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1 I. Takeuchi, Nagoya Institute of Technology 1/38
2 f(x) = w 1 x 1 + w 2 x w d x d f(x) = α 1 K(x, x 1 ) + α 2 K(x, x 2 ) α n K(x, x n ) {wj } d j=1 f {αi } n i=1 f I. Takeuchi, Nagoya Institute of Technology 2/38
3 LASSO L 1 w := arg min w R d λ w 1 }{{} + L 1 λ > 0 n (y i f(x i )) 2 i=1 SVM α 1 := arg min α R n 2 α Kα + C n max{0, 1 y i f(x i )} }{{} i=1 C > 0 K I. Takeuchi, Nagoya Institute of Technology 3/38
4 wj α i d n wj = 0 α i = 0 / I. Takeuchi, Nagoya Institute of Technology 4/38
5 w j = 0 α i = 0 Sure independence screening (Fan et al., 2007) Shrinking option in libsvm (Fan et al., 2005) safe screening w j = 0 α i = 0 (El Ghaoui et al., 2012) (Ogawa et al., 2013) I. Takeuchi, Nagoya Institute of Technology 5/38
6 Part 1 SVM Ogawa, Suzuki, and Takeuchi. Safe screening of non-support vectors in pathwise SVM computation. ICML2013. Part 2 Nakagawa, Suzumura, Karasuyama, Tsuda, and Takeuchi. Safe feature pruning for sparse high-order interaction models. arxiv: Part 3 Okumura, Suzuki, and Takeuchi. Quick sensitivity analysis for incremental data modification. KDD2015. Shibagaki, Suzuki, Karasuyama, and Takeuchi. Regularization Path of Cross-Validation Error Lower Bounds. NIPS2015. I. Takeuchi, Nagoya Institute of Technology 6/38
7 Part 1 SVM I. Takeuchi, Nagoya Institute of Technology 7/38
8 SVM 2 1 x x 1 Before safe screening (n = 1000 and d = 2) I. Takeuchi, Nagoya Institute of Technology 8/38
9 SVM Before safe screening (n = 1000 and d = 2) I. Takeuchi, Nagoya Institute of Technology 8/38
10 SVM 2 1 x x 1 Before safe screening After safe screening (n = 1000 and d = 2) I. Takeuchi, Nagoya Institute of Technology 8/38
11 SVM Before safe screening After safe screening (n = 1000 and d = 2) I. Takeuchi, Nagoya Institute of Technology 8/38
12 SVM Before safe screening After safe screening (n = 1000 and d = 2) I. Takeuchi, Nagoya Institute of Technology 8/38
13 SVM ŷ = { 1 if f(x) < 0, +1 if f(x) 0, f(x) = w x = n αi y i K(x, x i ) i=1 {(x i, y i )} n i=1 SVM α i = 0 f SV I. Takeuchi, Nagoya Institute of Technology 9/38
14 (SVs) : y i f(x i ) < 1 α i = C : y i f(x i ) = 1 α i [0, C] (non-svs) : y i f(x i ) > 1 α i = 0 }{{} X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * X1 * w y i f(x i ) = (y i x i ) w > 1 (x i, y i ) I. Takeuchi, Nagoya Institute of Technology 10/38
15 R d w B w B := {w w m r}, mr I. Takeuchi, Nagoya Institute of Technology 11/38
16 R d w B w B := {w w m r}, mr I. Takeuchi, Nagoya Institute of Technology 11/38
17 R d w B w B := {w w m r}, mr I. Takeuchi, Nagoya Institute of Technology 11/38
18 y i f(x i ) = (y i x i ) w (y i x i ) w min w B (y ix i ) w = (y i x i ) m y i x i r, (y i x i ) w max w B (y ix i ) w = (y i x i ) m + y i x i r I. Takeuchi, Nagoya Institute of Technology 12/38
19 y i f(x i ) = (y i x i ) w (y i x i ) w min w B (y ix i ) w = (y i x i ) m y i x i r, (y i x i ) w max w B (y ix i ) w = (y i x i ) m + y i x i r I. Takeuchi, Nagoya Institute of Technology 12/38
20 y i f(x i ) = (y i x i ) w (y i x i ) w min w B (y ix i ) w = (y i x i ) m y i x i r, (y i x i ) w max w B (y ix i ) w = (y i x i ) m + y i x i r I. Takeuchi, Nagoya Institute of Technology 12/38
21 w B w B min (y ix i ) w > 1 w B }{{} >1 (y i x i ) w > 1 } {{ } >1 α i = 0 }{{} SV w B SV I. Takeuchi, Nagoya Institute of Technology 13/38
22 SVM w := arg min w R d J(w) := 1 2 w 2 + C n l i (w). i=1 l i (w) := l(y i, x i w) R d w w B := { w } w m r. l i () R d l i ( ) m := 1 n C l i (), r := 1 n C l i (). i=1 i=1 I. Takeuchi, Nagoya Institute of Technology 14/38
23 SVM w := arg min w R d J(w) := 1 2 w 2 + C n l i (w). i=1 l i (w) := l(y i, x i w) w R d w w B := { w } w m r. l i ( w) R d l i w ( ) m := 1 n w C l i ( w), r := 1 n 2 2 w + C l i ( w). i=1 i=1 I. Takeuchi, Nagoya Institute of Technology 14/38
24 SVM w := arg min w R d J(w) := 1 2 w 2 + C n l i (w). i=1 l i (w) := l(y i, x i w) w R d w w B := { w } w m r. l i ( w) R d l i w ( ) m := 1 n w C l i ( w), r := 1 n 2 2 w + C l i ( w). i=1 i=1 I. Takeuchi, Nagoya Institute of Technology 14/38
25 w B w R d w w r SV MATLAB demo I. Takeuchi, Nagoya Institute of Technology 15/38
26 Before safe screening After safe screening I. Takeuchi, Nagoya Institute of Technology 16/38
27 w C C Part 3 I. Takeuchi, Nagoya Institute of Technology 17/38
28 Data Sample Size n LIBLINEAR Sc.Rule Sc.SVM Sc.Total acoustic 78, covtype 581, yahoo 1,036, url 2,396, kdd-a 8,407, kdd-b 19,264, C 0 = ( yy K ) 1 C = C 0 /0.8 SVM I. Takeuchi, Nagoya Institute of Technology 18/38
29 C 1 < C 2 <... Data Set Kernel (γ) LIBSVM/LIBLINEAR Sc.Rule Sc.SVM Sc.Total Linear dna RBF (0.1/d) n = 2, 000 RBF (1/d) d = 180 RBF (10/d) Linear DIGIT1 RBF (0.1/d) n = 1, 500 RBF (1/d) d = 241 RBF (10/d) Linear satimage RBF (0.1/d) n = 4, 435 RBF (1/d) d = 36 RBF (10/d) Linear gisette RBF (0.1/d) n = 6, 000 RBF (1/d) d = 5, 000 RBF (10/d) Linear mushrooms RBF (0.1/d) n = 8, 124 RBF (1/d) d = 112 RBF (10/d) Linear news20 RBF (0.1/d) n = 19, 996 RBF (1/d) d = 1, 355, 191 RBF (10/d) Linear shuttle RBF (0.1/d) n = 43, 500 RBF (1/d) d = 9 RBF (10/d) I. Takeuchi, Nagoya Institute of Technology 19/38
30 Part 2 I. Takeuchi, Nagoya Institute of Technology 20/38
31 L 1 LASSO LASSO LASSO w := arg min w R d λ w 1 + n (y i w x i ) 2 i=1 ( γ 1 := arg min γ i 1 2 i) γ R n 2 λ y s.t. n x ij γ i 1, j i=1 n x ij γi < 1 w j = 0, i=1 I. Takeuchi, Nagoya Institute of Technology 21/38
32 LASSO γ R (El Ghaoui et al., 2012) (Liu et al., 2014) (Fercoq et al., 2015) γ R n max x γ R ij γ i < 1 i=1 }{{} 1 n x ij γi < 1 i=1 }{{} 1 w j = 0 }{{} γ R I. Takeuchi, Nagoya Institute of Technology 22/38
33 d {(z i, y i )} n i=1, z i [0, 1], y i R D = r ρ=1 ( d ρ) f(z i ) = w 1 z 1 + w 2 z w d z d + w 1,2 z 1 z 2 + w 1,3 z 1 z w d 1,d z d 1 z d + w 1,2,3 z 1 z 2 z 3 + w 1,2,4 z 1 z 2 z w d 2,d 1,d z d 2 z d 1 z d X R n D LASSO X := n D (main effect) (2 nd order interactions) (r th order interactions) z zd 1 z1 1 z zd 1 1 z1 d... z1 1 z1 2 z1 r... zd r+1 1 z1 d r+2 z1 d z1 n... zd n z1 n zn 2... zd 1 n zn d... z1 n zn 2 zn r... zd r+1 n zn d r+2 zn d I. Takeuchi, Nagoya Institute of Technology 23/38
34 d = 5000, r = 5 D > n max γ i x ij, j = 1,..., D. γ R i=1 I. Takeuchi, Nagoya Institute of Technology 24/38
35 Safe Pruning Rule j spr(j) n spr(j) is true x ij γi < 1 w j = 0 for all j Des(j), i=1 Des(j) j I. Takeuchi, Nagoya Institute of Technology 25/38
36 spr(z 1 ) = false, A = {z 1 } I. Takeuchi, Nagoya Institute of Technology 26/38
37 spr(z 1 z 2 ) = true, A = {z 1 } I. Takeuchi, Nagoya Institute of Technology 26/38
38 spr(z 1 z 3 ) = false, A = {z 1, z 1 z 3 } I. Takeuchi, Nagoya Institute of Technology 26/38
39 spr(z 1 z 3 z 4 ) = false, A = {z 1, z 1 z 3, z 1 z 3 z 4 } I. Takeuchi, Nagoya Institute of Technology 26/38
40 spr(z 1 z 4 ) = true, A = {z 1, z 1 z 3, z 1 z 3 z 4 } I. Takeuchi, Nagoya Institute of Technology 26/38
41 spr(z 2 ) = true, A = {z 1, z 1 z 3, z 1 z 3 z 4 } I. Takeuchi, Nagoya Institute of Technology 26/38
42 spr(z 3 ) = true, A = {z 1, z 1 z 3, z 1 z 3 z 4 } I. Takeuchi, Nagoya Institute of Technology 26/38
43 spr(z 4 ) = false, A = {z 1, z 1 z 3, z 1 z 3 z 4, z 4 } I. Takeuchi, Nagoya Institute of Technology 26/38
44 A = {z 1, z 1 z 3, z 1 z 3 z 4, z 4 } I. Takeuchi, Nagoya Institute of Technology 26/38
45 λ 1 > λ 2 >... time IB SPR log Lam/LamMax traverse 1.8e e e e+08 1e+08 8e+07 6e+07 4e+07 2e+07 IB SPR log Lam/LamMax # of feature depth 1 depth 2 depth log Lam/LamMax protein time IB SPR log Lam/LamMax traverse 2.5e+08 2e e+08 1e+08 5e+07 IB SPR log Lam/LamMax # of feature depth 1 depth 2 depth log Lam/LamMax mnist Itemset Boosting (Saigo et al., 2006) I. Takeuchi, Nagoya Institute of Technology 27/38
46 Part 3 I. Takeuchi, Nagoya Institute of Technology 28/38
47 w w B θ w { ŷ = sign(f(x)) = sign(x w +1 if x ) = w > 0, 1 if x w < 0. min w B x w > 0 ŷ = +1, max w B x w < 0 ŷ = 1. I. Takeuchi, Nagoya Institute of Technology 29/38
48 (KDD2015) I. Takeuchi, Nagoya Institute of Technology 30/38
49 (KDD2015) I. Takeuchi, Nagoya Institute of Technology 30/38
50 (KDD2015) I. Takeuchi, Nagoya Institute of Technology 30/38
51 (KDD2015) The computational cost depends only on A + R. I. Takeuchi, Nagoya Institute of Technology 30/38
52 wold w wnew kdd2010 n train = D old > 8 million and n test > 0.5 million 0.01, 0.1, 1% % of updated instances 0.01% 0.1% 1% % of label identification % % % % of speed-ups % % % 99% 1/1000 I. Takeuchi, Nagoya Institute of Technology 31/38
53 (NIPS2015) Validation Error Regularization Parameter C Model selection can be done with approximation guarantee. I. Takeuchi, Nagoya Institute of Technology 32/38
54 (NIPS2015) Validation Error Regularization Parameter C Model selection can be done with approximation guarantee. I. Takeuchi, Nagoya Institute of Technology 32/38
55 (NIPS2015) Validation Error Regularization Parameter C Model selection can be done with approximation guarantee. I. Takeuchi, Nagoya Institute of Technology 32/38
56 (NIPS2015) Validation Error Regularization Parameter C Model selection can be done with approximation guarantee. I. Takeuchi, Nagoya Institute of Technology 32/38
57 (NIPS2015) Validation Error ε Regularization Parameter C Model selection can be done with approximation guarantee. I. Takeuchi, Nagoya Institute of Technology 32/38
58 + * ' & G + * ' & E x w C C x w C MNPO M RTSUVXWZY Q [ \^] _a` Vcb Q [ed V WfY [ \ ] Nhg KLNM K PRQTSVUNWYX O Z [/\ ]R^ Ù O Zba _ U WYX Z [ \ LTc ( ) % $# "!, > 2 43 ; < ; 9 < -/ :9 ;=< IH.?0 5@7 <BADC.FEB.?0 587 JLK ( ) % $# "!, -/ :9 ;< GF = >7:9 ; <@?BA.DC@ < 587:9 ; < HJI I. Takeuchi, Nagoya Institute of Technology 33/38
59 A lower and an upper bounds of wc x i is written as LB(w C x i ) = α(ŵ C, x i ) C C (β(ŵ C, x i ) + γ(g(ŵ C), x i )), UB(w C x i ) = β(ŵ C, x i ) + C C (α(ŵ C, x i ) + δ(g(ŵ C), x i )), where, for C C, and α(ŵ C, x i ) := 1 2 ( ŵ C x i + w C x i ) 0, β(ŵ C, x i ) := 1 2 ( ŵ C x i w C x i ) 0, γ(g(ŵ C), x i ) := 1 2 ( g(ŵ C), x i ) x i + g(ŵ C )x i ) 0, δ(g(ŵ C), x i ) := 1 2 ( g(ŵ C), x i ) x i g(ŵ C )x i ) 0, where g(ŵ C), x i ) is the gradient vector of the objective function at w = ŵ C. I. Takeuchi, Nagoya Institute of Technology 34/38
60 C 6 BDCFE?HG BJIKC 6 BDCFE?HG BJIKC ) /. - (,+* ')(!"# $ &% I. Takeuchi, Nagoya Institute of Technology 35/38
61 C 6 BDCFE?HG BJIKC 6 BDCFE?HG BJIKC ) /. - (,+* ')(!"# $ &% I. Takeuchi, Nagoya Institute of Technology 35/38
62 I. Takeuchi, Nagoya Institute of Technology 36/38
63 L. El Ghaoui, V. Viallon and T. Rabbani. Safe feature elimination in sparse supervised learning. Pacific Journal of Optimization, J. Liu, Z. Zhao, J. Wang and J. Ye. Safe Screening with Variational Inequalities and Its Application to Lasso. ICML2014. J. Fan and J. Lv. Sure independence screening for ultrahigh dimensional feature space. Journal of The Royal Statistical Society B, 70:849911, R. Fan, K. Chang, C. Hsieh, X. Wang and C. Lin. LIBLINEAR: A library for large linear classification. Journal of Machine Learning Research, vol. 9, pp , H. Saigo, T. Uno and K. Tsuda. Mining complex genotypic features for predicting hiv-1 drug resistance. Bioinformatics, 24: , K. Ogawa, Y. Suzuki and I. Takeuchi. Safe screening of non-support vectors in pathwise SVM computation. ICML2013. K. Nakagawa, S. Suzumura, M. Karasuyama, K. Tsuda and I. Takeuchi. Safe feature pruning for sparse high-order interaction models. arxiv: , S. Okumura, Y. Suzuki and I. Takeuchi. Quick sensitivity analysis for incremental data modification and its application to leave-one-out CV in linear classification problems. KDD2015. A. Shibagaki, Y. Suzuki, M. Karasuyama and I. Takeuchi. Regularization path of cross-validation error lower bounds. NIPS2015. I. Takeuchi, Nagoya Institute of Technology 37/38
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