# AMS 2000 subject classification. Primary 62G08, 62G20; secondary 62G99

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5 2 Adaptive group Lasso i oparametric additive models We describe a two-step approach that uses the group Lasso for variable selectio based o a splie represetatio of each compoet i additive models. I the first step, we use the stadard group Lasso to achieve a iitial reductio of the dimesio i the model ad obtai a iitial estimator of the oparametric compoets. I the secod step, we use the adaptive group Lasso to achieve cosistet selectio. Suppose that each X j takes values i [a, b] where a < b are fiite umbers. To esure uique idetificatio of the f j s, we assume that Ef j (X j = 0, 1 j p. Let a = ξ 0 < ξ 1 < < ξ K < ξ K+1 = b be a partitio of [a, b] ito K subitervals I Kt = [ξ t, ξ t+1, t = 0,..., K 1 ad I KK = [ξ K, ξ K+1 ], where K K = v with 0 < v < 0.5 is a positive iteger such that max 1 k K+1 ξ k ξ k 1 = O( v. Let S be the space of polyomial splies of degree l 1 cosistig of fuctios s satisfyig: (i the restrictio of s to I Kt is a polyomial of degree l for 1 t K; (ii for l 2 ad 0 l l 2, s is l times cotiuously differetiable o [a, b]. This defiitio is phrased after Stoe (1985, which is a descriptive versio of Schumaker (1981, page 108, Defiitio 4.1. There exists a ormalized B-splie basis {φ k, 1 k m } for S, where m (Schumaker Thus for ay f j S, we ca write K + l m f j (x = β jk φ k (x, 1 j p. (2 k=1 Uder suitable smoothess assumptios, the f j s ca be well approximated by fuctios i S. Accordigly, the variable selectio method described i this paper is based o the represetatio (2. Let a 2 ( m j=1 a j 2 1/2 deote the l2 orm of ay vector a IR m. Let β j = (β j1,..., β jm ad β = (β 1,..., β p. Let w = (w 1,..., w p be a give vector of weights, where 0 w j, 1 j p. Cosider the pealized least squares criterio L (µ, β = i=1 [ Y i µ p m ] 2 p β jk φ k (X ij + λ w j β j 2, (3 j=1 k=1 j=1 where λ is a pealty parameter. We study the estimators that miimize L (µ, β subject to the costraits m β jk φ k (X ij = 0, 1 j p. (4 i=1 k=1 5

6 These ceterig costraits are sample aalogs of the idetifyig restrictio Ef j (X j = 0, 1 j p. We ca covert (3-(4 to a ucostraied optimizatio problem by ceterig the respose ad the basis fuctios. Let φ jk = 1 φ k (X ij, ψ jk (x = φ k (x φ jk. (5 i=1 For simplicity ad without causig cofusio, we simply write ψ k (x = ψ jk (x. Defie Z ij = ( ψ 1 (X ij,..., ψ m (X ij. So Z ij cosists of values of the (cetered basis fuctios at the ith observatio of the jth covariate. Let Z j = (Z 1j,..., Z j be the m desig matrix correspodig to the jth covariate. The total desig matrix is Z = (Z 1,..., Z p. Let Y = (Y 1 Y,..., Y Y. With this otatio, we ca write L (β ; λ = Y Zβ λ p w j β j 2. (6 Here we have dropped µ i the argumet of L. With the ceterig, µ = Y. The miimizig (3 subject to (4 is equivalet to miimizig (6 with respect to β, but the ceterig costraits are ot eeded for (6. We ow describe the two-step approach to compoet selectio i the oparametric additive model (1. Step 1. Compute the group Lasso estimator. Let j=1 L 1 (β, λ 1 = Y Zβ λ 1 p β j 2. This objective fuctio is the special case of (6 that is obtaied by settig w j = 1, 1 j p. The group Lasso estimator is β β (λ 1 = arg mi β L 1 (β ; λ 1. Step 2. Use the group Lasso estimator β to obtai the weights by settig w j = j=1 { βj 1 2, if β j 2 > 0,, if β j 2 = 0. 6

9 Theorem 2 Suppose that (A1 to (A4 hold ad that λ 1 C log(pm for a sufficietly large costat C. The, (i Let Ãf = {j : f j 2 > 0, 1 j p}. There is a costat M 1 > 1 such that, with probability covergig to 1, Ãf M 1 q. (ii If (m log(pm / 0 ad (λ 2 1m / 2 0 as, the all the ozero additive compoets f j, 1 j q, are selected with probability covergig to oe. (iii f ( j f j 2 m log(pm ( 1 ( 1 2 = O p + O p + O + O m 2d where Ã2 = A 1 Ã1. ( 4m λ 2 1 2, j Ã2, Thus uder the coditios of Theorem 2, the group Lasso selects all the ozero additive compoets with high probability. Part (iii of the theorem gives the rate of covergece of the group Lasso estimator of the oparametric compoets. For ay two sequeces {a, b, = 1, 2,..., }, we write a b if there are costats 0 < c 1 < c 2 < such that c 1 a /b c 2 for all sufficietly large. We ow state a useful corollary of Theorem 2. Corollary 1 Suppose that (A1 to (A4 hold. If λ 1 log(pm ad m 1/(2d+1, the, (i If 2d/(2d+1 log(p 0 as, the with probability covergig to oe, all the ozero compoets f j, 1 j q, are selected ad the umber of selected compoets is o more tha M 1 q. (ii f j f j 2 2 = O p ( 2d/(2d+1 log(pm, j Ã2. For the λ 1 ad m give i Corollary 1, the umber of zero compoets ca be as large as exp(o( 2d/(2d+1. For example, if each f j has cotiuous secod derivative (d = 2, the it is exp(o( 4/5, which ca be much larger tha. 3.2 Selectio cosistecy of the adaptive group Lasso We ow cosider the properties of the adaptive group Lasso. We first state a geeral result cocerig the selectio cosistecy of the adaptive group Lasso, assumig a iitial cosistet estimator is available. We the apply to the case whe the group Lasso is used as the iitial estimator. We make the followig assumptios. 9

10 (B1 The iitial estimators β j are r -cosistet at zero: ad there exists a costat c b > 0 such that r max j A 0 β j 2 = O P (1, r, where b 1 = mi j A1 β j 2. P(mi j A 1 β j 2 c b b 1 1, (B2 Let q be the umber of ozero compoets ad s = p q be the umber of zero compoets. Suppose that (a (b 1/2 log 1/2 (s m λ 2 r + m + λ 2m 1/4 1/2 λ 2 r m (2d+1/2 = o(1, = o(1. We state coditio (B1 for a geeral iitial estimator, to highlight the poit that the availability of a r -cosistet estimator at zero is crucial for the adaptive group Lasso to be selectio cosistet. I other words, ay iitial estimator satisfyig (B1 will esure that the adaptive group Lasso (based o this iitial estimator is selectio cosistet, provided that certai regularity coditios are satisfied. We ote that it follows immediately from Theorem 1 that the group Lasso estimator satisfies (B1. We will come back to this poit below. For β ( β 1,..., β p ad β (β 1,..., β p, we say β = 0 β if sg 0 ( β j = sg 0 ( β j, 1 j p, where sg 0 ( x = 1 if x > 0 ad = 0 if x = 0. Theorem 3 Suppose that coditios (B1, (B2 ad (A1-(A4 hold. The (i P( β = 0 β 1. (ii q j=1 ( m β j β j = O p ( m ( 1 + O p + O + O m 2d 1 ( 4m 2 λ This theorem is cocered with the selectio ad estimatio properties of the adaptive group Lasso i terms of β. The followig theorem states the results i terms of the estimators of the oparametric compoets. 10

18 7 Appedix: Proofs We first prove the followig lemmas. Deote the cetered versios of S by S 0 j = { f j : f j (x = m k=1 b jk ψ k (x, (β j1,..., β jm IR m where ψ k s are the cetered splie bases defied i (5. }, 1 j p, Lemma 1 Suppose that f F ad Ef(X j = 0. The, uder (A3 ad (A4, there exists a f S 0 j satisfyig f f 2 = O p (m d + m 1/2 1/2. I particular, if we choose m = O( 1/(2d+1, the f f 2 = O p (m d = O p ( d/(2d+1. Proof of Lemma 1. By (A4, for f F, there is a f S such that f f 2 = O(m d. Let f = f 1 i=1 f (X ij. The f S 0 j ad f f f f + P f, where P is the empirical measure of iid radom variables X 1j,..., X j. Cosider P f = (P P f + P (f f. Here we use the liear fuctioal otatio, i.e., P f = fdp, where P is the probability measure of X 1j. For ay ε > 0, the bracketig umber N [] (ε, S 0 j, L 2 (P of S 0 j satisfies log N [] (ε, S 0 j, L 2 (P c 1 m log(1/ε for some costat c 1 > 0 (She ad Wog 1994, page 597. Thus by the maximal iequality, see e.g. Va der Vaart (1998, page 288, (P P f = O p ( 1/2 m 1/2. By (A4, P (f f C 2 f f 2 = O(m d for some costat C 2 > 0. The lemma follows from the triagle iequality. Lemma 2 Suppose that coditios (A2 ad (A4 hold. Let T jk = 1/2 m 1/2 ψ k (X ij ε i, 1 j p, 1 k m, i=1 ad T = max 1 j p,1 k m T jk. The E(T C 1 1/2 m 1/2 log(pm ( 2C2 m 1 log(pm + 4 log(2pm + C 2 m 1 1/2. 18

20 The lemma follows from (9 ad (13. Deote β A = (β j, j A ad Z A = (Z j, j A. Here β A is a A m 1 vector ad Z A is a A m matrix. Let C A = Z A Z A/. Whe A = {1,..., p}, we simply write C = Z Z/. Let ρ mi (C A ad ρ max (C A be the miimum ad maximum eigevalues of C A, respectively. Lemma 3 Let m = O( γ where 0 < γ < 0.5. Suppose that A is bouded by a fixed costat idepedet of ad p. Let h h m 1. The, uder (A3 ad (A4, with probability covergig to oe, where c 1 ad c 2 are two positive costats. c 1 h ρ mi (C A ρ max (C A c 2 h, Proof of Lemma 3. Without loss of geerality, suppose A = {1,..., k}. The Z A = (Z 1,..., Z q. Let b = (b 1,..., b q, where b j R m. By Lemma 3 of Stoe (1985, Z 1 b Z q b q 2 c 3 ( Z 1 b Z q b q 2 for a certai costat c 3 > 0. By the triagle iequality, Z 1 b Z q b q 2 Z 1 b Z q b q 2. Sice Z A b = Z 1 b Z q b q, the above two iequalities imply that c 3 ( Z 1 b Z q b q 2 Z A b 2 Z 1 b Z q b q 2. Therefore, c 2 3( Z 1 b Z q b q 2 2 Z A b 2 2 2( Z 1 b Z q b q 2 2. (14 Let C j = 1 Z jz j. By Lemma 6.2 of Zhou, She ad Wolf (1998, c 4 h ρ mi (C j ρ max (C j c 5 h, j A. (15 Sice C A = 1 Z A Z A, it follows from (14 that c 2 3 ( b 1 C 1 b b qc q b q b C A b 2 ( b 1C 1 b b qc q b q. 20

21 Therefore, by (15, b 1C 1 b 1 b b qc q b q b 2 2 = b 1C 1 b 1 b b b b qc q b q b q 2 2 b q 2 2 b 2 2 ρ mi (C 1 b b 2 2 c 4 h. + + ρ mi (C q b q 2 2 b 2 2 Similarly, Thus we have The lemma follows. b 1C 1 b 1 b b qc q b q b 2 2 c 2 3c 4 h b C A b b b 2c 5h. c 5 h. Proof of Theorem 1. The proof of parts (i ad (ii essetially follows the proof of Theorem 2.1 of Wei ad Huag (2008. The oly chage that must be made here is that we eed to cosider the approximatio error of the regressio fuctios by splies. Specifically, let ξ = ε + δ, where δ = (δ 1,..., δ with δ i = q j=1 (f 0j(X ij f j (X ij. Sice f 0j f j 2 = O(m d = O( d/(2d+1 for m = 1/(2d+1, we have for some costat C 1 > 0. For ay iteger t, let δ 2 C 1 qm 2d = C 1 q 1/(4d+2, χ t = max A =t max U Ak 2 =1,1 k t ξ V A (s V A (s 2 ad χ t = max A =t max U Ak 2 =1,1 k t ε V A (s V A (s 2 where V A (S A = ξ (Z A (Z A Z A 1 SA (I P A Xβ for N(A = q 1 = m 0, S A = (S A 1,, S A m, S Ak = λ d Ak U Ak ad U Ak 2 = 1. For a sufficietly large costat C 2 > 0, defie Ω t0 = {(Z, ε : x t σc 2 ((t 1m log(pm, t t 0 }, ad Ω t 0 = {(Z, ε : x t σc 2 (t 1m log(pm, t t 0 }, where t

22 As i the proof of Theorem 2.1 of Wei ad Huag (2008, (Z, ε Ω q Ã1 M 1 q, for a costat M 1 > 1. By the triagle ad Cauchy-Schwarz iequalities, ξ V A (s V A (s 2 = ε V A (s + δ V A (s V A (s 2 ε V A (s V A 2 + δ. (16 I the proof of Theorem 2.1 of Wei ad Huag (2008, it is show that P(Ω p 1+c 0 ( 2p exp 1. (17 p 1+c 0 Sice δ V A (s V A (s 2 δ 2 = C 1 q 1 2(2d+1 ad m = O( 1/(2d+1, we have for all t 0 ad sufficietly large, δ 2 C 1 q 1 2(2d+1 σc2 (t 1m log(p. (18 It follows from (16, (17 ad (18 that P(Ω 0 1. This completes the proof of part (i of Theorem 1. Before provig part (ii, we first prove part (iii of Theorem 1. By the defiitio of β ( β 1,..., β p, p p Y Z β λ 1 β j 2 Y Zβ λ 1 β j 2. (19 j=1 j=1 Let A 2 = {j : β j 2 0 or β j 2 0} ad d 2 = A 2. By part (i, d 2 = O p (q. By (19 ad the defiitio of A 2, Y Z A2 βa λ 1 β j 2 Y Z A2 β A λ 1 β j 2. (20 j A 2 j A 2 Let η = Y Zβ. Write Y Z A2 βa2 = Y Zβ Z A2 ( β A2 β A2 = η Z A2 ( β A2 β A2. 22

23 We have Y Z A2 βa2 2 2 = Z A2 ( β A2 β A η Z A2 ( β A2 β A2 + η η. We ca rewrite (20 as Z A2 ( β A2 β A η Z A2 ( β A2 β A2 λ 1 β j 2 λ 1 β j 2. (21 j A 1 j A 1 Now β j 2 β j 2 A 1 β A1 β A1 2 A 1 β A2 β A2 2. (22 j A 1 j A 1 Let ν = Z A2 ( β A2 β A2. Combiig (20, (21 ad (22 to get ν 2 2 2η ν λ 1 A1 β A2 β A2 2. (23 Let η be the projectio of η to the spa of Z A2, that is, η = Z A2 (Z A 2 Z A2 1 Z A 2 η. By the Cauchy-Schwartz iequality, 2 η ν 2 η 2 ν 2 2 η ν 2 2. (24 From (23 ad (24, we have ν η λ 1 A1 β A2 β A2 2. Let c be the smallest eigevalue of Z A 2 Z A2 /. By Lemma 3 ad part (i, c p m 1. Sice ν 2 2 c β A2 β A2 2 2 ad 2ab a 2 + b 2, It follows that c β A2 β A η (2λ 1 A c 2 c β A2 β A β A2 β A η λ2 1 A 1. (25 c 2 c 2 Let f 0 (X i = p j=1 f 0j(X ij ad f 0A (X i = j A f 0j(X ij. Write η i = Y i µ f 0 (X i + (µ Y + f 0 (X i j A 2 Z ijβ j = ε i + (µ Y + f A2 (X i f A2 (X i. 23

24 Sice µ Y 2 = O p ( 1 ad f 0j f j = O(m d, we have η ε O p (1 + O(d 2 m 2d, (26 where ε is the projectio of ε = (ε 1,..., ε to the spa of Z A2. We have Now max Z Aε 2 2 = A: A d 2 ε 2 2 = (Z A 2 Z A2 1/2 Z A 2 ε c Z A 2 ε 2 2. max Z jε 2 2 d 2 m max Z A: A d 2 1 j p,1 k m jkε 2, j A where Z jk = (ψ k (X 1j,..., ψ k (X j. By Lemma 2, max Z 1 j p,1 k m jkε 2 = m 1 max (m / 1/2 Z 1 j p,1 k m jkε 2 = O p (1m 1 log(pm. It follows that, Combiig (25, (26, ad (27, we get ε 2 2 = O p (1 d 2 log(pm c. (27 ( β A2 β A2 2 d2 log(pm ( 1 ( d2 m 2d 2 O p + O c 2 p + O c c Sice d 2 = O p (q, c p m 1 ad c p m 1, we have ( m β A2 β A O log(pm p This completes the proof of part (iii. ( m ( 1 + O p + O + O m 2d 1 + 4λ2 1 A 1. 2 c 2 ( 4m 2 λ 2 1 We ow prove part (ii. Sice f j 2 c f > 0, 1 j q, f j f j 2 = O(m d ad f j 2 f j 2 f j f j 2, we have f j 2 0.5c f for sufficietly large. By a result of de Boor (2001, see also (12 of Stoe (1986, there are positive costats c 6 ad c 7 such that 2. c 6 m 1 β 2 2 f j 2 2 c 7 m 1 β j 2 2. It follows that, β j 2 2 c 1 7 m f j c 1 7 c 2 f m. Therefore, if β j 2 0 but β j 2 = 0, the β j β j c 1 7 c 2 fm. (28 24

26 This is true if β = 0 β if β j 2 β j 2 < β j 2, j A 1, Z j(y Z A1 βa1 2 λ 2 w j /2, j A 1. Therefore, P( β 0 β P ( β j β j 2 β j 2, j A 1 +P ( Z j(y Z A1 βa1 2 > λ 2 w j /2, j A 1. have Let f 0j (X j = (f 0j (X 1j,..., f 0j (X j ad δ = j A 1 f 0j (X j Z A1 β A1. By Lemma 1, we Let H = I Z A1 (Z A 1 Z A1 1 Z A 1. By (32, 1 δ 2 = O p (qm 2d. (33 β A1 β A1 = 1 C 1 A 1 ( Z A1 (ε + δ λ 2 ν, (34 ad Y Z A1 βa1 = H ε + H δ + λ 2 Z A1 C 1 A 1 ν /. (35 Based o these two equatios, Lemma 5 below shows that P ( β j β j 2 β j 2, j A 1 0, ad Lemma 6 below shows that P ( Z j(y Z A1 βa1 2 > λ 2 w j /2, j A 1 0. These two equatios lead to part (i of the theorem. We ow prove part (ii of Theorem 3. As i (26, for η = Y Zβ ad η 1 = Z A1 (Z A 1 Z A1 1 Z A 1 η, we have η ε O p (1 + O(qm 2d, (36 26

27 where ε 1 is the projectio of ε = (ε 1,..., ε to the spa of Z A1. We have ε = (Z A 1 Z A1 1/2 Z A 1 ε ρ 1 Z A 1 ε 2 2 = O p (1 A 1 ρ 1. (37 Now similarly to the proof of (25, we ca show that Combiig (36, (37 ad (38, we get β A1 β A η λ2 2 A 1. (38 ρ 1 2 ρ 2 1 ( 8 ( 1 ( 1 ( 4λ β A1 β A = O p + O ρ 2 p + O + O 2. 1 ρ 1 m 2d 1 2 ρ 2 1 Sice ρ 1 p m 1, the result follows. The followig lemmas are eeded i the proof of Theorem 3. Lemma 4 For ν = (w j βj /(2 β j, j A 1, uder coditio (B1, Proof of Lemma 4. Write ν 2 = O p (h 2 = O p ( (b 2 1 c b 2 r 1 + qb 1 1. ν 2 = j A 1 w 2 j = j A 1 β j 2 = j A 1 β j 2 β j 2 β j 2 β j 2 + j A 1 β j 1. Uder (B2, j A 1 βj 2 β j 2 β j 2 β j 2 Mc 2 b b 4 1 β β, ad j A 1 β j 2 qb 2 1. The claim follows. Let ρ 3 be the maximum of the largest eigevalues of 1 Z jz j, j A 0, that is, ρ 3 = max j A0 1 Z jz j 2. By Lemma 3, b 1 O(m 1/2, ρ 1 p m 1, ρ 2 p m 1 ad ρ 3 p m 1. (39 Lemma 5 Uder coditios (B1, (B2, (A3 ad (A4, P ( β j β j 2 β j 2, j A 1 0. (40 27

28 Proof of Lemma 5. Let T j be a m qm matrix with the form T j = (0 m,..., 0 m, I m, 0 m,..., 0 m, where O m is a m m matrix of zeros ad I m is a m m idetity matrix, ad I m is at the jth block. By (34, β j β j = 1 T j C 1 A 1 (Z A 1 ε + Z A 1 δ λ 2 ν. By the triagle iequality, β j β j 2 1 T j C 1 A 1 Z A 1 ε T j C 1 A 1 Z A 1 δ λ 2 T j C 1 A 1 ν 2. (41 Let C be a geeric costat idepedet of. The first term o the right-had side By (33, the secod term max 1 T j C 1 A j A 1 Z A 1 ε 2 1 ρ 1 1 Z A 1 ε 2 1 = 1/2 ρ 1 1 1/2 Z A 1 ε 2 = O p (1 1/2 ρ 1 1 m 1/2 (qm 1/2 (42 By Lemma 4, the third term max 1 T j C 1 A j A 1 Z A 1 δ 2 C 1 A Z A 1 Z A1 1/2 2 1 δ 2 1 = O p (1ρ 1 1 ρ 1/2 2 q 1/2 m d. (43 max 1 λ 2 T j C 1 A j A 1 ν 2 λ 2 ρ 1 1 ν 2 = O p (1ρ λ 2 h. (44 1 Thus (40 follows from (39, (42, (43, (44 ad coditio (B2a. Lemma 6 Uder coditios (B1, (B2, (A3 ad (A4, P ( Z j(y Z A1 βa1 2 > λ 2 w j /2, j A 1 0. (45 Proof of Lemma 6. By (35, we have ( Z j Y ZA1 βa1 = Z j H ε + Z jh δ + λ 1 Z jz A1 C 1 A 1 ν. (46 28

29 Recall s = p q is the umber of zero compoets i the model. By Lemma 2, ( E max 1/2 Z j A 1 jh ε 2 O(1 { log(s m } 1/2. (47 Sice w j = β j 1 = O p (r for j A 1 ad by (47, for the first term o the right had side of (46 we have P ( Z jh ε 2 > λ 2 w j /6, j A 1 By (33, the secod term o the right had side of (46 P ( Z jh ε 2 > Cλ 2 r, j A 1 + o(1 ( = P max 1/2 Z j A 1 jh ε 2 > C 1/2 λ 2 r + o(1 O(1 1/2 {log(s m } 1/2 Cλ 2 r + o(1. (48 max Z j A 1 jh δ 2 1/2 max 1 Z j A 1 jz j 1/2 2 H 2 δ 2 = O(1ρ 1/2 3 q 1/2 m d. (49 By Lemma 4, the third term o the right had side of (46 max λ 2 1 Z j Z A1 C 1 A j ia 1 ν 2 λ 2 max 1/2 Z j 2 1/2 Z A1 C 1/2 A 1 j A 1 2 C 1/2 A 1 2 ν 2 1 = λ 2 ρ 1/2 3 ρ 1/2 1 O p (qb 1 1. (50 Therefore, (45 follows from (39, (48, (49, (50 ad coditio (B2b. Proof of Theorem 4. The proof is similar to that of Theorem 2 ad is omitted. Refereces [1] Bach, F. R. (2007. Cosistecy of the group Lasso ad multiple kerel learig. Joural of Machie Learig Research, 9, [2] Buea, F., Tsybakov, A. ad Wegkamp, M. (2007. Sparsity oracle iequalities for the lasso. Electroic Joural of Statististic, [3] Che, J. ad Che, Z. (2008. Exteded Bayesia iformatio criteria for model selectio with large model space. Biometrika, 95, [4] Che, J. ad Che, Z. (2009. Exteded BIC for small--large-p sparse GLM. Available from stachez/cheche.pdf. 29

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