Support Vector Machine Soft Margin Classifiers: Error Analysis

Transcription

1 Journal of Machine Learning Research? (2004)?-?? Subitted 9/03; Published??/04 Support Vector Machine Soft Margin Classifiers: Error Analysis Di-Rong Chen Departent of Applied Matheatics Beijing University of Aeronautics and Astronautics Beijing , P R CHINA Qiang Wu Yiing Ying Ding-Xuan Zhou Departent of Matheatics City University of Hong Kong Kowloon, Hong Kong, P R CHINA drchen@buaaeducn wuqiang@studentcityueduhk yying@cityueduhk azhou@athcityueduhk Editor: Peter Bartlett Abstract The purpose of this paper is to provide a PAC error analysis for the q-nor soft argin classifier, a support vector achine classification algorith It consists of two parts: regularization error and saple error While any techniques are available for treating the saple error, uch less is known for the regularization error and the corresponding approxiation error for reproducing kernel Hilbert spaces We are ainly concerned about the regularization error It is estiated for general distributions by a K-functional in weighted L q spaces For weakly separable distributions (ie, the argin ay be zero) satisfactory convergence rates are provided by eans of separating functions A projection operator is introduced, which leads to better saple error estiates especially for sall coplexity kernels The isclassification error is bounded by the V -risk associated with a general class of loss functions V The difficulty of bounding the offset is overcoe Polynoial kernels and Gaussian kernels are used to deonstrate the ain results The choice of the regularization paraeter plays an iportant role in our analysis Keywords: support vector achine classification, isclassification error, q-nor soft argin classifier, regularization error, approxiation error 1 Introduction In this paper we study support vector achine (SVM) classification algoriths and investigate the SVM q-nor soft argin classifier with 1 < q < Our purpose is to provide an error analysis for this algorith in the PAC fraework Let (X, d) be a copact etric space and Y = 1, 1 A binary classifier f : X 1, 1 is a function fro X to Y which divides the input space X into two classes Let ρ be a probability distribution on Z := X Y and (X, Y) be the corresponding rando variable The isclassification error for a classifier f : X Y is defined to be the c 2004 Di-Rong Chen, Qiang Wu, Yiing Ying, and Ding-Xuan Zhou

2 Chen, Wu, Ying, and Zhou probability of the event f(x ) Y: R(f) := Prob f(x ) Y = X P (Y = f(x) x)dρ X (x) (1) Here ρ X is the arginal distribution on X and P ( x) is the conditional probability easure given X = x The SVM q-nor soft argin classifier (Cortes & Vapnik 1995; Vapnik 1998) is constructed fro saples and depends on a reproducing kernel Hilbert space associated with a Mercer kernel Let K : X X R be continuous, syetric and positive seidefinite, ie, for any finite set of distinct points x 1,, x l X, the atrix (K(x i, x j )) l i,j=1 is positive seidefinite Such a kernel is called a Mercer kernel The Reproducing Kernel Hilbert Space (RKHS) H K associated with the kernel K is defined (Aronszajn, 1950) to be the closure of the linear span of the set of functions K x := K(x, ) : x X with the inner product, HK =, K satisfying K x, K y K = K(x, y) and K x, g K = g(x), x X, g H K Denote C(X) as the space of continuous functions on X with the nor Let κ := K Then the above reproducing property tells us that g κ g K, g H K (2) Define H K := H K + R For a function f = f 1 + b with f 1 H K and b R, we denote f = f 1 and b f = b R The constant ter b is called the offset For a function f : X R, the sign function is defined as sgn(f)(x) = 1 if f(x) 0 and sgn(f)(x) = 1 if f(x) < 0 Now the SVM q-nor soft argin classifier (SVM q-classifier) associated with the Mercer kernel K is defined as sgn (f z ), where f z is a iniizer of the following optiization proble involving a set of rando saples z = (x i, y i ) i=1 Z independently drawn according to ρ: 1 f z := arg in f H K 2 f 2 K + C ξ q i, i=1 subject to y i f(x i ) 1 ξ i, and ξ i 0 for i = 1,, Here C is a constant which depends on : C = C(), and often li C() = Throughout the paper, we assue 1 < q <, N, C > 0, and z = (x i, y i ) i=1 are rando saples independently drawn according to ρ Our target is to understand how sgn(f z ) converges (with respect to the isclassification error) to the best classifier, the Bayes rule, as and hence C() tend to infinity Recall the regression function of ρ: f ρ (x) = ydρ(y x) = P (Y = 1 x) P (Y = 1 x), x X (4) Y Then the Bayes rule is given (eg Devroye, L Györfi & G Lugosi 1997) by the sign of the regression function f c := sgn(f ρ ) Estiating the excess isclassification error (3) R(sgn(f z )) R(f c ) (5) 2

3 SVM Soft Margin Classifiers for the classification algorith (3) is our goal In particular, we try to understand how the choice of the regularization paraeter C affects the error To investigate the error bounds for general kernels, we rewrite (3) as a regularization schee Define the loss function V = V q as V (y, f(x)) := (1 yf(x)) q + = y f(x) q χ yf(x) 1, (6) where (t) + = ax0, t The corresponding V -risk is E(f) := E(V (y, f(x))) = V (y, f(x))dρ(x, y) (7) If we set the epirical error as E z (f) := 1 Z V (y i, f(x i )) = 1 i=1 (1 y i f(x i )) q +, (8) then the schee (3) can be written as (see Evgeniou, Pontil & Poggio 2000) f z = arg in E z (f) + 1 f H K 2C f 2 K (9) Notice that when H K is replaced by H K, the schee (9) is exactly the Tikhonov regularization schee (Tikhonov & Arsenin, 1977) associated with the loss function V So one ay hope that the ethod for analyzing regularization schees can be applied The definitions of the V -risk (7) and the epirical error (8) tell us that for a function f = f + b H K, the rando variable ξ = V (y, f(x)) on Z has the ean E(f) and 1 i=1 ξ(z i) = E z (f) Thus we ay expect by soe standard epirical risk iniization (ERM) arguent (eg Cucker & Sale 2001, Evgeniou, Pontil & Poggio 2000, Shawe- Taylor et al 1998, Vapnik 1998, Wahba 1990) to derive bounds for E(f z ) E(f q ), where f q is a iniizer of the V -risk (7) It was shown in Lin (2002) that for q > 1 such a iniizer is given by i=1 f q (x) = (1 + f ρ(x)) 1/(q 1) (1 f ρ (x)) 1/(q 1) (1 + f ρ (x)) 1/(q 1), x X (10) + (1 f ρ (x)) 1/(q 1) For q = 1 a iniizer is f c, see Wahba (1999) Note that sgn(f q ) = f c Recall that for the classification algorith (3), we are interested in the excess isclassification error (5), not the excess V -risk E(f z ) E(f q ) But we shall see in Section 3 that R(sgn(f)) R(f c ) 2 (E(f) E(f q )) One ight apply this to the function f z and get estiates for (5) However, special features of the loss function (6) enable us to do better: by restricting f z onto [ 1, 1], we can iprove the saple error estiates The idea of the following projection operator was introduced for this purpose in Bartlett (1998) Definition 1 The projection operator π is defined on the space of easurable functions f : X R as 1, if f(x) 1, π(f)(x) = 1, if f(x) 1, (11) f(x), if 1 < f(x) < 1 3

4 Chen, Wu, Ying, and Zhou It is trivial that sgn(π(f)) = sgn(f) Hence R(sgn(f z )) R(f c ) 2 (E(π(f z )) E(f q )) (12) The definition of the loss function (6) also tells us that V (y, π(f)(x)) V (y, f(x)), so E(π(f)) E(f) and E z (π(f)) E z (f) (13) According to (12), we need to estiate E(π(f z )) E(f q ) in order to bound (5) To this end, we introduce a regularizing function f K,C H K It is arbitrarily chosen and depends on C Proposition 2 Let f K,C H K, and f z be defined by (9) Then E(π(f z )) E(f q ) can be bounded by E(f K,C ) E(f q ) + 1 2C f K,C 2 K + E(π(f z )) E z (π(f z )) + E z (f K,C ) E(f K,C ) (14) Proof Decopose the difference E(π(f z )) E(f q ) as ( E(π(f z )) E z (π(f z )) + E z (π(f z )) E z (f K,C ) E(f K,C ) + ) ( E z (f K,C ) + 1 2C f z 2 K E(f K,C ) E(f q ) + 1 2C f K,C 2 K 2C f K,C 2 K 1 2C f z 2 K By the definition of f z and (13), the second ter is 0 Then the stateent follows ) The first ter in (14) is called the regularization error (Sale & Zhou 2004) It can be expressed as a generalized K-functional inf E(f) E(fq ) + 1 2C f 2 K when fk,c takes f H K a special choice f K,C (a standard choice in the literature, eg Steinwart 2001) defined as f K,C := arg in E(f) + 1 f H K 2C f 2 K (15) Definition 3 Let V be a general loss function and f V ρ be a iniizer of the V -risk (7) The regularization error for the regularizing function f K,C H K is defined as D(C) := E(f K,C ) E(f V ρ ) + 1 2C f K,C 2 K (16) It is called the regularization error of the schee (9) when f K,C = f K,C : D(C) := inf E(f) E(fρ V ) + 1 f H K 2C f 2 K 4

5 SVM Soft Margin Classifiers The ain concern of this paper is the regularization error We shall investigate its asyptotic behavior This investigation is not only iportant for bounding the first ter in (14), but also crucial for bounding the second ter (saple error): it is well known in structural risk iniization (Shawe-Taylor et al 1998) that the size of the hypothesis space is essential This is deterined by D(C) in our setting Once C is fixed, the saple error estiate becoes routine Therefore, we need to understand the choice of the paraeter C fro the bound for D(C) Proposition 4 For any C 1/2 and any f K,C H K, there holds where κ := E 0 /(1 + κq4 q 1 ), D(C) D(C) κ2 2C (17) E 0 := inf b R E(b) E(f q) (18) Moreover, κ = 0 if and only if for soe p 0 [0, 1], P (Y = 1 x) = p 0 in probability This proposition will be proved in Section 5 According to Proposition 4, the decay of D(C) cannot be faster than O(1/C) except for soe very special distributions This special case is caused by the offset in (9), for which D(C) 0 Throughout this paper we shall ignore this trivial case and assue κ > 0 When ρ is strictly separable, D(C) = O(1/C) But this is a very special phenoenon In general, one should not expect E(f) = E(f q ) for soe f H K Even for (weakly) separable distributions with zero argin, D(C) decays as O(C p ) for soe 0 < p < 1 To realize such a decay for these separable distributions, the regularizing function will be ultiples of a separating function For details and the concepts of strictly or weakly separable distributions, see Section 2 For general distributions, we shall choose f K,C = f K,C in Sections 6 and 7 and estiate the regularization error of the schee (9) associated with the loss function (6) by eans of the approxiation in the function space L q ρ X In particular, D(C) 1 K(fq, 2C ), where K(f q, t) is a K-functional defined as K(f q, t) := inf f f q q L q + t f 2 ρ K f H K X inf q2 q 1 (2 q 1 + 1) f f q L q ρx + t f 2 K f H K, if 1 < q 2,, if q > 2 In the case q = 1, the regularization error (Wu & Zhou, 2004) depends on the approxiation in L 1 ρ X of the function f c which is not continuous in general For q > 1, the regularization error depends on the approxiation in L q ρ X of the function f q When the regression function has good soothness, f q has uch higher regularity than f c Hence the convergence for q > 1 ay be faster than that for q = 1, which iproves the regularization error The second ter in (14) is called the saple error When C is fixed, the saple error E(f z ) E z (f z ) is well understood (except for the offset ter) In (14), the saple error is for the function π(f z ) instead of f z while the isclassification error (5) is kept: (19) 5

6 Chen, Wu, Ying, and Zhou R(sgn(π(f z ))) = R(sgn(f z )) Since the bound for V (y, π(f)(x)) is uch saller than that for V (y, f(x)), the projection iproves the saple error estiate Based on estiates for the regularization error and saple error above, our error analysis will provide ε(δ,, C, β) > 0 for any 0 < δ < 1 such that with confidence 1 δ, E(π(f z )) E(f q ) (1 + β)d(c) + ε(δ,, C, β) (20) Here 0 < β 1 is an arbitrarily fixed nuber Moreover, li ε(δ,, C, β) = 0 If f q lies in the L q ρ X -closure of H K, then li K(f q, t) = 0 by (19) Hence E(π(f z )) t 0 E(f q ) 0 with confidence as (and hence C = C()) becoes large This is the case when K is a universal kernel, ie, H K is dense in C(X), or when a sequence of kernels whose RKHS tends to be dense (eg polynoial kernels with increasing degrees) is used In suary, estiating the excess isclassification error R(sgn(f z )) R(f c ) consists of three parts: the coparison (12), the regularization error D(C) and the saple error ε(δ,, C, β) in (20) As functions of the variable C, D(C) decreases while ε(δ,, C, β) usually increases Choosing suitable values for the regularization paraeter C will give the optial convergence rate To this end, we need to consider the tradeoff between these two errors This can be done by iniizing the right side of (20), as shown in the following for Lea 5 Let p, α, τ > 0 Denote c p,α,τ := (p/τ) τ τ+p + (τ/p) p τ+p Then for any C > 0, ( ) αp C p + Cτ 1 α c τ+p p,α,τ (21) The equality holds if and only if C = (p/τ) 1 τ+p α τ+p This yields the optial power αp τ+p The goal of the regularization error estiates is to have p in D(C) = O(C p ), as large as possible But p 1 according to Proposition 4 Good ethods for saple error estiates provide large α and sall τ such that ε(δ,, C, β) = O( Cτ ) Notice that, as always, both α the approxiation properties (represented by the exponent p) and the estiation properties (represented by the exponents τ and α) are iportant to get good estiates for the learning rates (with the optial rate αp τ+p ) 2 Deonstrating with Weakly Separable Distributions With soe special cases let us deonstrate how our error analysis yields soe guidelines for choosing the regularization paraeter C For weakly separable distributions, we also copare our results with bounds in the literature To this end, we need the covering nuber of the unit ball B := f H K : f K 1 of H K (considered as a subset of C(X)) Definition 6 For a subset F of a etric space and η > 0, the covering nuber N (F, η) is defined to be the inial integer l N such that there exist l disks with radius η covering F 6

7 SVM Soft Margin Classifiers Denote the covering nuber of B in C(X) by N (B, η) Recall the constant κ defined in (18) Since the algorith involves an offset ter, we also need its covering nuber and set N (η) := (κ + 1 κ ) 1 η + 1 N (B, η) (22) Definition 7 We say that the Mercer kernel K has logarithic coplexity exponent s 1 if for soe c > 0, there holds log N (η) c (log(1/η)) s, η > 0 (23) The kernel K has polynoial coplexity exponent s > 0 if log N (η) c (1/η) s, η > 0 (24) The covering nuber N (B, η) has been extensively studied (see eg Bartlett 1998, Williason, Sola & Schölkopf 2001, Zhou 2002, Zhou 2003a) In particular, for convolution type kernels K(x, y) = k(x y) with ˆk 0 decaying exponentially fast, (23) holds (Zhou 2002, Theore 3) As an exaple, consider the Gaussian kernel K(x, y) = exp x y 2 /σ 2 with σ > 0 If X [0, 1] n and 0 < η exp90n 2 /σ 2 11n 3, then (23) is valid (Zhou 2002) with s = n + 1 A lower bound (Zhou 2003a) holds with s = n 2, which shows the upper bound is alost sharp It was also shown in (Zhou 2003a) that K has polynoial coplexity exponent 2n/p if K is C p To deonstrate our ain results concerning the regularization paraeter C, we shall only choose kernels having logarithic coplexity exponents or having polynoial coplexity exponents with s sall This eans, H K has sall coplexity The special case we consider here is the deterinistic case: R(f c ) = 0 Understanding how to find D(C) and ε(δ,, C, β) in (20) and then how to choose the paraeter C is our target For M 0, denote 0, if M = 0, θ M = (25) 1, if M > 0 Proposition 8 Suppose R(f c ) = 0 If f K,C H K satisfies V (y, f K,C (x)) M, then for every 0 < δ < 1, with confidence at least 1 δ, we have R(sgn(f z )) 2 ax ε 4M log(2/δ), + 4D(C), (26) where with M := 2CD(C) + 2Cεθ M, ε > 0 is the unique solution to the equation ( ) ε log N q2 q+3 M (a) If (23) is valid, then with c = 2 q c((q + 4) log 8) s, 3ε = log(δ/2) (27) 2q+9 ( (log + log(cd(c)) + ε log(cθm )) s ) + log(2/δ) c 7

8 Chen, Wu, Ying, and Zhou (b) If (24) is valid and C 1, then with a constant c (given explicitly in the proof), (2CD(C)) s/(2s+2) ε c log(2/δ) 1/(s+1) + (2C)s/(s+2) 1/( s 2 +1) (28) Note that the above constant c depends on c, q, and κ, but not on C, or δ The proof of Proposition 8 will be given in Section 5 We can now derive our error bound for weakly separable distributions Definition 9 We say that ρ is (weakly) separable by H K if there is a function fsp H K, called a separating function, satisfying f sp K = 1 and yfsp(x) > 0 alost everywhere It has separation exponent θ (0, + ] if there are positive constants γ, c such that ρ X x X : fsp(x) < γt c t θ, t > 0 (29) Observe that condition (29) with θ = + is equivalent to ρ X x X : fsp(x) < γt = 0, 0 < t < 1 That is, fsp(x) γ alost everywhere Thus, weakly separable distributions with separation exponent θ = + are exactly strictly separable distributions Recall (eg Vapnik 1998, Shawe-Taylor et al 1998) that ρ is said to be strictly separable by H K with argin γ > 0 if ρ is (weakly) separable together with the requireent yfsp(x) γ alost everywhere Fro the first part of Definition 9, we know that for a separable distribution ρ, the set x : fsp(x) = 0 has ρ X -easure zero, hence li ρ Xx X : fsp(x) < t = 0 t 0 The separation exponent θ easures the asyptotic behavior of ρ near the boundary of two classes It gives the convergence with a polynoial decay Theore 10 If ρ is separable and has separation exponent θ (0, + ] with (29) valid, then for every 0 < δ < 1, with confidence at least 1 δ, we have R(sgn(f z )) 2ε + 8 log(2/δ) where ε is solved by ( ε log N q2 q+3 2(2c ) 2 θ+2 γ 2θ 2 θ+2 C θ+2 + 2Cε + 4(2c ) 2 θ+2 (Cγ 2 ) θ θ+2, (30) ) 3ε = log(δ/2) (31) 2q+9 (a) If (23) is satisfied, with a constant c depending on γ we have ( log(2/δ) + (log + log C) s ) R(sgn(f z )) c + C θ θ+2 (32) 8

9 SVM Soft Margin Classifiers (b) If (24) is satisfied, there holds R(sgn(f z )) c log 2 ( C δ s (s+1)(θ+2) 1 s+1 + C s s+2 1 s/2+1 ) + C θ θ+2 (33) Proof Choose t = ( γθ 2c C )1/(θ+2) > 0 and the function f K,C = 1 t f sp H K Then we have 1 2C f K,C 2 K = 1 2Ct 2 Since yfsp(x) > 0 alost everywhere, we know that for alost every x X, y = sgn(fsp)(x) This eans f ρ = sgn(fsp) and then f q = sgn(fsp) It follows that R(f c ) = 0 and V (y, f K,C (x)) = (1 f sp(x) t ) q + [0, 1] alost everywhere Therefore, we ay take M = 1 in Proposition 8 Moreover, E(f K,C ) = (1 yf ) sp(x) q (1 t dρ = f ) sp(x) q + t dρ X + This can be bounded by (29) as (1 f sp(x) x: fsp(x) <t Z t ) q X dρ X ρ X x X : fsp(x) < t c ( t ) θ + γ It follows fro the choice of t that D(C) c ( t ) θ 1 + γ 2Ct 2 = (2c ) 2 θ+2 (Cγ 2 ) θ θ+2 (34) Then our conclusion follows fro Proposition 8 In case (a), the bound (32) tells us to choose C such that (log C) s / 0 and C as By (32) a reasonable choice of the regularization paraeter C is C = θ+2 θ, (log )s which yields R(sgn(f z )) = O( ) In case (b), when (24) is valid, C = θ+2 θ+s+θs = R(sgn(fz )) = O( θ θ+s+θs ) (35) The following is a typical exaple of separable distribution which is not strictly separable In this exaple, the separation exponent is θ = 1 Exaple 1 Let X = [ 1/2, 1/2] and ρ be the Borel probability easure on Z such that ρ X is the Lebesgue easure on X and 1, for 1/2 x 1/4 and 1/4 x 1/2, f ρ (x) = 1, for 1/4 x < 1/4 (a) If K is the linear polynoial kernel K(x, y) = x y, then R(sgn(f z )) R(f c ) 1/4 (b) If K is the quadratic polynoial kernel K(x, y) = (x y) 2, then with confidence 1 δ, R(sgn(f z )) q(q + 4)2q+11 (log + log C + 2 log(2/δ)) + 16 C 1/3 Hence one should take C such that log C/ 0 and C as 9

10 Chen, Wu, Ying, and Zhou Proof The first stateent is trivial To see (b), we note that dih K = 1 and κ = 1/4 Also, E 0 = inf E(b) = 1 Hence κ = b R 1/(1 + q4 q 2 ) Since H K = ax 2 : a R and ax 2 K = a, N (B, η) 1/(2η) Then (23) holds with s = 1 and c = 4q By Proposition 8, we see that ε q(q+4)2q+11 (log(c)+log(2/δ)) Take the function fsp(x) = x 2 1/16 H K with fsp K = 1 We see that yfsp(x) > 0 alost everywhere Moreover, [ 1 16t t x X : fsp(x) < t, 4 4 ] [ t, t The easure of this set is bounded by 8 2t Hence (29) holds with θ = 1, γ = 1 and c = 8 2 Then Theore 10 gives the stated bound for R(sgn(f z )) 4 ] In this paper, for the saple error estiates we only use the (unifor) covering nubers in C(X) Within the last a few years, the epirical covering nubers (in l or l 2 ), the leave-one-out error or stability analysis (eg Vapnik 1998, Bousquet & Ellisseeff 2002, Zhang 2004), and soe other advanced epirical process techniques such as the local Radeacher averages (van der Vaart & Wellner 1996, Bartlett, Bousquet & Mendelson 2004, and references therein) and the entropy integrals (van der Vaart & Wellner, 1996) have been developed to get better saple error estiates These techniques can be applied to various learning algoriths They are powerful to handle general hypothesis spaces even with large capacity In Zhang (2004) the leave-one-out technique was applied to iprove the saple error estiates given in Bousquet & Ellisseeff (2002): the saple error has a kernel-independent bound O( C C ), iproving the bound O( ) in Bousquet & Ellisseeff (2002); while the regularization error D(C) depends on K and ρ In particular, for q = 2 the bound (Zhang 2004, Corollary 42) takes the for: ( E(E(f z )) 1 + 4κ2 C ) 2 inf E(f) + 1 ( f H K 2C f 2 K = 1 + 4κ2 C ) 2 D(C) + E(fq ) (36) In Bartlett, Jordan & McAuliffe (2003) the epirical process techniques are used to iprove the saple error estiates for ERM In particular, Theore 12 there states that for a convex set F of functions on X, the iniizer f of the epirical error over F satisfies E( f) inf E(f) + K ax f F ( ε cr L 2 log(1/δ) ) 1/(2 β), BL log(1/δ), (37) Here K, c r, β are constants, and ε is solved by an inequality The constant B is a bound for differences, and L is the Lipschitz constant for the loss φ with respect to a pseudoetric on R For ore details, see Bartlett, Jordan & McAuliffe (2003) The definitions of B and L together tell us that φ(y 1 f(x 1 )) φ(y 2 f(x 2 )) LB, (x 1, y 1 ) Z, (x 2, y 2 ) Z, f F (38) Because of our iproveent for the bound of the rando variables V (y, f(x)) given by the projection operator, for the algorith (3) the error bounds we derive here are better 10

11 SVM Soft Margin Classifiers than existing results when the kernel has sall coplexity Let us confir this for separable distributions with separation exponent 0 < θ < and for kernels with polynoial coplexity exponent s > 0 satisfying (24) Recall that D(C) = O(C θ θ+2 ) Take q = 2 Consider the estiate (36) This together with (34) tells us that the derived bound for E(E(f z ) E(f q )) is at least of the order D(C) + C D(C) = O(C θ θ+2 + C θ+2 2 ) By Lea 5, the optial bound derived fro (36) is O( θ θ+2 ) Thus, our bound (35) is better than (36) when 0 < s < 2 θ+1 Turn to the estiate (37) Take F to be the ball of radius 2C D(C) of H K (the expected sallest ball where fz lies), we find that the constant LB in (38) should be at least κ 2C D(C) This tells us that one ter in the bound (37) for the saple error is ( 2C at least O D(C) ) = O( C θ+2 1 ), while the other two ters are ore involved Applying Lea 5 again, we find that the bound derived fro (37) is at least O( θ θ+1 ) So our bound (35) is better than (37) at least for 0 < s < 1 θ+1 Thus, our analysis with the help of the projection operator iproves existing error bounds when the kernel has sall coplexity Note that in (35), the values of s for which the projection operator gives an iproveent are values corresponding to rapidly diinishing regularization (C = β with β > 1 being large) For kernels with large coplexity, refined epirical process techniques (eg van der Vaart & Wellner 1996, Mendelson 2002, Zhang 2004, Bartlett, Jordan & McAuliffe 2003) should give better bounds: the capacity of the hypothesis space ay have ore influence on the saple error than the bound of the rando variable V (y, f(x)), and the entropy integral is powerful to reduce this influence To find the range of this large coplexity, one needs explicit rate analysis for both the saple error and regularization error This is out of our scope It would be interesting to get better error bounds by cobining the ideas of epirical process techniques and the projection operator Proble 11 How uch iproveent can we get for the total error when we apply the projection operator to epirical process techniques? Proble 12 Given θ > 0, s > 0, q 1, and 0 < δ < 1, what is the largest nuber α > 0 such that R(sgn(f z )) = O( α ) with confidence 1 δ whenever the distribution ρ is separable with separation exponent θ and the kernel K satisfies (24)? What is the corresponding optial choice of the regularization paraeter C? In particular, for Exaple 1 we have the following Conjecture 13 In Exaple 1, with confidence 1 δ there holds R(sgn(f z )) = O( log(2/δ) ) by choosing C = 3 Consider the well known setting (Vapnik 1998, Shawe-Taylor et al 1998, Steinwart 2001, Cristianini & Shawe-Taylor 2000) of strictly separable distributions with argin γ > 0 In this case, Shawe-Taylor et al (1998) shows that R(sgn(f)) 2 (log N (F, 2, γ/2) + log(2/δ)) (39) 11

12 Chen, Wu, Ying, and Zhou with confidence 1 δ for > 2/γ whenever f F satisfies y i f(x i ) γ Here F is a function set, N (F, 2, γ/2) = sup N (F, t, γ/2) and N (F, t, γ/2) denotes the covering t X 2 nuber of the set (f(t i )) 2 i=1 : f F in R2 with the l etric For a coparison of this covering nuber with that in C(X), see Pontil (2003) In our analysis, we can take f K,C = 1 γ f sp H K Then M = V (y, f K,C (x)) = 0, and D(C) = 1 We see fro Proposition 8 (a) that when (23) is satisfied, there holds 2Cγ 2 R(f z ) c( (log(1/γ)+log )s +log(1/δ) ) But the optial bound O( 1 ) is also valid for spaces with larger coplexity, which can be seen fro (39) by the relation of the covering nubers: N (F, 2, γ/2) N (F, γ/2) See also Steinwart (2001) We see the shortcoing of our approach for kernels with large coplexity This convinces the interest of Proble 11 raised above 3 Coparison of Errors In this section we consider how to bound the isclassification error by the V -risk A systeatic study of this proble was done for convex loss functions by Zhang (2004), and for ore general loss functions by Bartlett, Jordan, and McAuliffe (2003) Using these results, it can be shown that for the loss function V q, there holds ( ) ψ R(sgn(f)) R(f c ) E(f) E(f q ), where ψ : [0, 1] R + is a function defined in Bartlett, Jordan, and McAuliffe (2003) and can be explicitly coputed In fact, we have R(sgn(f)) R(f c ) c E(f) E(f q ) (40) Such a coparison of errors holds true even for a general convex loss function, see Theore 34 in Appendix The derived bound for the constant c in (40) need not be optial In the following we shall give an optial estiate in a sipler for The constant we derive depends on q and is given by C q = 1, if 1 q 2, 2(q 1)/q, if q > 2 (41) We can see that C q 2 Theore 14 Let f : X R be easurable Then R(sgn(f)) R(f c ) C q (E(f) E(f q )) 2 (E(f) E(f q )) Proof By the definition, only points with sgn(f)(x) f c (x) are involved for the isclassification error Hence R(sgn(f)) R(f c ) = f ρ (x) χ sgn(f)(x) sgn(f q)(x)dρ X (42) X 12

13 SVM Soft Margin Classifiers This in connection with the Schwartz inequality iplies that R(sgn(f)) R(f c ) X f ρ (x) 2 χ sgn(f)(x) sgn(f q)(x)dρ X 1/2 Thus it is sufficient to show for those x X with sgn(f)(x) sgn(f q )(x), where for x X, we have denoted E(f x) := By the definition of the loss function V q, we have It follows that f ρ (x) 2 C q (E(f x) E(f q x)), (43) Y V q (y, f(x))dρ(y x) (44) E(f x) = (1 f(x)) q 1 + f ρ (x) + + (1 + f(x)) q 1 f ρ (x) + (45) 2 2 E(f x) E(f q x) = f(x) fq(x) where F (u) is the function (depending on the paraeter f ρ (x)): F (u) = 1 f ρ(x) q(1 + f q (x) + u) q f ρ(x) 2 F (u)du, (46) q(1 f q (x) u) q 1 +, u R Since F (u) is nondecreasing and F (0) = 0, we see fro (46) that when sgn(f)(x) sgn(f q )(x), there holds But E(f x) E(f q x) E(f q x) = fq(x) 0 F (u)du = E(0 x) E(f q x) = 1 E(f q x) (47) 2 q 1 ( 1 f ρ (x) 2) (1 + fρ (x)) 1/(q 1) + (1 f ρ (x)) 1/(q 1) q 1 (48) Therefore, (43) is valid (with t = f ρ (x)) once the following inequality is verified: t 2 2 q 1 ( 1 t 2) 1 C q (1 + t) 1/(q 1) + (1 t) 1/(q 1) q 1, t [ 1, 1] This is the sae as the inequality ) 1/(q 1) (1 t2 (1 + t) 1/(q 1) + (1 t) 1/(q 1), t [ 1, 1] (49) 2 C q To prove (49), we use the Taylor expansion (1 + u) α = k=0 ( ) α u k = k k=0 k 1 l=0 (α l) k! 13 u k, u ( 1, 1)

14 Chen, Wu, Ying, and Zhou With α = 1/(q 1), we have ) 1/(q 1) (1 t2 = C q k=0 ( ) k 1 1 l=0 q 1 + l k! 1 C k q t 2k and (all the odd power ters vanish) (1 + t) 1/(q 1) + (1 t) 1/(q 1) 2 = k=0 ( ) k 1 1 l=0 q 1 + l k! ( k 1 l=0 1 q 1 + l + k 1 + l + k ) t 2k Note that k 1 l=0 This proves (49) and hence our conclusion 1 q 1 + l + k 1 + l + k 1 Cq k When R(f c ) = 0, the estiate in Theore 14 can be iproved (Zhang 2004, Bartlett, Jordan & McAuliffe 2003) to R(sgn(f)) E(f) (50) In fact, R(f c ) = 0 iplies f ρ (x) = 1 alost everywhere This in connection with (48) and (47) gives E(f q x) = 0 and E(f x) 1 when sgn(f)(x) f c (x) Then (43) can be iproved to the for f ρ (x) E(f x) Hence (50) follows fro (42) Theore 14 tells us that the isclassification error can be bounded by the V -risk associated with the loss function V So our next step is to study the convergence of the q-classifier with respect to the V -risk E 4 Bounding the Offset If the offset b is fixed in the schee (3), the saple error can be bounded by standard arguent using soe easureents of the capacity of the RKHS However, the offset is part of the optiization proble and even its boundedness can not be seen fro the definition This akes our setting here essentially different fro the standard Tikhonov regularization schee The difficulty of bounding the offset b has been realized in the literature (eg Steinwart 2002, Bousquet & Ellisseeff 2002) In this paper we shall overcoe this difficulty by eans of special features of the loss function V The difficulty raised by the offset can also be seen fro the stability analysis (Bousquet & Ellisseeff 2002) As shown in Bousquet & Ellisseeff (2002), the SVM 1-nor classifier without the offset is uniforly stable, eaning that sup z Z,z 0 Z V (y, f z (x)) V (y, f z (x)) L β with β 0 as, and z is the sae as z except that one eleent of z is replaced by z 0 The SVM 1-nor classifier with the offset is not uniforly stable To see this, we choose x 0 X and saples z = (x 0, y i ) 2n+1 i=1 with y i = 1 for i = 1,, n + 1, and y i = 1 for i = n+2,, 2n+1 Take z 0 = (x 0, 1) As x i are identical, one can see fro the definition 14

15 SVM Soft Margin Classifiers (9) that f z = 0 (since E z (f z ) = E z (b z + f z (x 0 ))) It follows that f z = 1 while f z = 1 Thus, f z f z = 2 which does not converge to zero as = 2n + 1 tends to infinity It is unknown whether the q-nor classifier is uniforly stable for q > 1 How to bound the offset is the ain goal of this section In Wu & Zhou (2004) a direct coputation is used to realize this point for q = 1 Here the index q > 1 akes a direct coputation very difficult, and we shall use two bounds to overcoe this difficulty By x (X, ρ X ) we ean that x lies in the support of the easure ρ X on X Lea 15 For any C > 0, N and z Z, a iniizer of (9) satisfies and a iniizer of (15) satisfies in f z(x i ) 1 and ax f z(x i ) 1 (51) 1 i 1 i inf f K,C (x) 1 and sup f K,C (x) 1 (52) x (X,ρ X ) x (X,ρ X ) Proof Suppose a iniizer of (9) f z satisfies r := in 1 i f z(x i ) > 1 Then f z (x i ) (r 1) 1 for each i We clai that y i = 1, i = 1,, In fact, if the set I := i 1,, : y i = 1 is not epty, we have E z (f z (r 1)) = 1 (1 + f z (x i ) (r 1)) q < 1 (1 + f z (x i )) q = E z (f z ), i I which is a contradiction to the definition of f z Hence our clai is verified Fro the clai we see that E z (f z (r 1)) = 0 = E z (f z ) This tells us that f z := f z (r 1) is a iniizer of (9) satisfying the first inequality, hence both inequalities of (51) In the sae way, if a iniizer of (9) f z satisfies r := ax f z(x i ) < 1 Then we 1 i can see that y i = 1 for each i Hence E z (f z r 1) = E z (f z ) and f z := f z r 1 is a iniizer of (9) satisfying the second inequality and hence both inequalities of (51) Therefore, we can always find a iniizer of (9) satisfying (51) We prove the second stateent in the sae way Suppose r := i I inf x (X,ρ X ) f K,C (x) > 1 for a iniizer f K,C of (15) Then f K,C (x) (r 1) 1 for alost every x (X, ρ X ) Hence ( ) ( E fk,c (r 1) = 1 + f q K,C (x) (r 1)) P (Y = 1 x)dρx E( f K,C ) X As f K,C is a iniizer of (15), the above equality ust hold It follows that P (Y = 1 x) = 0 for alost every x (X, ρ X ) Hence F K,C := f K,C (r 1) is a iniizer of (15) satisfying the first inequality and thereby both inequalities of (52) Siilarly, when r := sup f K,C (x) < 1 for a iniizer f K,C of (15) Then P (Y = x (X,ρ X ) 1 x) = 0 for alost every x (X, ρ X ) Hence F K,C := f K,C r 1 is a iniizer of (15) satisfying the second inequality and thereby both inequalities of (52) 15

16 Chen, Wu, Ying, and Zhou Thus, (52) can always be realized by a iniizer of (15) bk,c In what follows we always choose f z and f K,C to satisfy (51) and (52), respectively Lea 15 yields bounds for the H K -nor and offset for f z and f K,C Denote b fk,c Lea 16 For any C > 0, N, f K,C H K, and z Z, there hold (a) f K,C K 2C D(C) 2C, b K,C 1 + f K,C (b) f K,C 1 + 2κ 2C D(C) 1 + 2κ 2C, E( f K,C ) E(f q ) + D(C) 1 (c) b z 1 + κ f z K Proof By the definition (15), we see fro the choice f = that Then the first inequality in (a) follows Note that (52) gives and D(C) = E( f K,C ) E(f q ) + 1 2C f K,C 2 K 1 E(f q ) (53) 1 sup f K,C b K,C + f K,C x (X,ρ X ) bk,c f K,C inf f K,C 1 x (X,ρ X ) Thus, b K,C 1 + f K,C This proves the second inequality in (a) Since the first inequality in (a) and (2) lead to f K,C κ 2C D(C), we obtain f K,C f K,C 1 + 2κ 2C D(C) Hence the first inequality in (b) holds The second inequality in (b) is an easy consequence of (53) The inequality in (c) follows fro (51) in the sae way as the proof of the second inequality in (a) as 5 Convergence of the q-nor Soft Margin Classifier In this section, we apply the ERM technique to analyze the convergence of the q-classifier for q > 1 The situation here is ore coplicated than that for q = 1 We need the following lea concerning q > 1 Lea 17 For q > 1, there holds (x) q + (y)q + q (axx, y) q 1 + x y, x, y R If y [ 1, 1], then (1 x) q + (1 y)q + q4 q 1 x y + q2 q 1 x y q, x R 16

17 SVM Soft Margin Classifiers Proof We only need to prove the first inequality for x > y This is trivial: (x) q + (y)q + = x y q(u) q 1 + du q(x)q 1(x y) If y [ 1, 1], then 1 y 0 and the first inequality yields (1 x) q + (1 y)q + q (ax1 x, 1 y) q 1 + x y (54) When x 2y 1, we have (1 x) q + (1 y)q + q2 q 1 (1 y) q 1 x y q4 q 1 x y When x < 2y 1, we have 1 x < 2(y x) and x < 1 This in cobination with ax1 x, 1 y ax1 x, 2 and (54) iplies (1 x) q + (1 y)q + q(1 x) q q 1 x y q2 q 1 x y q + q2 q 1 x y This proves Lea 17 + The second part of Lea 17 will be used in Section 6 The first part can be used to verify Proposition 4 Proof of Proposition 4 It is trivial that D(C) D(C) To show the second inequality, apply the first inequality of Lea 17 to the two nubers 1 y f K,C (x) and 1 y b K,C We see that (1 y f K,C (x)) q + (1 y b K,C ) q + q(1 + f K,C (x) + b K,C ) q 1 f K,C(x) Notice that f K,C (x) κ f K,C K and by Lea 16, b K,C 1 + κ f K,C K Hence E( f K,C ) E( b K,C ) κq2 q 1 (1 + κ f K,C K ) q 1 f K,C K It follows that when f K,C K κ, we have E( f K,C ) E(f q ) E 0 κq2 q 1 (1 + κ κ) q 1 κ Since E 0 1, the definition (18) of κ yields κ 1/(1 + κ) in1, 1/κ Hence D(C) E( f K,C ) E(f q ) E 0 κq4 q 1 κ = κ κ 2 As C 1/2, we conclude (17) in this case When f K,C K > κ, we also have Thus in both case we have verified (17) D(C) 1 2C f K,C 2 K κ2 2C 17

18 Chen, Wu, Ying, and Zhou Note that κ = 0 if and only if E 0 = 0 This eans for soe b 0 [ 1, 1], f q(x) = b 0 in probability By the definition of f q, the last assertion of Proposition 4 follows The loss function V is not Lipschitz, but Lea 17 enables us to derive a bound for E(π(f z )) E z (π(f z )) with confidence, as done for Lipschitz loss functions in Mukherjee, Rifkin & Poggio (2002) Since the function π(f z ) changes and lies in a set of functions, we shall copare the error with the epirical error for functions fro a set F := π(f) : f B R + [ B, B] (55) Here B R = f H K : f K R and the constant B is a bound for the offset The following probability inequality was otivated by saple error estiates for the square loss (Barron 1990, Bartlett 1998, Cucker & Sale 2001, Lee, Bartlett & Williason 1998) and will be used in our estiates Lea 18 Suppose a rando variable ξ satisfies 0 ξ M, and z = (z i ) i=1 are independent saples Let µ = E(ξ) Then for every ε > 0 and 0 < α 1, there holds µ 1 i=1 Prob ξ(z i) z Z α ε exp 3α2 ε µ + ε 8M Proof As ξ satisfies ξ µ M, the one-side Bernstein inequality tells us that µ 1 i=1 Prob ξ(z i) z Z α α 2 (µ + ε)ε ε exp µ + ε 2 ( σ Mα µ + ε ε ) Here σ 2 E(ξ 2 ) ME(ξ) = Mµ since 0 ξ M Then we find that This yields the desired inequality σ Mα µ + ε ε Mµ M(µ + ε) 4 M(µ + ε) 3 Now we can turn to the error bound involving a function set Lea 17 yields the following bounds concerning the loss function V : E z (f) E z (g) q ax (1 + f ) q 1, (1 + g ) q 1 f g (56) and E(f) E(g) q ax (1 + f ) q 1, (1 + g ) q 1 f g (57) Lea 19 Let F be a subset of C(X) such that f 1 for each f F Then for every ε > 0 and 0 < α 1, we have E(f) E z (f) Prob z Z sup 4α ( ) αε ε N F, f F E(f) + ε q2 q 1 exp 3α2 ε 2 q+3 18

19 SVM Soft Margin Classifiers Proof Let f j J j=1 F with J = N (F, αε at f j with radius f F Then for each j, Lea 18 tells q2 q 1 ) such that F is covered by balls centered αε q2 q 1 Note that ξ = V (y, f(x)) satisfies 0 ξ (1 + f ) q 2 q for Prob z Z E(f j ) E z (f j ) α ε E(fj ) + ε exp 3α2 ε 8 2 q For each f F, there is soe j such that f f j αε This in connection with q2 q 1 (56) and (57) tells us that E z (f) E z (f j ) and E(f) E(f j ) are both bounded by αε Hence E z (f) E z (f j ) α E(f) E(f j ) ε and α ε E(f) + ε E(f) + ε The latter iplies that E(f j ) + ε 2 E(f) + ε Therefore, E(f) E z (f) Prob z Z sup 4α J E(f j ) E z (f j ) ε Prob α ε f F E(f) + ε E(fj ) + ε j=1 which is bounded by J exp 3α2 ε 2 q+3 Take F to be the set (55) The following covering nuber estiate will be used Lea 20 Let F be given by (55) with R κ and B = 1 + κr Its covering nuber in C(X) can be bounded as follows: ( η ) N (F, η) N, η > 0 2R Proof It follows fro the fact π(f) π(g) f g that N (F, η) N (B R + [ B, B], η) The latter is bounded by (κ + 1 κ ) 2R η + 1N ( B R, η 2) since 2B η (κ + 1 κ ) 2R η and (f + b f ) (g + b g ) f g + b f b g Note that an η 2R -covering of B is the sae as an η 2 -covering of B R Then our conclusion follows fro Definition 6 We are in a position to state our ain result on the error analysis Recall θ M in (25) Theore 21 Let f K,C H K, M V (y, f K,C (x)), and 0 < β 1 For every ε > 0, we have E(π(f z )) E(f q ) (1 + β) (ε + D(C)) with confidence at least 1 F (ε) where F : R + R + is defined by ε 2 ( ) β(ε + D(C)) 3/2 F (ε) := exp + N 2M( + ε) q2 q+3 exp 3β2 (D(C) + ε) 2 Σ 2 q+9 ( + ε) with := D(C) + E(f q ) and Σ := 2C( + ε)( + θ M ε) (58) 19

20 Chen, Wu, Ying, and Zhou Proof Let ε > 0 We prove our conclusion in three steps Step 1: Estiate E z (f K,C ) E(f K,C ) Consider the rando variable ξ = V (y, f K,C (x)) with 0 ξ M If M > 0, since σ 2 (ξ) ME(f K,C ) M(D(C) + E(f q )), by the one-side Bernstein inequality we obtain E z (f K,C ) E(f K,C ) ε with confidence at least ε 2 1 exp 2(σ 2 (ξ) Mε) 1 exp ε 2 2M( + ε) If M = 0, then ξ = 0 alost everywhere Hence E z (f K,C ) E(f K,C ) = 0 with probability 1 Thus, in both cases, there exists U 1 Z with easure at least 1 exp ε2 such that E z (f K,C ) E(f K,C ) θ M ε whenever z U 1 2M(ε+ ) Step 2: Estiate E(π(f z )) E z (π(f z )) Let F be given by (55) with R = 2C ( + θ M ε) and B = 1 + κr By Proposition 4, R 2CD(C) κ Applying Lea 19 to F with ε := D(C) + ε > 0 and α = β 8 ε/ ( ε + E(fq )) (0, 1/8], we have E(f) E z (f) 4α ε E(f) E(f q ) + ε + E(f q ), f F (59) for z U 2 where U 2 is a subset of Z with easure at least ( α ε ) 1 N F, q2 q 1 exp 3α2 ε ( β(ε + D(C)) 3/2 ) 2 q+3 1 N q2 q+3 exp Σ 3β2 (D(C) + ε) 2 2 q+9 ( + ε) In the above inequality we have used Lea 20 to bound the covering nuber For z U 1 U 2, we have 1 2C f z 2 K E z (f K,C ) + 1 2C f K,C 2 K E(f K,C ) + θ M ε + 1 2C f K,C 2 K which equals to D(C) + θ M ε + E(f q ) = + θ M ε It follows that f z K R By Lea 16, b z B This eans, π(f z ) F and (59) is valid for f = π(f z ) Step 3: Bound E(π(f z )) E(f q ) using (14) Let α and ε be the sae as in Step 2 For z U 1 U2, both E z (f K,C ) E(f K,C ) θ M ε ε and (59) with f = π(f z ) hold true Then (14) tells us that E(π(f z )) E(f q ) 4α ε (E(π(f z )) E(f q )) + ε + E(f q ) + ε Denote r := E(π(f z )) E(f q ) + ε + E(f q ) > 0, we see that It follows that Hence r ε + E(f q ) + 4α ε r + ε r 2α ε + 4α 2 ε + 2 ε + E(f q ) E(π(f z )) E(f q ) = r ( ε + E(f q )) ε + 8α 2 ε + 4α ε 4α E(f q )/ ε 20

21 SVM Soft Margin Classifiers Putting the choice of α into above, we find that β 2 ε 2 E(π(f z )) E(f q ) ε + 8 ( ε + E(f q )) + β ε ε 2 ε + E(f q ) β 2 ε 16 ( ε + E(f q )) E(f q) ε which is bounded by (1 + β) ε = (1 + β)(ε + D(C)) Finally, noting that the easure of U 1 U2 is at least 1 F (ε), the proof is finished Observe that the function F is strictly decreasing and F (0) > 1, li ε F (ε) = 0 Hence for every 0 < δ < 1 there exists a unique nuber ε > 0 satisfying F (ε) = δ Also, for a fixed ε > 0, F (ε) < δ for sufficiently large, since F (ε) can be written as a + cb with 0 < a, b < 1 Therefore, Theore 21 can be restated in the following for Corollary 22 Let F be given in Theore 21 For every 0 < δ < 1, define ε(δ,, C, β) > 0 to be the unique solution to the equation F (ε) = δ Then with confidence at least 1 δ, (20) holds Moreover, ε(δ,, C, β) = 0 li Now we can prove the stateents we ade in Section 2 for the deterinistic case Proof of Proposition 8 Take β = 1 Since R(f c ) = 0, we know that f q = f c, E(f q ) = 0 and = D(C) Then Σ 2C(D(C) + θ M ε) D(C) + ε and ( ) β(ε + D(C)) 3/2 ( ) ε N q2 q+3 N Σ q2 q+3 M The function value F (ε) can be bounded by ε 2 ( exp + N 2M(D(C) + ε) ε q2 q+3 M The first ter above is bounded by δ/2 when ε at ost δ/2 if ε ε Therefore 4M log(2/δ) ε(δ,, C, β) ax 4M log(2/δ) ) exp 3ε 2 q+9 + D(C) The second ter is + D(C), ε, and (26) follows fro Theore 21 (a) When (23) holds, noting that the left side of (27) is strictly decreasing, it is easy to check that (log + log(cd(c)) + ε 2 q c((q + 4) log 8) s log(cθm )) s + log(2/δ) This yields the stated bound for ε in the case (a) (b) If (24) is true, then the function defined by F (ε) := c (q2q+4 ) s (2CD(C)) s/2 + (2Cε) s/2 ε s 21 3ε 2 q+9

22 Chen, Wu, Ying, and Zhou satisfies F (ε ) log(δ/2) Since F is a decreasing function, ε ε whenever F ( ε ) log(δ/2) If ε r 1/(s+1) (2CD(C)) s/(2s+2) log 2 δ with r 2 q+8 κ s/(s+1) + q2 q+10 c 1/(s+1) / log 2, (60) then c(q2 q+4 ) s (2CD(C)) s/2 ( ε ) s ε rs/(s+1) 2q+8 2 q+8 (2CD(C)) s 2 2s+2 log δ c(q2 q+4 ) s 2 q+8 (r log(2/δ)) s+1 1 Since r q2 q+10 c 1/(s+1) / log 2, according to Proposition 4, this can be bounded by r 2 q+8 s/(s+1) (2CD(C)) s/(2s+2) log 2 1 r δ 2 2 q+9 κs/(s+1) log δ log δ 2 Here we have used the condition r 2 q+8 κ s/(s+1) In the sae way, if ε r 2/(s+2) (2C) s/(s+2) log 2 δ with we have for C 1/2, r 2 q+9 + q 2 4 q+5 c 2/(s+2) / log 2, (61) c(q2 q+4 ) s ( 2C ε ) s/2 ε 2 q log δ 2 Cobining the above two bounds, we obtain the desired estiate (28) with c deterined by the two conditions (60) and (61) The proof of Proposition 8 is coplete In the general case, the following bounds hold Corollary 23 For every 0 < δ 1, with confidence at least 1 δ there holds 2 (1 D(C)) q/2 q+1 + κ 2C log(2/δ) E(π(f z )) E(f q ) 2 ax, ε + 2 D(C), where ε is the solution to the equation ( ε 3/2 ) log N q4 q+2 3ε2 2C 4 q+5 = log δ 2 (62) Proof Take β = 1 and f K,C = f K,C By Lea 16, f K,C 1 + 2κ 2C D(C) and we can take M = 2 q (1 + κ 2C D(C)) q Also, = D(C) + E(f q ) 1 It follows that Σ 2C( + ε) 2C(1 + ε) Since E(π(f z )) 2 q, we only need to consider the range ε 2 q to bound ε(δ,, C, β) In this range, ( ) ( ) β(ε + D(C)) 3/2 ε 3/2 N q2 q+3 N Σ q4 q+2 2C 22

23 SVM Soft Margin Classifiers Then F (ε) can be bounded by ε 2 ( exp 4 q+1 (1 + κ 2C D(C)) + N q ε 3/2 q4 q+2 2C ) exp 3ε2 4 q+5 Thus F (ε) δ if 2 q+1 (1 + κ 2C D(C)) q/2 log(2/δ) ε ax, ε where ε is the solution to the equation (62) This together with Theore 21 yields the desired estiate The bound for the saple error derived in Corollary 23 ay be further iproved by the well developed epirical process techniques in the literature We shall discuss this elsewhere The total error (14) consists of two parts We shall not discuss the possibility of further iproving the saple error bound here, because it is of the sae iportance to understand the regularization error This becoes ore iportant when not uch estiate is available for the regularization error In the previous sections, we could copute D(C) explicitly only for special cases Most of the tie, ρ is not strictly separable, even not weakly separable Hence it is desirable to estiate D(C) explicitly for general distributions In the following we shall choose f K,C = f K,C and estiate D(C) 6 Error Analysis by Approxiation in L q Spaces The ain result on the convergence analysis given in Section 5 enables us to have soe nice observations These follow fro facts on approxiation in L q spaces Lea 24 If 1 < q 2, then (1 + u) q u q + qu, u R Proof Set the continuous function f(u) := 1 + u q + qu (1 + u) q + Then f(0) = 0 Since 0 < q 1 1, for u > 0 we have Hence f(u) 0 for u 0 For 1 < u < 0, we see that f (u) = q ( 1 + u q 1 (1 + u) q 1) 0 f (u) = q ( 1 ( u) q 1 (1 + u) q 1) = q ( 1 u q 1 (1 u ) q 1) 0 Hence f(u) 0 for 1 < u < 0 Finally, when u < 1, there holds f (u) = q ( 1 ( u) q 1) 0 23

24 Chen, Wu, Ying, and Zhou Therefore, we also have f(u) f( 1) 0 for u 1 Thus f(u) 0 on the whole real line and Lea 24 is proved Theore 25 Let f : X R be easurable Then f f q q L q, if 1 < q 2, ρ E(f) E(f q ) X ( ) q2 q 1 f f q L q ρx 2 q 1 + f f q q 1 L q, if q > 2 ρ X Proof Since f q (x) 1, by the second inequality of Lea 17, for each x X we have E(f x) E(f q x) = (1 yf(x)) q + (1 yf q(x)) q + dρ(y x) It follows that E(f) E(f q ) = X Y q4 q 1 f(x) f q (x) + q2 q 1 f(x) f q (x) q E(f x) E(f q x)dρ X q4 q 1 f f q L 1 ρx + q2 q 1 f f q q L q ρ X Then the inequality for the case q > 2 follows fro the Hölder inequality Turn to the case 1 < q 2 It is sufficient to show that for each x X, The definition (10) of f q tells us that and (1 + f q (x)) q 1 = (1 f q (x)) q 1 = These expressions in connection with (45) iply E(f x) = and together with (48) E(f x) E(f q x) f(x) f q (x) q (63) 2 q 1 (1 + f ρ (x)) (1 + f ρ (x)) 1/(q 1) + (1 f ρ (x)) 1/(q 1) q 1 2 q 1 (1 f ρ (x)) (1 + f ρ (x)) 1/(q 1) + (1 f ρ (x)) 1/(q 1) q 1 (1 + f(x))q + (1 f q(x)) q 1 (1 + f q (x)) q 1 + (1 f q (x)) q 1 + (1 f(x))q + (1 + f q(x)) q 1 (1 + f q (x)) q 1 + (1 f q (x)) q 1 E(f q x) = 2(1 f q(x)) q 1 (1 + f q (x)) q 1 (1 + f q (x)) q 1 + (1 f q (x)) q 1 Thus (63) follows fro the following inequality (by taking t = f(x) and θ = f q (x)): (1 + t) q + (1 θ)q 1 (1 + θ) q 1 + (1 θ) q 1 + (1 t)q + (1 + θ)q 1 (1 + θ) q 1 + (1 θ) q 1 2 (1 θ)q 1 (1 + θ) q 1 (1 + θ) q 1 + (1 θ) q 1 t θ q, t R, θ ( 1, 1) (64) 24

25 SVM Soft Margin Classifiers What is left is to verify the inequality (64) Since 1 < θ < 1, we have ( ) (1 + t) q 1 + t q ( + (1 θ)q 1 = (1 θ 2 ) q 1 (1 + θ) = (1 θ 2 ) q 1 (1 + θ) 1 + t θ 1 + θ 1 + θ By Lea 24 with u = (t θ)/(1 + θ), we see that ( (1 + t) q + (1 θ)q 1 (1 θ 2 ) q 1 (1 + θ) 1 + t θ 1 + θ In the sae way, by Lea 24 with u = (t θ)/(1 θ), we have ( (1 t) q + (1 + θ)q 1 (1 θ 2 ) q 1 (1 θ) 1 + t θ q 1 θ Cobining the above two estiates, we obtain + q + q t θ ) 1 + θ q t θ 1 θ (1 + t) q + (1 θ)q 1 + (1 t) q + (1 + θ)q 1 2(1 θ 2 ) q 1 + t θ q (1 θ) q 1 + (1 + θ) q 1 This proves our clai (64), thereby Theore 25 ) ) q + Recall the K-functional given by (19) Theore 26 For each C > 0, there holds D(C) K(f q, 1 2C ) Proof The case 1 < q 2 is an easy consequence of Theore 25 Turn to the case q > 2 The special choice f = H K and the fact f q L q ρx 1 tell us that for any t > 0, K(f q, t) = inf f H K f fq L q ρx 1 q2 q 1 (2 q 1 + 1) f f q L q ρx + t f 2 K According to Theore 25, for f H K with f f q L q ρx 1, we have Thus, D(C) = and the proof of Theore 26 is coplete E(f) E(f q ) q2 q 1 (2 q 1 + 1) f f q L q ρx inf E(f) E(f q ) + 1 f H K 2C f 2 1 K K(f q, 2C ) We can now derive several observations fro Theore 26 The first observation says that the SVM q-classifier converges when f q lies in the closure of H K in L q ρ X In particular, for any Borel probability easure ρ, this is always the case if K is a universal kernel since C(X) is dense in L q ρ X 25

26 Chen, Wu, Ying, and Zhou Corollary 27 If f q lies in the closure of H K in L q ρ X, then for every ε > 0 and 0 < δ < 1, there exist C ε > 0, 0 N and a sequence C with li C = such that Prob z Z R(sgn(f z )) R(f c ) ε 1 δ, 0, C ε C C Proof Since f q lies in the closure of H K in L q ρ X, for every ε > 0 there is soe f ε = fε +b ε H K such that q2 q (2 q 1 + 1) f ε f q L q ρx ε 2 /16 Take C ε = 8 fε 2 K /ε2 Then for any C C ε, we have 1 2C f ε 2 K ε2 /16 Theore 26 tells us that D(C) ε 2 /8 Take 0 such that and Also, we choose C such that 2 q+1 (1 + κ 2C ε ) q/2 log(2/δ) 0 ε2 8 ( ε 3 ) log N q4 q+4 3 0ε 4 2C ε 4 q+8 log δ 2 2 q+1 (1 + κ 2C ) q/2 log(2/δ) < ε2 8 and Then by Corollary 52, ε(δ,, C, β) ε2 8 Theore 14, our conclusion is proved ( ε 3 ) log N q4 q+4 3ε4 2C 4 q+8 log δ 2 when 0 and C ε C C Together with Our second observation fro Theore 26 concerns nonuniversal kernels which nonetheless ensures the convergence of the SVM q-classifier The point here is that H K is not dense in C(X), but after adding the offset the space H K becoes dense Exaple 2 Let K be a Mercer kernel on X = [0, 1]: K(x, y) = j J a j (x y) j, where J is a subset of N, a j > 0 for each j J, and j J a j < Note that this kernel satisfies K 0 (y) 0, hence f(0) = 0 for all f H K Hence the space H K is not dense in C(X) and K is not an universal kernel But if 1 j =, then H K is dense in C(X) (Zhou j J 2003a) and hence in L q ρ X Therefore, the SVM q-classifier associated with the (identical) kernel K converges Reark 28 In Section 4 and the proof of Theore 21, we have shown how the offset influences the saple error Proposition 4 and Exaple 2 tell that it ay also influence the approxiation error However, our analysis in the following two sections will not focus on this point and it ay be an interesting topic 26

27 SVM Soft Margin Classifiers In practical applications, one can use varying kernels for (3) Definition 29 Let K d d N be a sequence of Mercer kernels on X We say that the SVM q-classifier associated with the kernels K d converges if for a sequence C N of positive nubers, f z defined by (3) with K = K d and C = C satisfies the following: For every Borel probability easure ρ on Z, and 0 < δ < 1, ε > 0, for sufficiently large d there is soe d N such that Prob z Z R(sgn(f z )) R(f c ) ε 1 δ, d For a universal kernel K, one ay take K d to be identically K and the convergence holds But the kernels could change such as the polynoial kernels (Boser, Guyon & Vapnik 1992) Our third observation fro Theore 26 is to confir the convergence of the SVM q-classifiers with these kernels Proposition 30 For any 1 < q <, n N and X R n, the SVM q-classifier associated with K d d=1, the polynoial kernels K d(x, y) = (1 + x y) d, converges Proposition 30 is a consequence of a quantitative result below Let P d be the space of all polynoials of degree at ost d It is a RKHS H Kd with the Mercer kernel K d (x, y) = (1 + x y) d The rich knowledge fro approxiation theory tells us that for an arbitrary Borel probability easure on Z, there is a sequence of polynoials p d P d d=1 such that li f q p d L q d ρx = 0 The rate of this convergence depends on the regularity of the function f q (hence the function f ρ ) and the arginal distribution ρ X With this in hand, we can now state the result on the convergence of the q-classifier with polynoial kernels K d Corollary 31 Let X R n, and ρ be an arbitrary Borel probability easure on Z Let d N and K d (x, y) = (1 + x y) d be the polynoial kernel Set X := sup x Let x X p d P d d=1 satisfy E d := f q p d L q ρx 0 (as d ) Set N := (n + d)!/(n!d!) + 1 and 0 < σ < 2 q Then there exists q,σ N such that for q,σ and C = σ, for every 0 < δ < 1, with confidence 1 δ there holds R(sgn(f z )) R(f c ) q2 q+1 E d + p d Kd σ/2 + 2q+5 (N log ( 2 δ ))1/4 (1 + X 2 ) qd/8 1 4 qσ 8 Proof Take p d = p d with zero offset in the K-functional Then by Theore 26, D(C) K(f q, 1 2C ) q2q 1 (2 q 1 + 1)E d + 1 2C p d 2 K d The covering nubers of the finite diensional space P d (eg Cucker & Zhou 2004) and (22) give us the estiate: N (η) (κ + 1 κ ) ( 2 η + 1 ) N, where N 1 = (n + d)!/(n!d!) is the diension of the space P d (R n ) Also, κ = K d (1 + X 2 ) d/2 27

28 Chen, Wu, Ying, and Zhou Take C = σ By Corollary 23, solving the equation (62) yields ε(δ,, C, β) (2q+5 N + log(κ + 1 κ )(log + log(2/δ))1/2 + 4q+2 κ q 2 log( 2 δ ) 1 2 qσ 4 which is bounded by 4 5+q N log ( 2 δ )κq/2 qσ for q,σ Here q,σ is an integer depending on q, σ (but not on, d or δ) Then for each q,σ, with confidence 1 δ the desired estiate holds true Reark 32 Note that P d is a finite diension space Thus the nors Kd and L q ρx are equivalent for a fixed d when ρ X is non-degenerate It would be interesting to copare the nor p Kd with p L q ρx for p P d as d tends to infinity Proof of Proposition 30 For every ε > 0, there exists soe d 0 N such that q2 q 1 E d ε/2 for every d d 0 Then by Corollary 31 we can find soe q,σ d N such that σ/2 d p d Kd ε/4 and 2 q+5 (N log(2/δ)) 1/4 (1 + X 2 ) dq/8 qσ d ε/4 Then for any d we have R(sgn(f z )) R(f c ) ε with confidence 1 δ This proves Proposition 30 7 Rate of Convergence for the q-nor Soft Margin Classifier Corollary 23 and Theore 26 enable us to get the convergence rate for the SVM q-classifier The rate depends on the K-functional K(f q, t) It can be characterized by the quantity (Sale & Zhou 2003, Zhou 2003b) I q (g, R) := inf g f L qρx f H K (65) f K R Define J q (f q, R) := Then the following corollary holds true Corollary 33 For any t > 0 there holds (Iq (f q, R)) q, if 1 < q 2, q2 q 1 (2 q 1 + 1)I q (f q, R), if q > 2 K(f q, t) inf R>0 J q(f q, R) + tr 2 One ay choose appropriate R to estiate the convergence rate of K(f q, t), which together with Corollary 23 gives the convergence rate of the V -risk and a strategy of choosing the regularization paraeter C In general, the choice of R depends on the regularity of f q and the kernel K Let us deonstrate this by exaples In what follows let X R n have Lipschitz boundary and ρ be a probability easure such that dρ X = dx is the Lebesgue easure Consider q = 2 and thus f q = f ρ We use the 28

29 SVM Soft Margin Classifiers approxiation error studied in Sale & Zhou (2003) (see also Niyogi & Girosi (1996) for related discussion): I2(g, R) := inf g f f H L 2 K f K R to bound the ter I 2 (f ρ, R) With the choice b f = 0 we obtain I 2 (f ρ, R) I 2(f ρ, R) Note that we disregard the influence of the offset here and thus the rate need not be optial The first exaple includes spline kernels (Wahba 1990) Exaple 3 Let X R n and K be a Mercer kernel such that H K is the Sobolev space H r (X) with r > n/2 If f ρ lies in the Sobolev space H s (X) with 0 < s < r, then E(π(f z )) E(f ρ ) = O ( log + C ) ) + Cn/(4r+3n) 2r/(4r+3n) + O (C s/r Thus, C should be chosen such that C, C/ 0 as Proof It was shown in Sale & Zhou (2003, Theore 31) that for 0 < θ < 1, I2 (f ρ, R) = O(R θ/(1 θ) ) if and only if f ρ lies in the interpolation space (L 2 ρ X, H K ) θ, It is well known that H s (X) (L 2, H r (X)) s/r, for 0 < s < r Here H K = H r (X) and dρ X = dx Therefore, the assuption f ρ H s (X) tells us that f ρ (L 2 ρ X, H K ) s/r, Hence there holds I 2 (f ρ, R) I2(f ρ, R) C ρ R s/(r s) for soe constant C ρ Choose R = C ρ (r s)/r C (r s)/2r to obtain K(f ρ, 1 2C ) (I 2(f ρ, R)) 2 + R2 2C 3 2 C2(r s)/r ρ C s/r Using the well known covering nuber estiates for Sobolev spaces log N (B R, η) C r ( 1 η ) n/r and solving the equation (62), we see that ε 2 6 log κ + log + log C + log(2/δ) This proves the conclusion n/(4r) C r C n/(4r+3n) 2r/(4r+3n) Exaple 4 Let σ > 0, s > 0 and K be the Gaussian kernel K(x, y) = exp x y 2 σ 2 29

30 Chen, Wu, Ying, and Zhou (a) If f ρ lies in the interpolation space (L 2 ρ X, H K ) θ, for soe 0 < θ < 1, that is, K(f ρ, t) C θ t θ for soe constant C θ, then for any 0 < δ < 1, with confidence 1 δ, there holds ( ) ) C + (log ) n+1 E(π(f z )) E(f ρ ) = O + O (C θ This iplies the paraeter C should be taken to satisfy C and C/ 0 as An asyptotic optial choice is C = O( 1/(1+2θ) ) With this choice, the convergence rate is O( θ/(1+2θ) ) (b) If f ρ H s (X) with s > 0, then for any 0 < δ < 1, with confidence 1 δ, we have ( ) C + (log ) n+1 ( E(π(f z )) E(f ρ ) = O + O (log C) s/2) An asyptotically optial choice is C = O ( (log ) s ) which gives the convergence rate O((log ) s/2 ) Proof Solving the equation (62) with the covering nuber estiate (23) yields ( (1 + c)(log C + log ) ε n+1 = O ) Then the stateent in (a) follows easily fro Corollary 23, Theore 21 and Corollary 33 To see the conclusion in (b), we notice that the assuption f ρ H s (X) provides the approxiation error estiates (Sale & Zhou 2003, Zhou 2003b) I 2 (f ρ, R) I 2(f ρ, R) C s (log R) s/4 for every R > C s, where C s is a constant depending on s, σ, n and the Sobolev nor of f ρ Choose R = 2C (log C) s/4 to obtain K(f ρ, 1 2C ) (I 2(f ρ, R)) 2 + R2 2C (2s C 2 s + 1)(log C) s/2 Then the desired bound follows fro Theore 26 and the above established bound for ε It was shown in Sale & Zhou (2003) that for the Gaussian kernel in Exaple 4, I2 (g, R) = O(R ɛ ) with ɛ > 0 only if g is C Hence logarithic convergence rate is expected for a Sobolev function f ρ However, in practice, one often chooses the different variances σ of the Gaussian kernel according to the different saple size With this flexibility, the regularization error can be iproved greatly and polynoial convergence rates are possible Acknowledgents 30

31 SVM Soft Margin Classifiers The authors would like to thank Professor Zhen Wang for his helpful discussion on the best constants in Theore 14 and Theore 25 They are also grateful to the referees for the constructive suggestions and coents This work is supported partially by the Research Grants Council of Hong Kong [Project No CityU 1087/02P] and by City University of Hong Kong [Project No ] The corresponding author is Ding-Xuan Zhou Yiing Ying is on leave fro Institute of Matheatics, Chinese Acadey of Science, Beijing , P R CHINA Appendix Error Coparison for a General Loss Function In this appendix for a general convex loss function we bound the excess isclassification error by the excess V -risk Here the loss function takes the for V (y, f(x)) = φ(yf(x)) for a univariate function φ : R R + For each x X, we denote E(f x) := Y V (y, f(x))dρ(y x), (66) then E(f x) = Q(η(x), f(x)) Here Q : [0, 1] (R ± ) R + is given by Q(η, f) = ηφ(f) + (1 η)φ( f), and η : X R is defined by η(x) := P (Y = 1 x) Set f φ (η) := arg in f R ± Q(η, f) Then fρ V (x) = fφ (η(x)) The ain result of Zhang (2004) can be stated as follows Theore A Let φ be convex Assue fφ (η) > 0 when η > 05 Assue there exists c > 0 and s 1 such that for all η [0, 1], then for any easurable function f(x): 05 η s c s ( Q(η, 0) Q(η, f φ (η))), (67) R(sgn(f)) R(f c ) 2c ( E(f) E(f V ρ ) ) 1/s Further analysis was ade by Bartlett, Jordan & McAuliffe (2003) For exaple, it was proved in Bartlett, Jordan & McAuliffe (2003, Theore 6) that for a convex function φ, f φ (η) > 0 for any η > 05 if and only if φ is differentiable at 0 and φ (0) < 0 Borrowing soe ideas fro Bartlett, Jordan & McAuliffe (2003), we can derive a siple criterion for the condition (67) with s = 2 The existence of φ (0) eans that the function φ (x) is well defined in a neighborhood of 0 and is differentiable at 0 Note that the convexity of φ iplies φ (0) 0 31

32 Chen, Wu, Ying, and Zhou Theore 34 Let φ : R R + be a convex function such that φ (0) exists If φ (0) < 0 and φ (0) > 0, then (67) holds for s = 2 Hence for any easurable function f: R(sgn(f)) R(f c ) 2c E(f) E(fρ V ) Proof By the definition of φ (0), there exists soe 1/2 c 0 > 0 such that φ (f) φ (0) φ (0) f φ (0), f [ c 0, c 0 ] 2 This iplies φ (0) + φ (0)f φ (0) 2 If η > 1/2, then for 0 f c 0, f φ (f) φ (0) + φ (0)f + φ (0) f 2 Q f (η, f) = ηφ (f) (1 η)φ ( f) (2η 1)φ (0) + φ (0)f + φ (0) f 2 Thus for 0 f η := in φ (0) φ (0) (η 1 2 ), c 0, we have Q f (η, f) (2η 1)φ (0) φ (0) φ (0) φ (0) (η 1 2 ) φ (0) 2 (η 1 2 ) < 0 Therefore as a function of the variable f, Q(η, f) is strictly decreasing on the interval [0, η ] But fφ (η) > 0 is its inial point, hence Q(η, 0) Q(η, f φ (η)) Q(η, 0) Q(η, η) φ (0) 2 (η 1 2 ) η When φ (0) φ (0) (η 1 2 ) > c 0, we have η = c 0 2c 0 (η 1 2 ) Hence Q(η, 0) Q(η, f φ (η)) φ (0) 2 That is, (67) holds with s = 2 and in 2c 0, φ (0) φ (η 1 (0) 2 )2 2φ c = ax (0) φ (0), 1 φ (0)c 0 The proof for η < 1/2 is the sae by estiating the upper bound of Q f (η, f) for f < 0 Turn to the special loss function V = V q given in (6) by φ(t) = (1 t) q + Applying Theore 34, we see that the function φ satisfies φ (0) = q < 0 and φ (0) = q(q 1) > 0 This verifies (40) and the constant c can be obtained fro the proof of Theore 34 32

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