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学习笔记:神经网络反向推导

2018-05-23
AI

笔记说明

学习文章:深度学习—反向传播的具体案例

简单网络描述如下:

反向传播过程

首先正常传播,计算出总误差;反向传播就是为了计算总误差与待更新值得导数(权值)。而导数其实是反映权值对总误差的影响(变化率)。

1. 总误差公式
\[E_{total} = \frac{1}{2} [(0.01-out_{o1})^2 +(0.99 - out_{o2})^2] _{(0.01和0.99是预期值)}\]
2. 总误差对w5求导
\[\frac{\Delta E_{total}}{\Delta w5} = \frac{\Delta E_{total}}{\Delta out_{o1}} \times \frac{\Delta out_{o1}}{\Delta net_{o1}} \times \frac{\Delta net_{o1}}{\Delta w5}\]
  • 总误差对out_o1求导:
\[\frac{\Delta E_{total}}{\Delta out_{o1}} = 2*\frac{1}{2} *(0.01 - out_{o1}) *(-1) = out_{o1} - 0.01\]
  • out_o1对net_o1求导:
\[\frac{\Delta out_{o1}}{\Delta net_{o1}} = out_{o1} \times (1 - out_{o1})\]
  • net_o1对w5求导:
\[\frac{\Delta net_{o1}}{\Delta w5} = out_{h1}\]
  • 根据以上过程得出:
\[\frac{\Delta E_{total}}{\Delta w5}\]
3. 更新权值w5
\[w5_{new} = w5 - 0.5 \times \frac{\Delta E_{total}}{\Delta w5} _{(假如学习率为0.5 )}\]

以此类推,更新所有权值。之后重复。

注意:总误差对out_h1求导为
\(\frac{\Delta E_{total}}{\Delta out_{h1}} = \frac{\Delta E_{total}}{\Delta net_{o1}} \times \frac{\Delta net_{o1}}{\Delta out_{h1}} + \frac{\Delta E_{total}}{\Delta net_{o2}} \times \frac{\Delta net_{o2}}{\Delta out_{h1}}\)

草稿求导过程

ΔE/Δout_o1 = (target-out_o1)*(-1) = R1
ΔE/Δout_o2 = (target-out_o2)*(-1) = R2
Δout_o1/Δnet_o1 = out_o1*(1-out_o1) = R3
Δout_o2/Δnet_o2 = out_o2*(1-out_o2) = R4
Δnet_o1/Δw5 = out_h1 = R5
Δnet_o1/Δw6 = out_h2 = R6
Δnet_o2/Δw7 = out_h1 = R7
Δnet_o2/Δw8 = out_h2 = R8
=> ΔE/Δw5 = R1 * R3 * R5
=> ΔE/Δw6 = R1 * R3 * R6
=> ΔE/Δw7 = R2 * R4 * R7
=> ΔE/Δw8 = R2 * R4 * R8

Δnet_o1/Δout_h1 = w5 = R9
Δnet_o2/Δout_h1 = w7 = R10
Δnet_o1/Δout_h2 = w6 = R11
Δnet_o2/Δout_h2 = w8 = R12
=> ΔE/Δout_h1 = R1 * R3 * R9 + R2 * R4 * R10 = R13
=> ΔE/Δout_h2 = R1 * R3 * R11 + R2 * R4 * R12 = R14

Δout_h1/Δnet_h1 = out_h1*(1-out_h1) = R15
Δout_h2/Δnet_h2 = out_h2*(1-out_h2) = R16
Δnet_h1/Δw1 = i1 = R17
Δnet_h1/Δw2 = i2 = R18
Δnet_h2/Δw3 = i1 = R19
Δnet_h2/Δw4 = i2 = R20
=> ΔE/Δw1 = R13 * R15 * R17
=> ΔE/Δw2 = R13 * R15 * R18
=> ΔE/Δw3 = R14 * R16 * R19
=> ΔE/Δw4 = R14 * R16 * R20

RUBY 实现

Neuron.rb


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