Electronic Thesis and Dissertations UC Berkeley

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1 Electronic Thesis and Dissertations UC Berkeley Peer Reviewed Title: Compilation Techniques for Embedded Data Parallel Languages Author: Catanzaro, Bryan Christopher Acceptance Date: 2011 Series: UC Berkeley Electronic Theses and Dissertations Degree: Ph.D., Electrical Engineering & Computer SciencesUC Berkeley Advisor(s): Keutzer, Kurt W. Committee: Patterson, David A., McMains, Sara Permalink: Abstract: Copyright Information: All rights reserved unless otherwise indicated. Contact the author or original publisher for any necessary permissions. escholarship is not the copyright owner for deposited works. Learn more at escholarship provides open access, scholarly publishing services to the University of California and delivers a dynamic research platform to scholars worldwide.

10 Ax A x

11 ψ = 1 ψ = 10 ψ = 100 ψ = 1000 ψ = ψ =

13 P

21 3 5 9

28 1/5 1/3

34 E x E 1,...,E n A E E 1 + E 2 E 1 < E 2 E... E 1 E 2 E 1 E 2 E 1 E p E 2 F E 1,...,E n x 1,...,x n E F, A 1,...,A n E x A E x 1,...,x n A 1,...,A n E S F A

35 S E x 1,...,x n E E f x 1,...,x n S S

36 T S P S M V M S 1 S n P V 1 V n S V T S M P V M

37 S 1, S 2 S 1 S 2 S 1 S 1 S 1,..., S n n S 1,..., S n

44 ( x i x x i ) z

45 x i x x i { x[t] i indices, t = argi indices z[i] = y[i] i / indices

46 Ax A x arg i indices y = Ax A x

47 A = y i i A ij x j j x j x

48 ˆ

54 Φ Φ Φ

57 n i i

58 m n [m, n], Unit

62 y = Ax x

65 f (x i, y i )= x i y i 2 X = {x 0, x 1,...,x n } Y = {y 0, y 1,...,y n }

67 x i y i x i y i m s ζ(s) P(k; s) = k s ζ(s), k > 0, s > 1 s s s m

68 s min max max min (ψ) ψ = W max W min W max W min ψ ψ = 1

70 m m ψm ψ ψ = 1

71 ψ = 1 ψ = 1 m > 50

72 ψ = 10 ψ = 100

73 ψ = 1000 ψ = 10000

74 ψ = m < 10 ψ = 1 ψ = m < 50 ψ = 100 ψ = 1000 ψ = ψ = ψ 1000

76 O P

77 P P { } { } G = N, E N n N n o, n h O P n o n n h n h E n i n j n i n j e E e b, e c, e d {False, True} C D e b e b e c n i e d n i C c C C = {None, Local, Global}

78 None Local local local local local Global global global D = {None, Backward, Forward} None None Backward n 1 0 Backward Forward 0 n 1 Forward Global None n n

79 (N, L,...,L) L (N,...,N) N (N, L, G) N (N, N, N) N (N, L) N (N, N, N) N (N, L) N (N, N, N) N (G, L) L (N, N) N (L, L, G) N (N, N, N) N (N, L,...,L) L (N,...,N) N (N, L, G) N (N, N, N) N (N, L) N (N, F, N) F (N, L) N (N, B, N) B (G, L) L (N, N) N (L, L, G) N (N, N, N) N

80 n n o n h n n e n n ec n ed n e c < n ec (n ed = None) (e d = n ed ) e b = True n n

81 k 2 k

93 !"#"$$$%&'(")&'*("$+&'**("$$*("$,&'("-&'***" "!./012.)"#"$'(")(",*" "!./012.%"#"$'(")("+("+("-*" "!./343"""#"$%&'(")&'("+&'(",&'("-&'*" [1, 2, 3] [1, 2, 3.0] [[1, 2], [3]] [[1, 2],3]

94 n m ψm (ψ + n 1)m ψmn ψ

95 3, 3, Unit [[1, 2, 3], [4, 5, 6], [7, 8, 9]] (3, 3) (3, 1) [1, 2, 3, 4, 5, 6, 7, 8, 9] (1, 3) [1, 4, 7, 2, 5, 8, 3, 6, 9] (1, 4) [1, 4, 7,, 2, 5, 8,, 3, 6, 9, ]

97 6.39s 0.52s 195µs 6.04s 0.49s 202µs 5.82s 0.49s 171µs 7.46s 0.63s 360µs

100 { } { }

101 A = M K K K M = 4 K = 3 K M

102

103 Ax = b A

104

105 A M 1 x 0 b n r 0 = b Ax 0 z 0 = M 1 r 0 p 0 = z 0 k = 0 rt k z k α k = pk T Ap k x k+1 = x k + α k p k r k+1 = r k α k Ap k z k+1 = M 1 r k+1 β k = zt k+1 r k+1 z T k r k p k+1 = z k+1 + β k p k k = k + 1 k > n M 1 a, b, c [ ] ai b m i = i [ ] 1 [ ] p i = m 1 ai b i = i 1 ci b = i b i c i a i c i b i b i b i a i b i c i

106 v i V v i V i v 1 n i V n i α i β i ff fi

107

108

109 A v 1 V i = 1 z i = Av i α i = z T i V i z i = z i VV T z i β i = z i V i+1 = z i /β i i = i + 1

110 T j j j α 1, α 2,...,α j β 1, β 2,...,β j 1 SΘS = T j

111 x R n y { 1, 1} x i, i {1,..., l} y i, i {1,..., l} max α F(α) = subject to l i=1 α i 1 2 αt Qα 0 α i C, i 1...l y T α = 0 x i R n i y i { 1, 1} x i α i C Q ij = y i y j Φ(x i, x j ) Φ(x i, x j ) Φ(x i, x j ; γ) =exp { γ x i x j 2} γ

112 α i x i y i i {1..l} τ α i f i = y i i {1..l} b high b low i high i low α ihigh α ilow f i i {1..l} b high i high b low i low α ihigh α ilow b low b high + 2τ b high = 1 i high = min{i : y i = 1} b low = 1 i low = min{i : y i = 1} i high i low α i low = α ilow + y ilow (b high b low )/η α i high = α ihigh + y ilow y ihigh (α ilow α i low ) η = Φ(x ihigh, x ihigh )+Φ(x ilow, x ilow ) 2Φ(x ihigh, x ilow ) α i low α i high 0 α i C f i = l j=1 α jy j Φ(x i, x j ) y i α f f i = f i +(α i high α ihigh )y ihigh Φ(x ihigh, x i ) +(α i low α ilow )y ilow Φ(x ilow, x i )

113 I high = {i :0< α i < C} {i : y i > 0, α i = 0} {i : y i < 0, α i = C} I low = {i :0< α i < C} {i : y i > 0, α i = C} {i : y i < 0, α i = 0} ɛ {i : ɛ < α i < (C ɛ)} b high = min{ f i : i I high } b low = max{ f i : i I low } b low b high + 2τ i high i low α i high = arg min{ f i : i I high } i low = arg max{ f i : i I low }

114

115

116

117

118

119

120 1 ψ ψ > % 100% 10% 33% 9

121 45 100%

122 {Distributed, Sequential}

123

124 45 100%

125

126

127

128

129

130

131

Fast Support Vector Machine Training and Classification on Graphics Processors

Fast Support Vector Machine Training and Classification on Graphics Processors Bryan Christopher Catanzaro Narayanan Sundaram Kurt Keutzer Electrical Engineering and Computer Sciences University of California