11. 13
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12. 14
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§
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MS
53. 55
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29
𝑧8
0.83
data
data data
data
Forward
54. 56
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29
𝑧8
0.83
𝑧%
0.88
data
data
data data
data
Forward
55. 57
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29
𝑧8
0.83
𝑧%
0.88
𝑦
2.88
data
data
data
data data
data
Forward
56. 58
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29
𝑧8
0.83
𝑧%
0.88
𝑦
2.88 1.00
data grad
data grad
data grad
data grad
𝜕𝑦
𝜕𝑦
= 1
data grad
data grad
Backward
57. 59
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29 0.88
𝑧8
0.83
𝑧%
0.88 3.29
𝑦
2.88 1.00
data grad
data grad
data grad
data grad
𝜕𝑦
𝜕𝑦
= 1
𝜕𝑦
𝜕𝑧"
= 𝑧#
𝜕𝑦
𝜕𝑧#
= 𝑧"
data grad
data grad
= 0.88
= 3.29
Backward
58. 60
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29 0.88
𝑧8
0.83
𝑧%
0.88 3.29
𝑦
2.88 1.00
data grad
data grad
data grad
data grad
𝜕𝑦
𝜕𝑦
= 1
𝜕𝑦
𝜕𝑧"
= 𝑧#
𝜕𝑦
𝜕𝑧#
= 𝑧"
𝜕𝑧"
𝜕𝑧$
= 1
𝜕𝑧"
𝜕𝑧0
= 2 𝑧0
data grad
data grad
= 1.67
𝜕𝑦
𝜕𝑧$
=
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑦
𝜕𝑧0
=
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
= 0.88
= 3.29
Backward
59. 61
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.00
data grad
data grad
data grad
data grad
𝜕𝑦
𝜕𝑦
= 1
𝜕𝑦
𝜕𝑧"
= 𝑧#
𝜕𝑦
𝜕𝑧#
= 𝑧"
𝜕𝑧"
𝜕𝑧$
= 1
𝜕𝑧"
𝜕𝑧0
= 2 𝑧0
data grad
data grad
= 1.67
𝜕𝑦
𝜕𝑧$
=
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑦
𝜕𝑧0
=
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
= 0.88
= 3.29
Backward
60. 62
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.00
data grad
data grad
data grad
data grad
𝜕𝑦
𝜕𝑦
= 1
𝜕𝑦
𝜕𝑧"
= 𝑧#
𝜕𝑦
𝜕𝑧#
= 𝑧"
𝜕𝑧"
𝜕𝑧$
= 1
𝜕𝑧"
𝜕𝑧0
= 2 𝑧0
data grad
data grad
= 1.67
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
𝜕𝑧#
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑧$
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
𝜕𝑧0
𝜕𝑥
𝜕𝑧#
𝜕𝑥
=
3
2 𝑥
= 1.37
= 0.88
= 3.29
Backward
61. 63
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.00
data grad
data grad
data grad
data grad
𝜕𝑦
𝜕𝑦
= 1
𝜕𝑦
𝜕𝑧"
= 𝑧#
𝜕𝑦
𝜕𝑧#
= 𝑧"
𝜕𝑧"
𝜕𝑧$
= 1
𝜕𝑧"
𝜕𝑧0
= 2 𝑧0
data grad
data grad
= 1.67
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
𝜕𝑧#
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑧$
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
𝜕𝑧0
𝜕𝑥
𝜕𝑧#
𝜕𝑥
=
3
2 𝑥
= 1.37
𝜕𝑧$
𝜕𝑥
=
1
𝑥
= 0.83
= 0.88
= 3.29
Backward
63. 65
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 0.14
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.00
data grad
data grad
data grad
data grad
𝜕𝑦
𝜕𝑦
= 1
data grad
data grad
1.67
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
𝜕𝑧#
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑧$
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
𝜕𝑧0
𝜕𝑥
1.37
0.83
-0.69
0.88
3.29
1
Backward
64. 66
𝑥
1.2 0.14
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝜕𝑧#
𝜕𝑦
𝜕𝑧$
𝜕𝑦
𝜕𝑧0
𝜕𝑦
𝜕𝑧"
𝜕𝑦
𝜕𝑦
Forward Backward y
( )
65. 67
𝑧. = 3 𝑥 𝑥
1.2
data grad
𝑧.
3.29
data grad
Forward
Backward
z2.data
← 3 * sqrt(x.data)
𝑥
1.2
data grad
𝑧.
3.29
data grad
backward
x.grad
← 3/(2*sqrt(x.data)) * z2.grad
𝜕𝑧#
𝜕𝑥
=
3
2 𝑥
x.grad
← 3/(2*sqrt(x.data))
* z2.grad
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
×
𝜕𝑧#
𝜕𝑥
66. 68
𝑧. = 3 𝑥 𝑥
1.2
data grad
𝑧.
3.29
data grad
Forward
Backward
z2.data
← 3 * sqrt(x.data)
𝑥
1.2
data grad
𝑧.
3.29
data grad
backward
x.grad
← 3/(2*sqrt(x.data)) * z2.grad
𝜕𝑧#
𝜕𝑥
=
3
2 𝑥
x.grad
← 3/(2*sqrt(x.data))
* z2.grad
𝜕𝑦
𝜕𝑧#
= 0.88
1.20 0.88
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
×
𝜕𝑧#
𝜕𝑥
67. 69
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29
𝑧8
0.83
𝑧%
0.88
𝑦
2.88
data
data grad
data grad
data grad
grad data grad
data grad
( )
1. Forward ( & )
68. 70
𝜕𝑧+
𝜕𝑥
= −
1
𝑥,
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29
𝑧8
0.83
𝑧%
0.88
𝑦
2.88
data
data
data
data data
data
grad
grad
grad
grad grad
grad
⾒ Backward
⾒
𝜕𝑧-
𝜕𝑥
=
1
𝑥
𝜕𝑧,
𝜕𝑥
=
3
2 𝑥
𝜕𝑦
𝜕𝑧,
= 𝑧.
𝜕𝑧.
𝜕𝑧-
= 1
𝜕𝑧.
𝜕𝑧+
= 2 𝑧+
𝜕𝑦
𝜕𝑧.
= 𝑧,
1. Forward ( & )
69. 71
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2
𝑧7
0.18
𝑧.
3.29
𝑧8
0.83
𝑧%
0.88
𝑦
2.88
data
data
data
data data
data
grad
grad
grad
grad grad
grad
x.grad
← 3/(2*sqrt(x.data)) * z2.grad
x.grad
← (-1/x.data**2) * z4.grad
z2.grad
← z1.data* y.grad
z1.grad
← z2.data* y.grad
z3.grad
← 1.0 * z1.grad
z4.grad
← (2*z4.data)* z1.grad
x.grad
← 1/x.data * z3.grad
1. Forward ( & )
⾒ Backward
⾒
70. 72
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 0.0
𝑧7
0.18 0.0
𝑧.
3.29 0.0
𝑧8
0.83 0.0
𝑧%
0.88 0.0
𝑦
2.88 0.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
2. Backward (grad 0 )
71. 73
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 0.0
𝑧7
0.18 0.0
𝑧.
3.29 0.0
𝑧8
0.83 0.0
𝑧%
0.88 0.0
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
3. Backward ( grad 1 )
72. 74
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 0.0
𝑧7
0.18 0.0
𝑧.
3.29 0.88
𝑧8
0.83 0.0
𝑧%
0.88 3.29
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
z2.grad
← z1.data* y.grad
z1.grad
← z2.data* y.grad
4. Backward (y grad )
73. 75
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 0.0
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
z3.grad
← 1.0 * z1.grad
z4.grad
← (2*z4.data)* z1.grad
𝜕𝑦
𝜕𝑧$
=
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑦
𝜕𝑧0
=
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
5. Backward (z1 grad )
74. 76
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 1.20
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
x.grad
← 3/(2*sqrt(x.data)) * z2.grad
1.20
grad=0.0
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
𝜕𝑧#
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑧$
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
𝜕𝑧0
𝜕𝑥
6. Backward (z2 grad )
75. 77
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 3.94
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
x.grad
← 1/x.data * z3.grad
2.74
grad=1.20
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
𝜕𝑧#
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑧$
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
𝜕𝑧0
𝜕𝑥
𝜕𝑦
𝜕𝑧$
6. Backward (z3 grad )
76. 78
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 0.14
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
-3.80
grad=3.94
x.grad
← (-1/x.data**2) * z4.grad
𝜕𝑦
𝜕𝑥
=
𝜕𝑦
𝜕𝑧#
𝜕𝑧#
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧$
𝜕𝑧$
𝜕𝑥
+
𝜕𝑦
𝜕𝑧"
𝜕𝑧"
𝜕𝑧0
𝜕𝑧0
𝜕𝑥
𝜕𝑦
𝜕𝑧0
6. Backward (z4 grad )
77. 79
𝑦 = 𝑧%𝑧.
𝑧% = 𝑧8
.
+ 𝑧7
𝑧. = 3 𝑥
𝑧7 = log 𝑥
𝑧8 = 1/𝑥
𝑥
1.2 0.14
𝑧7
0.18 3.29
𝑧.
3.29 0.88
𝑧8
0.83 5.48
𝑧%
0.88 3.29
𝑦
2.88 1.0
data
data
data
data data
data
grad
grad
grad
grad grad
grad
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝜕𝑧#
𝜕𝑦
𝜕𝑧$
𝜕𝑦
𝜕𝑧0
𝜕𝑦
𝜕𝑧"
𝜕𝑦
𝜕𝑦
( )
y