • We are studying upper bounds of the approximation, estimation, and optimization errors (2). It is often accepted that these upper bounds give a realistic idea of the actual convergence rates [9, 10, 11, 12]. Another way to ﬁnd comfort in this approach is to say that we study guaranteed convergence rates instead of the possibly pathological special cases. • We are studying the asymptotic properties of the tradeoff when the problem size increases. Instead of carefully balancing the three terms, we write E = O(Eapp ) + O(Eest ) + O(Eopt ) and only need to ensure that the three terms decrease with the same asymptotic rate. • We are considering a ﬁxed family of functions F and therefore avoid taking into account the approximation error Eapp . This part of the tradeoff covers a wide spectrum of practical realities such as choosing models and choosing features. In the context of this work, we do not believe we can meaningfully address this without discussing, for instance, the thorny issue of feature selection. Instead we focus on the choice of optimization algorithm. • Finally, in order to keep this paper short, we consider that the family of functions F is linearly parametrized by a vector w ∈ Rd . We also assume that x, y and w are bounded, ensuring that there is a constant B such that 0 ≤ ℓ(fw (x), y) ≤ B and ℓ(·, y) is Lipschitz. We ﬁrst explain how the uniform convergence bounds provide convergence rates that take the optimization error into account. Then we discuss and compare the asymptotic learning properties of several optimization algorithms. 3.1 Convergence of the Estimation and Optimization Errors The optimization error Eopt depends directly on the optimization accuracy ρ. However, the accuracy ˜ ρ involves the empirical quantity En (fn ) − En (fn ), whereas the optimization error Eopt involves ˜ its expected counterpart E(fn ) − E(fn ). This section discusses the impact on the optimization error Eopt and of the optimization accuracy ρ on generalization bounds that leverage the uniform convergence concepts pioneered by Vapnik and Chervonenkis (e.g., [2].) In this discussion, we use the letter c to refer to any positive constant. Multiple occurences of the letter c do not necessarily imply that the constants have identical values. 3.1.1 Simple Uniform Convergence Bounds Recall that we assume that F is linearly parametrized by w ∈ Rd . Elementary uniform convergence results then state that r » – E sup |E(f ) − En (f )| ≤ c f ∈F d , n where the expectation is taken with respect to the random choice of the training set.2 This result immediately provides a bound on the estimation error: Eest = ≤ ´ ` ` ∗ ´ ∗ ∗ ´˜ E(fn ) − En (fn ) + En (fn ) − En (fF ) + En (fF ) − E(fF ) r » – d . 2 E sup |E(f ) − En (f )| ≤ c n f ∈F E ˆ` This same result also provides a combined bound for the estimation and optimization errors: Eest + Eopt = + ≤ ˆ ˜ ˆ ˜ ˜ ˜ ˜ E E(fn ) − En (fn ) + E En (fn ) − En (fn ) ∗ ∗ ∗ E [En (fn ) − En (fF )] + E [En (fF ) − E(fF )] r r r ! d d d c +ρ+0+c = c ρ+ . n n n Unfortunately, this convergence rate is known to be pessimistic in many important cases. More sophisticated bounds are required. 3.1.2 Faster Rates in the Realizable Case When the loss functions ℓ(ˆ, y) is positive, with probability 1 − e−τ for any τ > 0, relative uniform y convergence bounds state that r E(f ) − En (f ) d n τ p sup log + . ≤c n d n f ∈F E(f ) This result is very useful because it provides faster convergence rates O(log n/n) in the realizable case, that is when ℓ(fn (xi ), yi ) = 0 for all training examples (xi , yi ). We have then En (fn ) = 0, ˜ En (fn ) ≤ ρ, and we can write r q n d τ ˜ ˜ E(fn ) − ρ ≤ c E(fn ) log + . n d n q 2 d Although the original Vapnik-Chervonenkis bounds have the form c n log n , the logarithmic term can d be eliminated using the “chaining” technique (e.g., [10].) Viewing this as a second degree polynomial inequality in variable ˜ E(fn ), we obtain „ « d n τ ˜ E(fn ) ≤ c ρ + log + . n d n Integrating this inequality using a standard technique (see, e.g., [13]), we obtain a better convergence rate of the combined estimation and optimization error: „ « h i h i d n ∗ ˜ ˜ Eest + Eopt = E E(fn ) − E(fF ) ≤ E E(fn ) = c ρ + log . n d 3.1.3 Fast Rate Bounds Many authors (e.g., [10, 4, 12]) obtain fast statistical estimation rates in more general conditions. These bounds have the general form α n 1 d log for ≤ α ≤ 1. n d 2 This result holds when one can establish the following variance condition: Eapp + Eest ≤ c ∀f ∈ F Eapp + ∗ ℓ(f (X), Y ) − ℓ(fF (X), Y ) E 2 ≤ c ∗ E(f ) − E(fF ) (4) 1 2− α . (5) The convergence rate of (4) is described by the exponent α which is determined by the quality of the variance bound (5). Works on fast statistical estimation identify two main ways to establish such a variance condition. • Exploiting the strict convexity of certain loss functions [12, theorem 12]. For instance, Lee et al. [14] establish a O(log n/n) rate using the squared loss ℓ(ˆ, y) = (ˆ − y)2 . y y • Making assumptions on the data distribution. In the case of pattern recognition problems, for instance, the “Tsybakov condition” indicates how cleanly the posterior distributions P (y|x) cross near the optimal decision boundary [11, 12]. The realizable case discussed in section 3.1.2 can be viewed as an extreme case of this. Despite their much greater complexity, fast rate estimation results can accomodate the optimization accuracy ρ using essentially the methods illustrated in sections 3.1.1 and 3.1.2. We then obtain a bound of the form α d n ˜ E = Eapp + Eest + Eopt = E E(fn ) − E(f ∗ ) ≤ c Eapp + log +ρ . (6) n d For instance, a general result with α = 1 is provided by Massart [13, theorem 4.2]. Combining this result with standard bounds on the complexity of classes of linear functions (e.g., [10]) yields the following result: d n ˜ E = Eapp + Eest + Eopt = E E(fn ) − E(f ∗ ) ≤ c Eapp + log + ρ . (7) n d See also [15, 4] for more bounds taking into account the optimization accuracy. 3.2 Gradient Optimization Algorithms We now discuss and compare the asymptotic learning properties of four gradient optimization algo∗ rithms. Recall that the family of function F is linearly parametrized by w ∈ Rd . Let wF and wn ∗ correspond to the functions fF and fn deﬁned in section 2.1. In this section, we assume that the functions w → ℓ(fw (x), y) are convex and twice differentiable with continuous second derivatives. Convexity ensures that the empirical const function C(w) = En (fw ) has a single minimum. Two matrices play an important role in the analysis: the Hessian matrix H and the gradient covariance matrix G, both measured at the empirical optimum wn . ∂ 2 ℓ(fwn (x), y) ∂2C (wn ) = En , 2 ∂w ∂w2 H = G = En ∂ℓ(fwn (x), y) ∂w ∂ℓ(fwn (x), y) ∂w (8) ′ . (9) The relation between these two matrices depends on the chosen loss function. In order to summarize them, we assume that there are constants λmax ≥ λmin > 0 and ν > 0 such that, for any η > 0, we can choose the number of examples n large enough to ensure that the following assertion is true with probability greater than 1 − η : tr(G H −1 ) ≤ ν EigenSpectrum(H) ⊂ [ λmin , λmax ] and (10) The condition number κ = λmax /λmin is a good indicator of the difﬁculty of the optimization [16]. The condition λmin > 0 avoids complications with stochastic gradient algorithms. Note that this condition only implies strict convexity around the optimum. For instance, consider the loss function ℓ is obtained by smoothing the well known hinge loss ℓ(z, y) = max{0, 1 − yz} in a small neighborhood of its non-differentiable points. Function C(w) is then piecewise linear with smoothed edges and vertices. It is not strictly convex. However its minimum is likely to be on a smoothed vertex with a non singular Hessian. When we have strict convexity, the argument of [12, theorem 12] yields fast estimation rates α ≈ 1 in (4) and (6). This is not necessarily the case here. The four algorithm considered in this paper use information about the gradient of the cost function to iteratively update their current estimate w(t) of the parameter vector. • Gradient Descent (GD) iterates w(t + 1) = w(t) − η ∂C 1 (w(t)) = w(t) − η ∂w n n i=1 ∂ ℓ fw(t) (xi ), yi ∂w where η > 0 is a small enough gain. GD is an algorithm with linear convergence [16]. When η = 1/λmax , this algorithm requires O(κ log(1/ρ)) iterations to reach accuracy ρ. The exact number of iterations depends on the choice of the initial parameter vector. • Second Order Gradient Descent (2GD) iterates n w(t + 1) = w(t) − H −1 1 ∂ ∂C (w(t)) = w(t) − H −1 ℓ fw(t) (xi ), yi ∂w n ∂w i=1 where matrix H −1 is the inverse of the Hessian matrix (8). This is more favorable than Newton’s algorithm because we do not evaluate the local Hessian at each iteration but simply assume that we know in advance the Hessian at the optimum. 2GD is a superlinear optimization algorithm with quadratic convergence [16]. When the cost is quadratic, a single iteration is sufﬁcient. In the general case, O(log log(1/ρ)) iterations are required to reach accuracy ρ. • Stochastic Gradient Descent (SGD) picks a random training example (xt , yt ) at each iteration and updates the parameter w on the basis of this example only, w(t + 1) = w(t) − η ∂ ℓ fw(t) (xt ), yt . t ∂w Murata [17, section 2.2], characterizes the mean ES [w(t)] and variance VarS [w(t)] with respect to the distribution implied by the random examples drawn from the training set S at each iteration. Applying this result to the discrete training set distribution for η = 1/λmin , we have δw(t)2 = O(1/t) where δw(t) is a shorthand notation for w(t) − wn . We can then write ES [ C(w(t)) − inf C ] = = ≤ ˆ ` ´˜ ` ´ ES tr H δw(t) δw(t)′ + o 1 t ` ´ ` ´ tr H ES [δw(t)] ES [δw(t)]′ + H VarS [w(t)] + o 1 t `1´ `1´ 2 tr(GH) + o t ≤ νκ + o t . t t (11) Therefore the SGD algorithm reaches accuracy ρ after less than νκ2/ρ + o(1/ρ) iterations on average. The SGD convergence is essentially limited by the stochastic noise induced by the random choice of one example at each iteration. Neither the initial value of the parameter vector w nor the total number of examples n appear in the dominant term of this bound! When the training set is large, one could reach the desired accuracy ρ measured on the whole training set without even visiting all the training examples. This is in fact a kind of generalization bound. Table 2: Asymptotic results for gradient algorithms (with probability 1). Compare the second last column (time to optimize) with the last column (time to reach the excess test error ǫ). Legend: n number of examples; d parameter dimension; κ, ν see equation (10). Algorithm Cost of one iteration GD O(nd) 2GD O d2 + nd SGD O(d) 2SGD O d2 Iterations to reach ρ O κ log 1 ρ 1 O log log ρ νκ2 ρ ν ρ O ndκ log O 1 ρ +o +o Time to reach accuracy ρ 1 ρ 1 ρ d2 + nd log log O O Time to reach E ≤ c (Eapp + ε) d2 κ ε1/α O 1 ρ O d2 ε1/α dνκ2 ρ d2 ν ρ log2 1 ε log 1 log log 1 ε ε O O d ν κ2 ε d2 ν ε • Second Order Stochastic Gradient Descent (2SGD) replaces the gain η by the inverse of the Hessian matrix H: w(t + 1) = w(t) − 1 −1 ∂ H ℓ fw(t) (xt ), yt . t ∂w Unlike standard gradient algorithms, using the second order information does not change the inﬂuence of ρ on the convergence rate but improves the constants. Using again [17, theorem 4], accuracy ρ is reached after ν/ρ + o(1/ρ) iterations. For each of the four gradient algorithms, the ﬁrst three columns of table 2 report the time for a single iteration, the number of iterations needed to reach a predeﬁned accuracy ρ, and their product, the time needed to reach accuracy ρ. These asymptotic results are valid with probability 1, since the probability of their complement is smaller than η for any η > 0. The fourth column bounds the time necessary to reduce the excess error E below c (Eapp +ε) where c `d ´α is the constant from (6). This is computed by observing that choosing ρ ∼ n log n in (6) achieves d the fastest rate for ε, with minimal computation time. We can then use the asymptotic equivalences d ρ ∼ ε and n ∼ ε1/α log 1 . Setting the fourth column expressions to Tmax and solving for ǫ yields ε the best excess error achieved by each algorithm within the limited time Tmax . This provides the asymptotic solution of the Estimation–Optimization tradeoff (3) for large scale problems satisfying our assumptions. These results clearly show that the generalization performance of large-scale learning systems depends on both the statistical properties of the estimation procedure and the computational properties of the chosen optimization algorithm. Their combination leads to surprising consequences: • The SGD and 2SGD results do not depend on the estimation rate α. When the estimation rate is poor, there is less need to optimize accurately. That leaves time to process more examples. A potentially more useful interpretation leverages the fact that (11) is already a kind of generalization bound: its fast rate trumps the slower rate assumed for the estimation error. • Second order algorithms bring little asymptotical improvements in ε. Although the superlinear 2GD algorithm improves the logarithmic term, all four algorithms are dominated by the polynomial term in (1/ε). However, there are important variations in the inﬂuence of the constants d, κ and ν. These constants are very important in practice. • Stochastic algorithms (SGD, 2SGD) yield the best generalization performance despite being the worst optimization algorithms. This had been described before [18] and observed in experiments. In contrast, since the optimization error Eopt of small-scale learning systems can be reduced to insigniﬁcant levels, their generalization performance is solely determined by the statistical properties of their estimation procedure. 4 Conclusion Taking in account budget constraints on both the number of examples and the computation time, we ﬁnd qualitative differences between the generalization performance of small-scale learning systems and large-scale learning systems. The generalization properties of large-scale learning systems depend on both the statistical properties of the estimation procedure and the computational properties of the optimization algorithm. We illustrate this fact with some asymptotic results on gradient algorithms. Considerable reﬁnements of this framework can be expected. Extending the analysis to regularized risk formulations would make results on the complexity of primal and dual optimization algorithms [19, 20] directly exploitable. The choice of surrogate loss function [7, 12] could also have a non-trivial impact in the large-scale case. Acknowledgments Part of this work was funded by NSF grant CCR-0325463. References [1] Leslie G. Valiant. A theory of learnable. Proc. of the 1984 STOC, pages 436–445, 1984. [2] Vladimir N. Vapnik. Estimation of Dependences Based on Empirical Data. Springer Series in Statistics. Springer-Verlag, Berlin, 1982. [3] St´ phane Boucheron, Olivier Bousquet, and G´ bor Lugosi. Theory of classiﬁcation: a survey of recent e a advances. ESAIM: Probability and Statistics, 9:323–375, 2005. [4] Peter L. Bartlett and Shahar Mendelson. Empirical minimization. Probability Theory and Related Fields, 135(3):311–334, 2006. [5] J. Stephen Judd. On the complexity of loading shallow neural networks. Journal of Complexity, 4(3):177– 192, 1988. [6] Richard O. Duda and Peter E. Hart. Pattern Classiﬁcation And Scene Analysis. Wiley and Son, 1973. [7] Tong Zhang. Statistical behavior and consistency of classiﬁcation methods based on convex risk minimization. The Annals of Statistics, 32:56–85, 2004. [8] Clint Scovel and Ingo Steinwart. Fast rates for support vector machines. In Peter Auer and Ron Meir, editors, Proceedings of the 18th Conference on Learning Theory (COLT 2005), volume 3559 of Lecture Notes in Computer Science, pages 279–294, Bertinoro, Italy, June 2005. Springer-Verlag. [9] Vladimir N. Vapnik, Esther Levin, and Yann LeCun. Measuring the VC-dimension of a learning machine. Neural Computation, 6(5):851–876, 1994. [10] Olivier Bousquet. Concentration Inequalities and Empirical Processes Theory Applied to the Analysis of Learning Algorithms. PhD thesis, Ecole Polytechnique, 2002. [11] Alexandre B. Tsybakov. Optimal aggregation of classiﬁers in statistical learning. Annals of Statististics, 32(1), 2004. [12] Peter L. Bartlett, Michael I. Jordan, and Jon D. McAuliffe. Convexity, classiﬁcation and risk bounds. Journal of the American Statistical Association, 101(473):138–156, March 2006. [13] Pascal Massart. Some applications of concentration inequalities to statistics. Annales de la Facult´ des e Sciences de Toulouse, series 6, 9(2):245–303, 2000. [14] Wee S. Lee, Peter L. Bartlett, and Robert C. Williamson. The importance of convexity in learning with squared loss. IEEE Transactions on Information Theory, 44(5):1974–1980, 1998. [15] Shahar Mendelson. A few notes on statistical learning theory. In Shahar Mendelson and Alexander J. Smola, editors, Advanced Lectures in Machine Learning, volume 2600 of Lecture Notes in Computer Science, pages 1–40. Springer-Verlag, Berlin, 2003. [16] John E. Dennis, Jr. and Robert B. Schnabel. Numerical Methods For Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983. [17] Noboru Murata. A statistical study of on-line learning. In David Saad, editor, Online Learning and Neural Networks. Cambridge University Press, Cambridge, UK, 1998. [18] L´ on Bottou and Yann Le Cun. Large scale online learning. In Sebastian Thrun, Lawrence K. Saul, e and Bernhard Sch¨ lkopf, editors, Advances in Neural Information Processing Systems 16. MIT Press, o Cambridge, MA, 2004. [19] Thorsten Joachims. Training linear SVMs in linear time. In Proceedings of KDD’06, Philadelphia, PA, USA, August 20-23 2006. ACM. [20] Don Hush, Patrick Kelly, Clint Scovel, and Ingo Steinwart. QP algorithms with guaranteed accuracy and run time for support vector machines. Journal of Machine Learning Research, 7:733–769, 2006.

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