Support Vector Machine Soft Margin Classifiers: Error Analysis


 Nigel Gibson
 2 years ago
 Views:
Transcription
1 Journal of Machine Learning Research? (2004)??? Subitted 9/03; Published??/04 Support Vector Machine Soft Margin Classifiers: Error Analysis DiRong Chen Departent of Applied Matheatics Beijing University of Aeronautics and Astronautics Beijing , P R CHINA Qiang Wu Yiing Ying DingXuan Zhou Departent of Matheatics City University of Hong Kong Kowloon, Hong Kong, P R CHINA Editor: Peter Bartlett Abstract The purpose of this paper is to provide a PAC error analysis for the qnor 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 Kfunctional 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, qnor soft argin classifier, regularization error, approxiation error 1 Introduction In this paper we study support vector achine (SVM) classification algoriths and investigate the SVM qnor 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 DiRong Chen, Qiang Wu, Yiing Ying, and DingXuan 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 qnor 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 qnor soft argin classifier (SVM qclassifier) 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 Kfunctional 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 (ShaweTaylor 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 Kfunctional 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, ShaweTaylor 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 leaveoneout 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 leaveoneout technique was applied to iprove the saple error estiates given in Bousquet & Ellisseeff (2002): the saple error has a kernelindependent 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, ShaweTaylor et al 1998, Steinwart 2001, Cristianini & ShaweTaylor 2000) of strictly separable distributions with argin γ > 0 In this case, ShaweTaylor 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 qclassifier 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 1nor 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 1nor 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 qnor 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 qnor Soft Margin Classifier In this section, we apply the ERM technique to analyze the convergence of the qclassifier 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 oneside 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 oneside 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
Machine Learning Applications in Grid Computing
Machine Learning Applications in Grid Coputing George Cybenko, Guofei Jiang and Daniel Bilar Thayer School of Engineering Dartouth College Hanover, NH 03755, USA gvc@dartouth.edu, guofei.jiang@dartouth.edu
More informationReliability Constrained Packetsizing for Linear Multihop Wireless Networks
Reliability Constrained acketsizing for inear Multihop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois
More informationMINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS
MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Abstract. We show that for sufficiently large n, every 3unifor hypergraph on n vertices with iniu
More informationarxiv:0805.1434v1 [math.pr] 9 May 2008
Degreedistribution stability of scalefree networs Zhenting Hou, Xiangxing Kong, Dinghua Shi,2, and Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of
More informationThis paper studies a rental firm that offers reusable products to price and qualityofservice sensitive
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol., No. 3, Suer 28, pp. 429 447 issn 523464 eissn 5265498 8 3 429 infors doi.287/so.7.8 28 INFORMS INFORMS holds copyright to this article and distributed
More informationData Set Generation for Rectangular Placement Problems
Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George
More informationON SELFROUTING IN CLOS CONNECTION NETWORKS. BARRY G. DOUGLASS Electrical Engineering Department Texas A&M University College Station, TX 778433128
ON SELFROUTING IN CLOS CONNECTION NETWORKS BARRY G. DOUGLASS Electrical Engineering Departent Texas A&M University College Station, TX 7788 A. YAVUZ ORUÇ Electrical Engineering Departent and Institute
More informationApplying Multiple Neural Networks on Large Scale Data
0 International Conference on Inforation and Electronics Engineering IPCSIT vol6 (0) (0) IACSIT Press, Singapore Applying Multiple Neural Networks on Large Scale Data Kritsanatt Boonkiatpong and Sukree
More informationTrading Regret for Efficiency: Online Convex Optimization with Long Term Constraints
Journal of Machine Learning Research 13 2012) 25032528 Subitted 8/11; Revised 3/12; Published 9/12 rading Regret for Efficiency: Online Convex Optiization with Long er Constraints Mehrdad Mahdavi Rong
More informationOnline Bagging and Boosting
Abstract Bagging and boosting are two of the ost wellknown enseble learning ethods due to their theoretical perforance guarantees and strong experiental results. However, these algoriths have been used
More informationOSCILLATION OF DIFFERENCE EQUATIONS WITH SEVERAL POSITIVE AND NEGATIVE COEFFICIENTS
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Hasan Öğünez and Özkan Öcalan OSCILLATION OF DIFFERENCE EQUATIONS WITH SEVERAL POSITIVE AND NEGATIVE COEFFICIENTS Abstract. Our ai in this paper is to
More informationAnalyzing Spatiotemporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy
Vol. 9, No. 5 (2016), pp.303312 http://dx.doi.org/10.14257/ijgdc.2016.9.5.26 Analyzing Spatioteporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy Chen Yang, Renjie Zhou
More informationBinary Embedding: Fundamental Limits and Fast Algorithm
Binary Ebedding: Fundaental Liits and Fast Algorith Xinyang Yi The University of Texas at Austin yixy@utexas.edu Eric Price The University of Texas at Austin ecprice@cs.utexas.edu Constantine Caraanis
More informationMultiClass Deep Boosting
MultiClass Deep Boosting Vitaly Kuznetsov Courant Institute 25 Mercer Street New York, NY 002 vitaly@cis.nyu.edu Mehryar Mohri Courant Institute & Google Research 25 Mercer Street New York, NY 002 ohri@cis.nyu.edu
More informationFactored Models for Probabilistic Modal Logic
Proceedings of the TwentyThird AAAI Conference on Artificial Intelligence (2008 Factored Models for Probabilistic Modal Logic Afsaneh Shirazi and Eyal Air Coputer Science Departent, University of Illinois
More informationExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona Network
2013 European Control Conference (ECC) July 1719, 2013, Zürich, Switzerland. ExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona
More informationPERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO
Bulletin of the Transilvania University of Braşov Series I: Engineering Sciences Vol. 4 (53) No.  0 PERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO V. CAZACU I. SZÉKELY F. SANDU 3 T. BĂLAN Abstract:
More informationUse of extrapolation to forecast the working capital in the mechanical engineering companies
ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2014. Vol. 1. No. 1. 23 28 Use of extrapolation to forecast the working capital in the echanical engineering copanies A. Cherep, Y. Shvets Departent of finance
More informationAlgorithmica 2001 SpringerVerlag New York Inc.
Algorithica 2001) 30: 101 139 DOI: 101007/s0045300100030 Algorithica 2001 SpringerVerlag New York Inc Optial Search and OneWay Trading Online Algoriths R ElYaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin
More informationInformation Processing Letters
Inforation Processing Letters 111 2011) 178 183 Contents lists available at ScienceDirect Inforation Processing Letters www.elsevier.co/locate/ipl Offline file assignents for online load balancing Paul
More information2. FINDING A SOLUTION
The 7 th Balan Conference on Operational Research BACOR 5 Constanta, May 5, Roania OPTIMAL TIME AND SPACE COMPLEXITY ALGORITHM FOR CONSTRUCTION OF ALL BINARY TREES FROM PREORDER AND POSTORDER TRAVERSALS
More informationModels and Algorithms for Stochastic Online Scheduling 1
Models and Algoriths for Stochastic Online Scheduling 1 Nicole Megow Technische Universität Berlin, Institut für Matheatik, Strasse des 17. Juni 136, 10623 Berlin, Gerany. eail: negow@ath.tuberlin.de
More informationCRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS
641 CRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS Marketa Zajarosova 1* *Ph.D. VSB  Technical University of Ostrava, THE CZECH REPUBLIC arketa.zajarosova@vsb.cz Abstract Custoer relationship
More information6. Time (or Space) Series Analysis
ATM 55 otes: Tie Series Analysis  Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction,
More informationStochastic Online Scheduling on Parallel Machines
Stochastic Online Scheduling on Parallel Machines Nicole Megow 1, Marc Uetz 2, and Tark Vredeveld 3 1 Technische Universit at Berlin, Institut f ur Matheatik, Strasse des 17. Juni 136, 10623 Berlin, Gerany
More informationFactor Model. Arbitrage Pricing Theory. Systematic Versus NonSystematic Risk. Intuitive Argument
Ross [1],[]) presents the aritrage pricing theory. The idea is that the structure of asset returns leads naturally to a odel of risk preia, for otherwise there would exist an opportunity for aritrage profit.
More informationPricing Asian Options using Monte Carlo Methods
U.U.D.M. Project Report 9:7 Pricing Asian Options using Monte Carlo Methods Hongbin Zhang Exaensarbete i ateatik, 3 hp Handledare och exainator: Johan Tysk Juni 9 Departent of Matheatics Uppsala University
More informationExact Matrix Completion via Convex Optimization
Exact Matrix Copletion via Convex Optiization Eanuel J. Candès and Benjain Recht Applied and Coputational Matheatics, Caltech, Pasadena, CA 91125 Center for the Matheatics of Inforation, Caltech, Pasadena,
More informationMedia Adaptation Framework in Biofeedback System for Stroke Patient Rehabilitation
Media Adaptation Fraework in Biofeedback Syste for Stroke Patient Rehabilitation Yinpeng Chen, Weiwei Xu, Hari Sundara, Thanassis Rikakis, ShengMin Liu Arts, Media and Engineering Progra Arizona State
More informationSTRONGLY CONSISTENT ESTIMATES FOR FINITES MIX'IURES OF DISTRIBUTION FUNCTIONS ABSTRACT. An estimator for the mixing measure
STRONGLY CONSISTENT ESTIMATES FOR FINITES MIX'IURES OF DISTRIBUTION FUNCTIONS Keewhan Choi Cornell University ABSTRACT The probles with which we are concerned in this note are those of identifiability
More informationData Streaming Algorithms for Estimating Entropy of Network Traffic
Data Streaing Algoriths for Estiating Entropy of Network Traffic Ashwin Lall University of Rochester Vyas Sekar Carnegie Mellon University Mitsunori Ogihara University of Rochester Jun (Ji) Xu Georgia
More informationSearching strategy for multitarget discovery in wireless networks
Searching strategy for ultitarget discovery in wireless networks Zhao Cheng, Wendi B. Heinzelan Departent of Electrical and Coputer Engineering University of Rochester Rochester, NY 467 (585) 75{878,
More informationCapacity of MultipleAntenna Systems With Both Receiver and Transmitter Channel State Information
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 2697 Capacity of MultipleAntenna Systes With Both Receiver and Transitter Channel State Inforation Sudharan K. Jayaweera, Student Meber,
More informationQuality evaluation of the modelbased forecasts of implied volatility index
Quality evaluation of the odelbased forecasts of iplied volatility index Katarzyna Łęczycka 1 Abstract Influence of volatility on financial arket forecasts is very high. It appears as a specific factor
More informationBayes Point Machines
Journal of Machine Learning Research (2) 245 279 Subitted 2/; Published 8/ Bayes Point Machines Ralf Herbrich Microsoft Research, St George House, Guildhall Street, CB2 3NH Cabridge, United Kingdo Thore
More informationPartitioned EliasFano Indexes
Partitioned Eliasano Indexes Giuseppe Ottaviano ISTICNR, Pisa giuseppe.ottaviano@isti.cnr.it Rossano Venturini Dept. of Coputer Science, University of Pisa rossano@di.unipi.it ABSTRACT The Eliasano
More informationAUC Optimization vs. Error Rate Minimization
AUC Optiization vs. Error Rate Miniization Corinna Cortes and Mehryar Mohri AT&T Labs Research 180 Park Avenue, Florha Park, NJ 0793, USA {corinna, ohri}@research.att.co Abstract The area under an ROC
More informationAN ALGORITHM FOR REDUCING THE DIMENSION AND SIZE OF A SAMPLE FOR DATA EXPLORATION PROCEDURES
Int. J. Appl. Math. Coput. Sci., 2014, Vol. 24, No. 1, 133 149 DOI: 10.2478/acs20140011 AN ALGORITHM FOR REDUCING THE DIMENSION AND SIZE OF A SAMPLE FOR DATA EXPLORATION PROCEDURES PIOTR KULCZYCKI,,
More informationPreferencebased Search and Multicriteria Optimization
Fro: AAAI02 Proceedings. Copyright 2002, AAAI (www.aaai.org). All rights reserved. Preferencebased Search and Multicriteria Optiization Ulrich Junker ILOG 1681, route des Dolines F06560 Valbonne ujunker@ilog.fr
More informationImage restoration for a rectangular poorpixels detector
Iage restoration for a rectangular poorpixels detector Pengcheng Wen 1, Xiangjun Wang 1, Hong Wei 2 1 State Key Laboratory of Precision Measuring Technology and Instruents, Tianjin University, China 2
More informationAn Innovate Dynamic Load Balancing Algorithm Based on Task
An Innovate Dynaic Load Balancing Algorith Based on Task Classification Hongbin Wang,,a, Zhiyi Fang, b, Guannan Qu,*,c, Xiaodan Ren,d College of Coputer Science and Technology, Jilin University, Changchun
More informationComment on On Discriminative vs. Generative Classifiers: A Comparison of Logistic Regression and Naive Bayes
Coent on On Discriinative vs. Generative Classifiers: A Coparison of Logistic Regression and Naive Bayes JingHao Xue (jinghao@stats.gla.ac.uk) and D. Michael Titterington (ike@stats.gla.ac.uk) Departent
More informationABSTRACT KEYWORDS. Comonotonicity, dependence, correlation, concordance, copula, multivariate. 1. INTRODUCTION
MEASURING COMONOTONICITY IN MDIMENSIONAL VECTORS BY INGE KOCH AND ANN DE SCHEPPER ABSTRACT In this contribution, a new easure of coonotonicity for diensional vectors is introduced, with values between
More informationSoftware Quality Characteristics Tested For Mobile Application Development
Thesis no: MGSE201502 Software Quality Characteristics Tested For Mobile Application Developent Literature Review and Epirical Survey WALEED ANWAR Faculty of Coputing Blekinge Institute of Technology
More informationCrossDomain Metric Learning Based on Information Theory
Proceedings of the TwentyEighth AAAI Conference on Artificial Intelligence CrossDoain Metric Learning Based on Inforation Theory Hao Wang,2, Wei Wang 2,3, Chen Zhang 2, Fanjiang Xu 2. State Key Laboratory
More informationEfficient Key Management for Secure Group Communications with Bursty Behavior
Efficient Key Manageent for Secure Group Counications with Bursty Behavior Xukai Zou, Byrav Raaurthy Departent of Coputer Science and Engineering University of NebraskaLincoln Lincoln, NE68588, USA Eail:
More informationASIC Design Project Management Supported by Multi Agent Simulation
ASIC Design Project Manageent Supported by Multi Agent Siulation Jana Blaschke, Christian Sebeke, Wolfgang Rosenstiel Abstract The coplexity of Application Specific Integrated Circuits (ASICs) is continuously
More informationFast large scale Gaussian process regression using approximate matrixvector products
Fast large scale Gaussian process regression using approxiate atrixvector products Vikas C. Raykar and Raani Duraiswai Departent of Coputer Science and Institute for advanced coputer studies University
More informationModeling operational risk data reported above a timevarying threshold
Modeling operational risk data reported above a tievarying threshold Pavel V. Shevchenko CSIRO Matheatical and Inforation Sciences, Sydney, Locked bag 7, North Ryde, NSW, 670, Australia. eail: Pavel.Shevchenko@csiro.au
More informationCooperative Caching for Adaptive Bit Rate Streaming in Content Delivery Networks
Cooperative Caching for Adaptive Bit Rate Streaing in Content Delivery Networs Phuong Luu Vo Departent of Coputer Science and Engineering, International University  VNUHCM, Vietna vtlphuong@hciu.edu.vn
More informationPartitioning Data on Features or Samples in CommunicationEfficient Distributed Optimization?
Partitioning Data on Features or Saples in CounicationEfficient Distributed Optiization? Chenxin Ma Industrial and Systes Engineering Lehigh University, USA ch54@lehigh.edu Martin Taáč Industrial and
More informationAn Integrated Approach for Monitoring Service Level Parameters of SoftwareDefined Networking
International Journal of Future Generation Counication and Networking Vol. 8, No. 6 (15), pp. 1974 http://d.doi.org/1.1457/ijfgcn.15.8.6.19 An Integrated Approach for Monitoring Service Level Paraeters
More informationPosition Auctions and Nonuniform Conversion Rates
Position Auctions and Nonunifor Conversion Rates Liad Blurosen Microsoft Research Mountain View, CA 944 liadbl@icrosoft.co Jason D. Hartline Shuzhen Nong Electrical Engineering and Microsoft AdCenter
More informationReal Time Target Tracking with Binary Sensor Networks and Parallel Computing
Real Tie Target Tracking with Binary Sensor Networks and Parallel Coputing Hong Lin, John Rushing, Sara J. Graves, Steve Tanner, and Evans Criswell Abstract A parallel real tie data fusion and target tracking
More informationAirline Yield Management with Overbooking, Cancellations, and NoShows JANAKIRAM SUBRAMANIAN
Airline Yield Manageent with Overbooking, Cancellations, and NoShows JANAKIRAM SUBRAMANIAN Integral Developent Corporation, 301 University Avenue, Suite 200, Palo Alto, California 94301 SHALER STIDHAM
More informationRECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION. Henrik Kure
RECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION Henrik Kure Dina, Danish Inforatics Network In the Agricultural Sciences Royal Veterinary and Agricultural University
More informationThe Research of Measuring Approach and Energy Efficiency for Hadoop Periodic Jobs
Send Orders for Reprints to reprints@benthascience.ae 206 The Open Fuels & Energy Science Journal, 2015, 8, 206210 Open Access The Research of Measuring Approach and Energy Efficiency for Hadoop Periodic
More informationStable Learning in Coding Space for MultiClass Decoding and Its Extension for MultiClass Hypothesis Transfer Learning
Stable Learning in Coding Space for MultiClass Decoding and Its Extension for MultiClass Hypothesis Transfer Learning Bang Zhang, Yi Wang 2, Yang Wang, Fang Chen 2 National ICT Australia 2 School of
More informationINTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE SYSTEMS
Artificial Intelligence Methods and Techniques for Business and Engineering Applications 210 INTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE
More informationSAMPLING METHODS LEARNING OBJECTIVES
6 SAMPLING METHODS 6 Using Statistics 66 2 Nonprobability Sapling and Bias 66 Stratified Rando Sapling 62 6 4 Cluster Sapling 64 6 5 Systeatic Sapling 69 6 6 Nonresponse 62 6 7 Suary and Review of
More informationLecture L263D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L63D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More informationPerformance Evaluation of Machine Learning Techniques using Software Cost Drivers
Perforance Evaluation of Machine Learning Techniques using Software Cost Drivers Manas Gaur Departent of Coputer Engineering, Delhi Technological University Delhi, India ABSTRACT There is a treendous rise
More informationEvaluating Software Quality of Vendors using Fuzzy Analytic Hierarchy Process
IMECS 2008 92 March 2008 Hong Kong Evaluating Software Quality of Vendors using Fuzzy Analytic Hierarchy Process Kevin K.F. Yuen* Henry C.W. au Abstract This paper proposes a fuzzy Analytic Hierarchy
More informationSupport Vector Machines with Clustering for Training with Very Large Datasets
Support Vector Machines with Clustering for Training with Very Large Datasets Theodoros Evgeniou Technology Management INSEAD Bd de Constance, Fontainebleau 77300, France theodoros.evgeniou@insead.fr Massimiliano
More informationFuzzy Sets in HR Management
Acta Polytechnica Hungarica Vol. 8, No. 3, 2011 Fuzzy Sets in HR Manageent Blanka Zeková AXIOM SW, s.r.o., 760 01 Zlín, Czech Republic blanka.zekova@sezna.cz Jana Talašová Faculty of Science, Palacký Univerzity,
More informationModified Latin Hypercube Sampling Monte Carlo (MLHSMC) Estimation for Average Quality Index
Analog Integrated Circuits and Signal Processing, vol. 9, no., April 999. Abstract Modified Latin Hypercube Sapling Monte Carlo (MLHSMC) Estiation for Average Quality Index Mansour Keraat and Richard Kielbasa
More informationModel Selection and Stability in kmeans Clustering
Model Selection and Stability in keans Clustering Ohad Shair and Naftali Tishby School of Coputer Science and Engineering Interdisciplinary Center for Neural Coputation The Hebrew University, Jerusale
More informationDensity Level Detection is Classification
Density Level Detection is Classification Ingo Steinwart, Don Hush and Clint Scovel Modeling, Algorithms and Informatics Group, CCS3 Los Alamos National Laboratory {ingo,dhush,jcs}@lanl.gov Abstract We
More informationOnline Appendix I: A Model of Household Bargaining with Violence. In this appendix I develop a simple model of household bargaining that
Online Appendix I: A Model of Household Bargaining ith Violence In this appendix I develop a siple odel of household bargaining that incorporates violence and shos under hat assuptions an increase in oen
More informationOn Computing Nearest Neighbors with Applications to Decoding of Binary Linear Codes
On Coputing Nearest Neighbors with Applications to Decoding of Binary Linear Codes Alexander May and Ilya Ozerov Horst Görtz Institute for ITSecurity RuhrUniversity Bochu, Gerany Faculty of Matheatics
More informationThe Velocities of Gas Molecules
he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes
More informationEvaluating the Effectiveness of Task Overlapping as a Risk Response Strategy in Engineering Projects
Evaluating the Effectiveness of Task Overlapping as a Risk Response Strategy in Engineering Projects Lucas Grèze Robert Pellerin Nathalie Perrier Patrice Leclaire February 2011 CIRRELT201111 Bureaux
More informationMarkovian inventory policy with application to the paper industry
Coputers and Cheical Engineering 26 (2002) 1399 1413 www.elsevier.co/locate/copcheeng Markovian inventory policy with application to the paper industry K. Karen Yin a, *, Hu Liu a,1, Neil E. Johnson b,2
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Abstract. We copare the option pricing forulas of Louis Bachelier and BlackMertonScholes
More informationResearch Article Performance Evaluation of Human Resource Outsourcing in Food Processing Enterprises
Advance Journal of Food Science and Technology 9(2): 964969, 205 ISSN: 20424868; eissn: 20424876 205 Maxwell Scientific Publication Corp. Subitted: August 0, 205 Accepted: Septeber 3, 205 Published:
More informationOptimal ResourceConstraint Project Scheduling with Overlapping Modes
Optial ResourceConstraint Proect Scheduling with Overlapping Modes François Berthaut Lucas Grèze Robert Pellerin Nathalie Perrier Adnène Hai February 20 CIRRELT2009 Bureaux de Montréal : Bureaux de
More informationEquivalent Tapped Delay Line Channel Responses with Reduced Taps
Equivalent Tapped Delay Line Channel Responses with Reduced Taps Shweta Sagari, Wade Trappe, Larry Greenstein {shsagari, trappe, ljg}@winlab.rutgers.edu WINLAB, Rutgers University, North Brunswick, NJ
More informationExample: Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationA Comparative Study of Parametric and Nonparametric Regressions
Iranian Econoic Review, Vol.16, No.3, Fall 11 A Coparative Study of Paraetric and Nonparaetric Regressions Shahra Fattahi Received: 1/8/8 Accepted: 11/4/4 Abstract his paper evaluates inflation forecasts
More informationThe Application of Bandwidth Optimization Technique in SLA Negotiation Process
The Application of Bandwidth Optiization Technique in SLA egotiation Process Srecko Krile University of Dubrovnik Departent of Electrical Engineering and Coputing Cira Carica 4, 20000 Dubrovnik, Croatia
More informationEnergy Efficient VM Scheduling for Cloud Data Centers: Exact allocation and migration algorithms
Energy Efficient VM Scheduling for Cloud Data Centers: Exact allocation and igration algoriths Chaia Ghribi, Makhlouf Hadji and Djaal Zeghlache Institut MinesTéléco, Téléco SudParis UMR CNRS 5157 9, Rue
More informationAn Optimal Task Allocation Model for System Cost Analysis in Heterogeneous Distributed Computing Systems: A Heuristic Approach
An Optial Tas Allocation Model for Syste Cost Analysis in Heterogeneous Distributed Coputing Systes: A Heuristic Approach P. K. Yadav Central Building Research Institute, Rooree 247667, Uttarahand (INDIA)
More informationAn improved TFIDF approach for text classification *
Zhang et al. / J Zheiang Univ SCI 2005 6A(1:4955 49 Journal of Zheiang University SCIECE ISS 10093095 http://www.zu.edu.cn/zus Eail: zus@zu.edu.cn An iproved TFIDF approach for text classification
More informationA framework for performance monitoring, load balancing, adaptive timeouts and quality of service in digital libraries
Int J Digit Libr (2000) 3: 9 35 INTERNATIONAL JOURNAL ON Digital Libraries SpringerVerlag 2000 A fraework for perforance onitoring, load balancing, adaptive tieouts and quality of service in digital libraries
More informationAn Approach to Combating Freeriding in PeertoPeer Networks
An Approach to Cobating Freeriding in PeertoPeer Networks Victor Ponce, Jie Wu, and Xiuqi Li Departent of Coputer Science and Engineering Florida Atlantic University Boca Raton, FL 33431 April 7, 2008
More informationSOME APPLICATIONS OF FORECASTING Prof. Thomas B. Fomby Department of Economics Southern Methodist University May 2008
SOME APPLCATONS OF FORECASTNG Prof. Thoas B. Foby Departent of Econoics Southern Methodist University May 8 To deonstrate the usefulness of forecasting ethods this note discusses four applications of forecasting
More informationESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET
ESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET Francisco Alonso, Roberto Blanco, Ana del Río and Alicia Sanchis Banco de España Banco de España Servicio de Estudios Docuento de
More informationA Fast Algorithm for Online Placement and Reorganization of Replicated Data
A Fast Algorith for Online Placeent and Reorganization of Replicated Data R. J. Honicky Storage Systes Research Center University of California, Santa Cruz Ethan L. Miller Storage Systes Research Center
More informationExploiting Hardware Heterogeneity within the Same Instance Type of Amazon EC2
Exploiting Hardware Heterogeneity within the Sae Instance Type of Aazon EC2 Zhonghong Ou, Hao Zhuang, Jukka K. Nurinen, Antti YläJääski, Pan Hui Aalto University, Finland; Deutsch Teleko Laboratories,
More informationOnline Learning, Stability, and Stochastic Gradient Descent
Online Learning, Stability, and Stochastic Gradient Descent arxiv:1105.4701v3 [cs.lg] 8 Sep 2011 September 9, 2011 Tomaso Poggio, Stephen Voinea, Lorenzo Rosasco CBCL, McGovern Institute, CSAIL, Brain
More informationManaging Complex Network Operation with Predictive Analytics
Managing Coplex Network Operation with Predictive Analytics Zhenyu Huang, Pak Chung Wong, Patrick Mackey, Yousu Chen, Jian Ma, Kevin Schneider, and Frank L. Greitzer Pacific Northwest National Laboratory
More informationCLOSEDLOOP SUPPLY CHAIN NETWORK OPTIMIZATION FOR HONG KONG CARTRIDGE RECYCLING INDUSTRY
CLOSEDLOOP SUPPLY CHAIN NETWORK OPTIMIZATION FOR HONG KONG CARTRIDGE RECYCLING INDUSTRY Y. T. Chen Departent of Industrial and Systes Engineering Hong Kong Polytechnic University, Hong Kong yongtong.chen@connect.polyu.hk
More informationA Scalable Application Placement Controller for Enterprise Data Centers
W WWW 7 / Track: Perforance and Scalability A Scalable Application Placeent Controller for Enterprise Data Centers Chunqiang Tang, Malgorzata Steinder, Michael Spreitzer, and Giovanni Pacifici IBM T.J.
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationLearning, Regularization and IllPosed Inverse Problems
Learning, Regularization and IllPosed Inverse Problems Lorenzo Rosasco DISI, Università di Genova rosasco@disi.unige.it Andrea Caponnetto DISI, Università di Genova caponnetto@disi.unige.it Ernesto De
More informationGenerating Certification Authority Authenticated Public Keys in Ad Hoc Networks
SECURITY AND COMMUNICATION NETWORKS Published online in Wiley InterScience (www.interscience.wiley.co). Generating Certification Authority Authenticated Public Keys in Ad Hoc Networks G. Kounga 1, C. J.
More informationConsiderations on Distributed Load Balancing for Fully Heterogeneous Machines: Two Particular Cases
Considerations on Distributed Load Balancing for Fully Heterogeneous Machines: Two Particular Cases Nathanaël Cheriere Departent of Coputer Science ENS Rennes Rennes, France nathanael.cheriere@ensrennes.fr
More informationCalculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance
Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 453167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21
More informationDesign of Model Reference Self Tuning Mechanism for PID like Fuzzy Controller
Research Article International Journal of Current Engineering and Technology EISSN 77 46, PISSN 347 56 4 INPRESSCO, All Rights Reserved Available at http://inpressco.co/category/ijcet Design of Model Reference
More informationModeling Cooperative Gene Regulation Using Fast Orthogonal Search
8 The Open Bioinforatics Journal, 28, 2, 889 Open Access odeling Cooperative Gene Regulation Using Fast Orthogonal Search Ian inz* and ichael J. Korenberg* Departent of Electrical and Coputer Engineering,
More information