1. 反向传播算法介绍
误差反向传播(Error Back Propagation)算法,简称BP算法。BP算法由信号正向传播和误差反向传播组成。它的主要思想是由后一级的误差计算前一级的误差,从而极大减少运算量。
设训练数据为({m{(x^{(1)},y^{(1)}),cdots,(x^{(N)}),y^{(N)}}})共(N)个,输出为(n_L)维,即(m y^{(i)} = (y_1^{(i)},cdots,y_{n_L}^{(i)}))。
2. 信息前向传播
以第2层为例:
[z_1^{(2)} = w_{11}^{(2)} x_1 + w_{12}^{(2)} x_2 + w_{13}^{(2)} x_3 + b_1^{(2)} \
z_2^{(2)} = w_{21}^{(2)} x_1 + w_{22}^{(2)} x_2 + w_{23}^{(2)} x_3 + b_2^{(2)} \
z_3^{(2)} = w_{11}^{(2)} x_1 + w_{12}^{(2)} x_2 + w_{13}^{(2)} x_3 + b_1^{(2)} \
a_1^{(2)} = f(z_1^{(2)}) \
a_2^{(2)} = f(z_2^{(2)}) \
a_3^{(2)} = f(z_3^{(2)}) \
]
上述等式用向量化可表示为
[m z^{(2)} = m W^{(2)} cdot m a^{(1)} + b^{(2)} \
m a^{(2)} = f(m z^{(2)})
]
类似地,可归纳出
[m z^{(l)} = m W^{(l)} cdot m a^{(l)} + b^{(l)} quad (2 leq l leq L) \
m a^{(l)} = f(m z^{(l)})
]
对L层神经网络,最终输出为(m a^{(l)})。
从输入层到输出层,信息前向传播的流向为
[m {x = a^{(1)} o z^{(2)} o cdots o a^{(L-1)} o z^{(L)} o a^{(L)} = y}
]
- 误差反向传播
对单独一个训练数据((m{x^{(i)},y^{(i)}}))来说,代价函数(cost function)为
[E^{(i)} = frac 1 2 ||m{y^{(i)}-a^{(i)}}||^2 = frac 1 2 sum _{k=1}^{n_L} (y_k^{(i)}-a_k^{(i)})^2
]
为了描述方便,省去上标(^{(i)})(打公式的上标也很辛苦),将代价函数记为(E)。
总的损失函数为
[E_{total} = frac 1 N sum _{i=1}^N E^{(i)}
]
采用梯度下降法更新参数(w_{ij}^{(l)},b_i^{l}, 2 leq l leq L)。
采用梯度下降法更新参数的公式为:
[m W^{(l)} = m W^{(l)} - mu frac {partial E_{total}}{partial m W^{(l)}} = m W^{(l)} - frac mu N sum _{i=1}^N frac{partial E^{(i)}}{partial m W^{(l)}} \
m b^{(l)} = m b^{(l)} - mu frac {partial E_{total}}{partial m b^{(l)}} = m b^{(l)} - frac mu N sum _{i=1}^N frac{partial E^{(i)}}{partial m b^{(l)}}
]
3.1 输出层的权重参数更新
将(E)在隐藏层展开:
[E = frac 1 2 ||m y - m a^{(3)}|| = frac 1 2 [(y_1-a_1^{(3)})^2+(y_2-a_2^{(3)})^2]
= frac 1 2 [(y_1-f(z_1^{(3)}))^2+(y_2-f(z_2^{(3)}))^2] \
= frac 1 2 [(y_1-f(w_{11}^{(3)} a_1^{(2)} + w_{12}^{(3)} a_2^{(2)} + w_{13}^{(3)} a_3^{(2)} + b_1^{(3)}))^2+(y_2-f(w_{21}^{(3)} a_1^{(2)} + w_{22}^{(3)} a_2^{(2)} + w_{23}^{(3)} a_3^{(2)} + b_2^{(3)})))^2]
]
由求导的链式法则,对隐藏层到输出层神经元的权重参数求偏导,有:
[frac {partial E}{partial w_{11}^{(3)}} = frac 1 2 cdot 2 cdot (y_1-a_1^{(3)})(-frac{partial a_1^{(3)}}{partial w_{11}^{(3)}})=-(y_1-a_1^{(3)})f'(z_1^{(3)})a_1^{(2)}
]
记(frac{partial E}{partial z_i^{(l)}})记为(delta _i^{(l)}),即(delta _i^{(l)} = frac{partial E}{partial z_i^{(l)}}),称为误差项(灵敏度),代表该层对最终总误差的影响大小。
(frac {partial E}{partial w_{11}^{(3)}})可写为:
[frac {partial E}{partial w_{11}^{(3)}} = frac {partial E}{partial z_1^{(3)}} cdot frac {partial z_1^{(3)}}{partial w_{11}^{(3)}} = delta _1^{(3)} a_1^{(2)}
]
同理可求得
[frac {partial E}{partial w_{12}^{(3)}} = delta _1^{(3)} a_2^{(2)}, quad frac {partial E}{partial w_{13}^{(3)}} = delta _1^{(3)} a_3^{(2)}, quad frac {partial E}{partial w_{21}^{(3)}} = delta _2^{(3)} a_1^{(2)}, quad frac {partial E}{partial w_{22}^{(3)}} = delta _2^{(3)} a_2^{(2)}, quad frac {partial E}{partial w_{23}^{(3)}} = delta _2^{(3)} a_3^{(2)}
]
引入(delta _i^{(l)})一个很重要的原因是可通过(delta _{i+1}^{(l)})来求解(delta _i^{(l)}),这样可以充分利用之前计算过的结果来加快整个计算过程,这也是反向传播算法的核心思想。
推广:
[delta _i^{(L)} = -(y_i-a_i^{(L)})f'(z_i^{(L)}) (1 leq i leq n_L) \
frac {partial E}{partial w_{ij}^{(L)}} = delta _i^{(L)} cdot a_j^{(L-1)} (1 leq i leq n_L, 1 leq j leq n_{L-1})
]
表示成向量形式:
[m delta ^{(L)} = -(m y-m a^{(L)}) odot f'(m z^{(L)}) \
riangledown _{m W^{(L)}} E = m delta ^{(L)} cdot (m a^{(L-1)})^T
]
其中,(odot)表示哈达玛积(Hadamard Product)或称Element-wise Product,即2个矩阵对应位置的元素相乘。( riangledown _{m W^{(L)}} E)得到一个新的矩阵,这个矩阵中第(i)行第(j)列的元素由(E)对(m W^{(L)})中的元素(w_{ij}^{(L)})求偏导得到。
先求出最后一行的误差,再通过反向传播一层一层向前传导,更新前面层的误差值。
3.2 输出层与隐藏层的权重参数更新
[frac {partial E}{partial w_{ij}^{(l)}} = delta _i^{(l)} cdot a_j^{(l-1)}
]
其中,(delta _i^{(l)})与(m delta ^{(l+1)})(注意与下一层的所有误差项均有关,因此写成向量)的关系推导如下:
[delta _i^{(l)} = frac {partial E}{partial z_i^{(l)}} = sum _{j=1}^{n_{l+1}}frac {partial E}{partial z_j^{(l+1)}} frac{partial z_j^{(l+1)}}{partial z_i^{(l)}} = sum _{j=1}^{n_{l+1}} delta _j^{(l+1)}frac{partial z_j^{(l+1)}}{partial z_i^{(l)}} \
z_j^{(l+1)} = sum _{j=1}^{n_{l}} w_{ji}^{(l+1)} a_i^{(l)} + b_j^{(l+1)} = sum _{j=1}^{n_{l}} w_{ji}^{(l+1)} f(z_i^{(l)}) + b_j^{(l+1)}\
herefore frac {partial z_j^{(l+1)}}{partial z_i^{(l)}} = frac{partial z_j^{(l+1)}}{partial a_i^{(l)}} frac{partial a_i^{(l)}}{partial z_i^{(l)}} = w_{ji}^{(l+1)} f'(z_i^{(l)})
]
代入
[delta _i^{(l)} = sum _{j=1}^{n_{l+1}} delta _j^{(l+1)} w_{ji}^{(l+1)} f'(z_i^{(l)}) = (sum _{j=1}^{n_{l+1}} delta _j^{(l+1)} w_{ji}^{(l+1)}) cdot f'(z_i^{(l)})
]
表示成矩阵(向量)形式为:
[m delta^{(l)} = ((m W^{(l+1)})^T m delta^{(l+1)}) odot m f'(m z^{(l)})
]
(f(x))的一个重要性质就是
[f'(x) = f(x)(1-f(x))
]
3.3 输出层与隐藏层的偏执参数更新
[frac{partial E}{partial b_i^{(l)}} = frac{partial E}{partial z_i^{(l)}} frac{partial z_i^{(l)}}{partial b_i^{(l)}} = delta _i^{(l)}
]
表示成矩阵形式为:
[ riangledown _{m b^{(l)}} E = m delta^{(l)}
]
3.4 4个核心公式
[egin{aligned}
& delta _i^{(L)} = -(y_i-a_i^{(L)})f'(z_i^{(l)}) \
& delta _i^{(l)} = (sum _{j=1}^{n_{l+1}} delta _j^{(l+1)}w_{ji}^{(l+1)})f'(z_i^{(l)}) \
& frac{partial E}{partial w_{ij}^{(l)}} = delta _i^{(l)} a_j^{(l-1)} \
& frac{partial E}{partial b_i^{(l)}} = delta _i^{(l)}
end{aligned}
]
表示成矩阵形式为:
[egin{aligned}
& m delta^{(L)} = -(m y - m a^{(L)}) odot m f'(m z^{(L)}) \
& m delta^{(l)} = ((m W^{(l+1)})^T m delta^{(l+1)}) odot m f'(m z^{(l)}) \
& riangledown_{m W^{(l)}} E = frac{partial E}{partial m W^{(l)}} = m delta^{(l)}(m a^{(l-1)})^T \
& riangledown_{m b^{(l)}} E = frac{partial E}{partial m b^{(l)}} = m delta^{(l)}
end{aligned}
]