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- ありかつ ちゃわんや
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1 CV PR 1, 1, PRMU/CVIM UT / Tokyo Tech DAL PRMU/CVIM 1 / 58
2 Outline 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 2 / 58
3 Outline Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 3 / 58
4 Outline Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 4 / 58
5 Lasso 1/3 - x 1, y 1, x 2, y 2,..., x m, y m x i R n y i = w, x i + ϵ i, ϵ i N 0, σ 2. Lw = 1 2 m i=1 y i w, x i 2 = 1 2 y Xw 2. 1 m < n 2 w w 0 1: minimize w 0, subject to Lw C. 2: minimize Lw, subject to w 0 C. NP!! UT / Tokyo Tech DAL PRMU/CVIM 5 / 58
6 Lasso 1/3 - x 1, y 1, x 2, y 2,..., x m, y m x i R n y i = w, x i + ϵ i, ϵ i N 0, σ 2. Lw = 1 2 m i=1 y i w, x i 2 = 1 2 y Xw 2. 1 m < n 2 w w 0 1: minimize w 0, subject to Lw C. 2: minimize Lw, subject to w 0 C. NP!! UT / Tokyo Tech DAL PRMU/CVIM 5 / 58
7 Lasso 1/3 - x 1, y 1, x 2, y 2,..., x m, y m x i R n y i = w, x i + ϵ i, ϵ i N 0, σ 2. Lw = 1 2 m i=1 y i w, x i 2 = 1 2 y Xw 2. 1 m < n 2 w w 0 1: minimize w 0, subject to Lw C. 2: minimize Lw, subject to w 0 C. NP!! UT / Tokyo Tech DAL PRMU/CVIM 5 / 58
8 Lasso 1/3 - x 1, y 1, x 2, y 2,..., x m, y m x i R n y i = w, x i + ϵ i, ϵ i N 0, σ 2. Lw = 1 2 m i=1 y i w, x i 2 = 1 2 y Xw 2. 1 m < n 2 w w 0 1: minimize w 0, subject to Lw C. 2: minimize Lw, subject to w 0 C. NP!! UT / Tokyo Tech DAL PRMU/CVIM 5 / 58
9 Lasso 2/3 - p- p w p p = n w j p : { p 1 p < x 0.01 x 0.5 x x w 1 w 0 UT / Tokyo Tech DAL PRMU/CVIM 6 / 58
10 Lasso 2/3 - p- p w p p = n w j p : { p 1 p < x 0.01 x 0.5 x x w 1 w 0 UT / Tokyo Tech DAL PRMU/CVIM 6 / 58
11 Lasso 3/3-1: minimize w 1, subject to Lw C. 2: minimize Lw, subject to w 1 C. 3: minimize Lw + λ w [From Efron et al. 2003] UT / Tokyo Tech DAL PRMU/CVIM 7 / 58
12 - l 1-0 λ l 1 - l 1 - Wipf&Nagarajan, UT / Tokyo Tech DAL PRMU/CVIM 8 / 58
13 - l 1 - minimize w R n m i=1 log Py i x i ; w+λ w 1 Py x; w = σ y w, x 0.5 y { 1, +1} σ u 1 h u i σu = 1 1+exp u Yuan&Lin,06 minimize w R n Lw + λ g G w g 2 w G {1,..., n} w = g2 q = G. w g q w g1 UT / Tokyo Tech DAL PRMU/CVIM 9 / 58
14 - Multiple Kernel Learning, MKL: Lanckriet, Bach, et al., 04 H 1, H 2,..., H n RKHS K 1, K 2,..., K n f = f 1 }{{} minimize f j H j,b R minimize α j R m,b R + f }{{} H 1 H 2 f n Lf 1 + f f n + b + λ f l K j α j + b1 + λ α j K j = α j K j α j }{{} H n f j Hj α j K j UT / Tokyo Tech DAL PRMU/CVIM 10 / 58
15 - Multiple Kernel Learning, MKL: Lanckriet, Bach, et al., 04 H 1, H 2,..., H n RKHS K 1, K 2,..., K n f = f 1 }{{} minimize f j H j,b R minimize α j R m,b R + f }{{} H 1 H 2 f n Lf 1 + f f n + b + λ f l K j α j + b1 + λ α j K j = α j K j α j }{{} H n f j Hj α j K j UT / Tokyo Tech DAL PRMU/CVIM 10 / 58
16 - f l A 2 f Q l z = 1 2 y z 2, A = fl Q Aw = 1 y Xw 2 2 x1. x m m fl L z = log1 + exp y i z i, A = i=1 m fl L Aw = log σy i w, x i i=1 x1. x m UT / Tokyo Tech DAL PRMU/CVIM 11 / 58
17 Outline Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 12 / 58
18 - minimize w R n f l Aw + φ λ w. A R m n m: n: f l 2 φ λ w φ λ w = λ w 1 ηφ λ = φ ηλ f l LARS No Free Lunch m n A UT / Tokyo Tech DAL PRMU/CVIM 13 / 58
19 - : φ λ w FOCUSS Rao & Kreutz-Delgado, 99 Majorization-Minimization Figueiredo et al., 07 Sub-gradient L-BFGS Andrew & Gao, 07; Yu et al., 08 : A UT / Tokyo Tech DAL PRMU/CVIM 14 / 58
20 - : A A = I n 1 min w R n 2 y w λ w 1 = 1 min w j R 2 y j w j 2 + λ w j. w j = ST λ y j y j λ λ y j, = 0 λ y j λ, y j + λ y j λ. λ λ φ λ UT / Tokyo Tech DAL PRMU/CVIM 15 / 58
21 - Iterative Shrinkage/Thresholding IST Figueiredo&Nowak, 03; Daubechies et al., 04,...: 1 w 1 2 w t+1 argmin w R n Qηt w; w t + φ λ w Q η w; w t = Lw t + L w t w w t + 1 }{{} 2η 1 1 w w t 2 2 }{{} 2 2. Q η w; w t η UT / Tokyo Tech DAL PRMU/CVIM 16 / 58
22 - IST argmin Qηt w; w t + φ λ w w R n = argmin w R n const. + L w t w w t = argmin w w t 2 2 w R n 2η + φ λw =: ST ηt λ w t t }{{} w t = w t η t Lw t w t+1 ST ηt λ w t η t Lw t }{{}}{{} A w w t 2 2 2η + φ λw t UT / Tokyo Tech DAL PRMU/CVIM 17 / 58
23 - IST argmin Qηt w; w t + φ λ w w R n = argmin w R n const. + L w t w w t = argmin w w t 2 2 w R n 2η + φ λw =: ST ηt λ w t t }{{} w t = w t η t Lw t w t+1 ST ηt λ w t η t Lw t }{{}}{{} A w w t 2 2 2η + φ λw t UT / Tokyo Tech DAL PRMU/CVIM 17 / 58
24 - IST argmin Qηt w; w t + φ λ w w R n = argmin w R n const. + L w t w w t = argmin w w t 2 2 w R n 2η + φ λw =: ST ηt λ w t t }{{} w t = w t η t Lw t w t+1 ST ηt λ w t η t Lw t }{{}}{{} A w w t 2 2 2η + φ λw t UT / Tokyo Tech DAL PRMU/CVIM 17 / 58
25 - IST argmin Qηt w; w t + φ λ w w R n = argmin w R n const. + L w t w w t = argmin w w t 2 2 w R n 2η + φ λw =: ST ηt λ w t t }{{} w t = w t η t Lw t w t+1 ST ηt λ w t η t Lw t }{{}}{{} A w w t 2 2 2η + φ λw t UT / Tokyo Tech DAL PRMU/CVIM 17 / 58
26 - minimize w R n f l Aw + φ λ w. f l 2 φ λ w φ λ w = λ w 1 1 ST λ z = argmin w R n 2 w z φ λw STz λ λ z φ λ ST A UT / Tokyo Tech DAL PRMU/CVIM 18 / 58
27 Outline Dual Augmented Lagrangian 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 19 / 58
28 Outline Dual Augmented Lagrangian Proximal minimization 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 20 / 58
29 Dual Augmented Lagrangian Proximal minimization Proximal Minimization Rockafellar, w 1 2 w t+1 argmin w R n f l Aw }{{} +φ λ w + 1 2η t w w t 2 2 }{{} 2 f η w=min w R n f l A w + φ λ w + 1 2η w w 2 2 1: f η w f w = f l Aw + φ λ w 2: f η w = f w fw f η w λ 0 λ UT / Tokyo Tech DAL PRMU/CVIM 21 / 58
30 Dual Augmented Lagrangian Proximal minimization IST 1 w 1 2 w t+1 ST ηt λ w t + η t A f l Aw t Dual Augmented Lagrangian 1 w 1 2 w t+1 ST ηt λ w t + η t A α t α t = argmin α R m fl α + 1 ST ηt λw t + η t A α 2 2 2η t UT / Tokyo Tech DAL PRMU/CVIM 22 / 58
31 Dual Augmented Lagrangian Proximal minimization IST 2 2 DAL UT / Tokyo Tech DAL PRMU/CVIM 23 / 58
32 Dual Augmented Lagrangian Proximal minimization Dual Augmented Lagrangian 1 w 1 2 w t+1 ST ηt λ w t + η t A α t α t = argmin α R m Legendre fl } α + 1 ST {{} ηt λw t + η t A α 2 2 2η t f l f y = sup x y x f x UT / Tokyo Tech DAL PRMU/CVIM 24 / 58
33 Outline Dual Augmented Lagrangian Legendre 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 25 / 58
34 Legendre Dual Augmented Lagrangian Legendre f x f y f y = sup y x f x x f x y -f*y UT / Tokyo Tech DAL PRMU/CVIM 26 / 58
35 Legendre Dual Augmented Lagrangian Legendre f x f y f y = sup y x f x x f x -f*y y UT / Tokyo Tech DAL PRMU/CVIM 26 / 58
36 Legendre Dual Augmented Lagrangian Legendre f x f y f y = sup y x f x x f x -f*y y UT / Tokyo Tech DAL PRMU/CVIM 26 / 58
37 Legendre Dual Augmented Lagrangian Legendre f x f y f y = sup y x f x x f x -f*y y UT / Tokyo Tech DAL PRMU/CVIM 26 / 58
38 Legendre Dual Augmented Lagrangian Legendre f x f y f y = sup y x f x x -f*y y 1 2 f x = x 2 2σ 2 f y = sup xy x 2 x 2σ 2 = σ 2 yy σ2 y 2 2σ 2 = σ2 y 2 2 fx f*y UT / Tokyo Tech DAL PRMU/CVIM 27 / 58
39 Dual Augmented Lagrangian Legendre Legendre f y fx f*y f y xy f x }{{} f f x = f x f x xy f y UT / Tokyo Tech DAL PRMU/CVIM 28 / 58
40 Dual Augmented Lagrangian Legendre Legendre : f y = sup x y x f x 2 f x = log1+ exp x f y = sup xy log1 + exp x x = log 1 y y y y log1 + 1 y = y logy + 1 y log1 y fx f*y 0 1 UT / Tokyo Tech DAL PRMU/CVIM 29 / 58
41 Dual Augmented Lagrangian Legendre Legendre : f y = sup x y x f x 3 l 1 : f x = x f y = sup xy x = x { 0 y 1, + otherwise. fx f*y UT / Tokyo Tech DAL PRMU/CVIM 30 / 58
42 Dual Augmented Lagrangian Legendre f l Aw + φ λ w = max w v A α f v R n,α R m l α + φ λ }{{ v } w max w v A α f v R n,α R m l α φ λ v = max w A α f α R m l α + max w v φ v R n λ v = f l Aw + φ λ w UT / Tokyo Tech DAL PRMU/CVIM 31 / 58
43 Dual Augmented Lagrangian Legendre Proximal minimization w t+1 argmin w R n { = argmin w R n f l Aw + φ λ w + 1 2η t w w t 2 2 max f v R n,α R m l α φ λ v + w v A α min max } + 1 w w t 2 2 2η t UT / Tokyo Tech DAL PRMU/CVIM 32 / 58
44 Dual Augmented Lagrangian Legendre Proximal minimization w t+1 argmin w R n { = argmin w R n f l Aw + φ λ w + 1 2η t w w t 2 2 max f v R n,α R m l α φ λ v + w v A α min max } + 1 w w t 2 2 2η t UT / Tokyo Tech DAL PRMU/CVIM 32 / 58
45 Dual Augmented Lagrangian Legendre Proximal minimization w t+1 argmin w R n { = argmin w R n f l Aw + φ λ w + 1 2η t w w t 2 2 max f v R n,α R m l α φ λ v + w v A α min max } + 1 w w t 2 2 2η t UT / Tokyo Tech DAL PRMU/CVIM 32 / 58
46 Dual Augmented Lagrangian Legendre Dual Augmented Lagrangian 1 w t+1 ST ηt λ w t + A α t λ λ UT / Tokyo Tech DAL PRMU/CVIM 33 / 58
47 Dual Augmented Lagrangian Legendre Dual Augmented Lagrangian 1 w t+1 ST ηt λ w t + A α t 2 α t = argmin fl α + 1 ST ηt λw t + η t A α 2 2 α R m 2η t φ λ w Φ λ w λ 0 λ UT / Tokyo Tech DAL PRMU/CVIM 34 / 58
48 Dual Augmented Lagrangian Legendre Dual Augmented Lagrangian 1 w t+1 ST ηt λ w t + A α t 2 α t = argmin α R m f l α }{{} A + 1 ST ηt λw t + η t A α 2 2 2η } t {{} α : AST η t λw t + η t A α : η t A + A + 3 A α 2 A + A ; l 1 - UT / Tokyo Tech DAL PRMU/CVIM 35 / 58
49 Outline Dual Augmented Lagrangian 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 36 / 58
50 Dual Augmented Lagrangian LASSO 2 l 1 l1_ls SpaRSA IST A R m n m: n: A=randnm,n; A=U*diag1./1:m*V ; 2 n = 4m, n < m = 1024, n < 1e + 6 UT / Tokyo Tech DAL PRMU/CVIM 37 / 58
51 Dual Augmented Lagrangian CPU time secs #iterations Sparsity % DALchol η 1 =1000 DALcg η 1 =1000 SpaRSA l1_ls Number of observations a Normal conditioning CPU time secs #iterations Sparsity % DALchol η 1 = DALcg η 1 = SpaRSA l1_ls Number of samples b Poor conditioning UT / Tokyo Tech DAL PRMU/CVIM 38 / 58
52 Dual Augmented Lagrangian CPU time secs #iterations Sparsity % Number of unknown variables DALchol η 1 =1000 DALcg η 1 =1000 SpaRSA l1_ls UT / Tokyo Tech DAL PRMU/CVIM 39 / 58
53 Outline Dual Augmented Lagrangian 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 40 / 58
54 Dual Augmented Lagrangian minimize f H,b R,d R n m ξ i + λ 2 f 2 Hd i=1 subject to y i f x i + b 1 ξ i i = 1,..., m K d = K j, 0, 1. i=1 j UT / Tokyo Tech DAL PRMU/CVIM 41 / 58
55 Dual Augmented Lagrangian minimize α R m,b R,d R n m ξ i + λ 2 α K dα i=1 subject to y i K dα i + b 1 ξ i i = 1,..., m K d = K j, 0, 1. i=1 j max0, 1 yf UT / Tokyo Tech DAL PRMU/CVIM 42 / 58
56 Dual Augmented Lagrangian minimize α R m,b R,d R n subject to K d = LK dα + b1 + λ 2 α K dα K j, 0, i=1 1. j max0, 1 yf UT / Tokyo Tech DAL PRMU/CVIM 43 / 58
57 Dual Augmented Lagrangian minimize α R m,b R,γ R n subject to K d = LK dα + b1 + λ 2 α K dα K j, 0, i=1 1. α j j = 1,..., n j α K dα = min α j R m α j K j α j subject to K j α j = K dα β 1 α j K j α j + β K dα 2 α j = β β = α K j α j UT / Tokyo Tech DAL PRMU/CVIM 44 / 58
58 Dual Augmented Lagrangian minimize α R m,b R,γ R n subject to K d = LK dα + b1 + λ 2 α K dα K j, 0, i=1 1. α j j = 1,..., n j α K dα = min α j R m α j K j α j subject to K j α j = K dα β 1 α j K j α j + β K dα 2 α j = β β = α K j α j UT / Tokyo Tech DAL PRMU/CVIM 44 / 58
59 Dual Augmented Lagrangian minimize α R m,b R,γ R n subject to K d = LK dα + b1 + λ 2 α K dα K j, 0, i=1 1. α j j = 1,..., n j α K dα = min α j R m α j K j α j subject to K j α j = K dα β 1 α j K j α j + β K dα 2 α j = β β = α K j α j UT / Tokyo Tech DAL PRMU/CVIM 44 / 58
60 Dual Augmented Lagrangian L K j α j + b1 + λ 2 subject to 0, 1. minimize α j R m,b R,γ R n α j K j α j = α j K j α j d 2 j α j K j 2 = α j K j j = α j K j α j αj K j 2 2 j = 1 Jensen UT / Tokyo Tech DAL PRMU/CVIM 45 / 58
61 Dual Augmented Lagrangian L K j α j + b1 + λ 2 subject to 0, 1. minimize α j R m,b R,γ R n α j K j α j = α j K j α j d 2 j α j K j 2 = α j K j j = α j K j α j αj K j 2 2 j = 1 Jensen UT / Tokyo Tech DAL PRMU/CVIM 45 / 58
62 Dual Augmented Lagrangian L K j α j + b1 + λ 2 subject to 0, 1. minimize α j R m,b R,γ R n α j K j α j = α j K j α j d 2 j α j K j 2 = α j K j j = α j K j α j αj K j 2 2 j = 1 Jensen UT / Tokyo Tech DAL PRMU/CVIM 45 / 58
63 Dual Augmented Lagrangian L K j α j + b1 + λ 2 subject to 0, 1. minimize α j R m,b R,γ R n α j K j α j = α j K j α j d 2 j α j K j 2 = α j K j j = α j K j α j αj K j 2 2 j = 1 Jensen UT / Tokyo Tech DAL PRMU/CVIM 45 / 58
64 Dual Augmented Lagrangian L K j α j + b1 + λ 2 subject to 0, 1. minimize α j R m,b R,γ R n α j K j α j = α j K j α j d 2 j α j K j 2 = α j K j j = α j K j α j αj K j 2 2 j = 1 Jensen UT / Tokyo Tech DAL PRMU/CVIM 45 / 58
65 Dual Augmented Lagrangian L K j α j + b1 + λ 2 subject to 0, 1. minimize α j R m,b R,γ R n α j K j α j = α j K j α j d 2 j α j K j 2 = α j K j j = α j K j α j αj K j 2 2 j = 1 Jensen UT / Tokyo Tech DAL PRMU/CVIM 45 / 58
66 Dual Augmented Lagrangian 2 2 minimize L K j α j + b1 + λ 2 α j K α j R n,b R 2 j minimize L K j α j + b1 + λ α j K α j R n j,b R A B A αj L + λ α j K j αj α j K j 0 B αj L + λ αj α j K j 0 UT / Tokyo Tech DAL PRMU/CVIM 46 / 58
67 SpicyMKL Dual Augmented Lagrangian DAL + MKL = SpicyMKL Sparse Iterative MKL DAL. Soft-thresholding 0 ST λ α j = α j K j λ αj α j K j λ otherwise α j K j UT / Tokyo Tech DAL PRMU/CVIM 47 / 58
68 Outline 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 48 / 58
69 2 Demo1 True + Noise True + Estimated x128 σ = 5 [ x x, s t a t ] = d a l s q l 1 z e r o s m * n, 1, H, Y :, l a m b d a, ' e t a ', 5 0 0, ' s o l v e r ', ' c g ' ; { { 乗ロス + L 1 正則化初期値 ペナルティーの強さ畳み込み行列入力画像正則化定数 の初期値 インナーループの最適化に C G 法を使う UT / Tokyo Tech DAL PRMU/CVIM 49 / 58
70 Demo2 β t=0, 3, 6, 9, 12, 18, MKL UT / Tokyo Tech DAL PRMU/CVIM 50 / 58
71 Demo : minimize w R 3 70,b R m 70 l L w, x i + b + λ w j 2 Soft-threholding: ST λ w j = max0, w j 2 λ w j 2 [ w w, b b, s t a t ] = d a l l r g l z e r o s n s, n c, F :, :, Y :, l a m b d a ; i=1 w j 重み バイアス ロジスティック損失 + グループラッソー正則化 初期値 n s = 3, n c = 7 0 特徴量ラベル正則化定数 UT / Tokyo Tech DAL PRMU/CVIM 51 / 58
72 Demo2.2 MKL 0 Baranzini K x i, x j = 1 + x i x j 2 o p t = s t r u c t ' l o s s ', ' l o g i t ' ; ロジスティック損失を指定 [ a l p h a, d, b, a c t s e t ] = S p i c y M K L K, Y, l a m b d a, o p t ; サンプル重み カーネル重み バイアスアクティブセット カーネル m x m x n ラベル 正則化定数 UT / Tokyo Tech DAL PRMU/CVIM 52 / 58
73 Demo3 Caltech101 Fei-Fei et al., 2004 anchor, ant, cannon, chair, cup ,760 = : van de Sande hsvsift, sift sift 4px, sift 8px visual words spatial pyramid 22 : χ 2 10 UT / Tokyo Tech DAL PRMU/CVIM 53 / 58
74 Outline 1-2 Dual Augmented Lagrangian Proximal minimization Legendre 3 4 UT / Tokyo Tech DAL PRMU/CVIM 54 / 58
75 l 1 - DAL Legendre Legendre MKL UT / Tokyo Tech DAL PRMU/CVIM 55 / 58
76 w t+1 argmin f w + 1 w 2η t w w t 2 2 f w t η t w t+1 w t 0 w t+1 w t η t f w t+1 }{{} η=0 w t+1 = w t 1 + η t η=1 η=2 η=100 UT / Tokyo Tech DAL PRMU/CVIM 56 / 58
77 Convolution Inf-convolution: f gx = inf f x y + gy y Legendre f g α = f α + g α f g α = sup αx inff x y + gx x y = sup x sup αx f x y gy y = f α + sup αy gy y = f α + g α UT / Tokyo Tech DAL PRMU/CVIM 57 / 58
78 Proximity Operation f x f y = sup x x y f x Proximity operator 1 prox f z = inf x 2 z x f x Moreau s decomposition Moreau, 65; Combettes&Wajs, 05 prox f z + prox f z = z UT / Tokyo Tech DAL PRMU/CVIM 58 / 58
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