知能科学:ニューラルネットワーク

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1

2 2 3 4

3 (Neural Network) (Deep Learning) (Deep Learning) (

4 x

5 x

6 = ax + b x

7 x

8 x

9 ? x

10 x

11 x w σ b = σ(wx + b) x w b w b

12 σ(x) = + e x

13 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x, x 2, x 3 w, w 2, w 3 b w, w 2, w 3 b

14 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x, x 2, x 3 w, w 2, w 3 b w, w 2, w 3 b

15 x w x2 w2 σ x3 w3 b = σ(w T x + b) x x = x 2 x 3 w = w w 2 w 3

16 ( ) x

17 (3 2 ) x x2 x3 2

18 (3 2 ) x x2 x3 2

19 (3 2 ) x x2 x3 2

20 (3 2 ) x x2 x3 2

21 (3 2 ) x x2 x3 2

22 (3 2 ) x x2 x3 2

23 x x2 x3 2, 2 [, ]

24 x w x2 w2 Σ x3 w3 b = w x + w 2 x 2 + w 3 x 3 + b x, x 2, x 3 w, w 2, w 3 b w, w 2, w 3 b

25 σ σ x σ σ x σ Σ σ σ [, ] (, )

26 x u = σ(5x + 5) σ -3 u 2 = σ( 3x + 6) u 3 = σ( 2x + 4) = σ(2u + u 2 + 4u 3 3) x

27 x Σ -3 u = σ(5x + 5) u 2 = σ( 3x + 6) u 3 = σ( 2x + 4) = 2u + u 2 + 4u x

28 x x x2 x3 2

29 = σ(wx + b) w =. b =

30 = σ(wx + b) w = 5. b =

31 = σ(wx + b) w =. b =

32 = σ(wx + b) w = 5. b =

33 = σ(wx + b) w = 5. b =

34 = σ(wx + b) w = 5. b =

35 = σ(wx + b) w = 5. b =

36 = σ(wx + b) wx + b = b/w =

37 = (-)

38 (-7) =

39 x 5 σ σ -7 Σ =

40 (-4) =

41 x 5-5 σ 4-5 σ -4 Σ =

42 =

43 σ x σ σ Σ = σ

44 x

45 x x

46 x x x x

47 (, ) [, ] x σ = = σ x

48 (, ) [, ] x x Σ -3 σ = = σ x

49 (, ) [, ] x x Σ -3 σ = = σ x x σ -3

50

51 = σ(a x + a 2 x 2 + c) σ(a x + a 2 x 2 c) a = r cos θ a 2 = r sin θ r = 2 θ = c =

52 = σ(a x + a 2 x 2 + c) σ(a x + a 2 x 2 c) a = r cos θ a 2 = r sin θ r = 2 θ = π/8 c =

53 = σ(a x + a 2 x 2 + c) σ(a x + a 2 x 2 c) a = r cos θ a 2 = r sin θ r = 2 θ = 2π/8 c =

54 = σ(a x + a 2 x 2 + c) σ(a x + a 2 x 2 c) a = r cos θ a 2 = r sin θ r = 2 θ = 3π/8 c =

55 N =

56 N =

57 N =

58 N =

59 b w b2 u w 2 x w 2 b3 u2 w 2 2 b w 3 u3 w 2 3 u = σ(w x + b ) u 2 = σ(w2 x + b 2 ) u 3 = σ(w3 x + b 3 ) = σ(w 2 u + w2 2 u 2 + w3 2 u 3 + b)

60 x b w b2 w 2 w 3 b3 u u2 u3 w 2 w 2 2 b w t E t E = ( t)2 2

61 x b w b2 w 2 w 3 b3 u u2 u3 w 2 w 2 2 b w t E E

62 f (x, ) (x, ) min f (x, ) (x n, n ) (x n+, n+ ) x n+ = x n α f x (x n, n ) n+ = n α f (x n, n ) α

63 σ(x) = + e x σ(x) = e x + e x σ (x) = ( e x ) ( + e x ) 2 = + e e x x + e x = σ(x)( σ(x))

64 x w σ b = σ( + wx + + b) x = σ ( ) w = σ( ) { σ( )} w = ( )w

65 x w σ b = σ( + wx + + b) w = σ ( ) x = ( )x b = σ ( ) = ( )

66 x w σ b = σ( + wx + + b) = ( )w x = ( )x w b = ( )

67 E w 2 = ( t) w 2 = σ( + w 2 u + ) w 2 = ( )u E w 2 = ( t)( )u

68 E b = ( t) b = σ( + b) b = ( ) E b = ( t)( )

69 E = ( t) w w w u u w = u u w = σ( + w 2 u + ) u = ( )w 2

70 u = σ( + w x + ) u w = u ( u )x E w = ( t)( )w 2 u ( u )x = E w 2 w 2 ( u )x

71 E = ( t) b b b u u = u b u b = σ( + w 2 u + ) u = ( )w 2

72 u = σ( + b) u b = u ( u ) E b = ( t)( )w 2 u ( u ) = E b w 2 u ( u )

73 E w 2 k E b = ( t)( )u k = ( t)( ) E = E w 2 wk wk 2 k ( u k )x E = E b k b w k 2 u k ( u k ) (Back Propagation)

74 w 2 k := w 2 k α E w 2 k b := b α E b w k := w k α E w k b k := b k α E b k (Back Propagation)

75 x 2 2 t α =.

76

77

78

79

80

81

82

83

84 8

85

86 %

87 (Back Propagation)

知能科学:ニューラルネットワーク

知能科学:ニューラルネットワーク 2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,

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