最近看Coursera的机器学习课程，第五周讲解反向传播算法（Backpropagation Algorithm）时，部分结论没有给出推导，我这里推导一下。
1、损失函数
1.1、逻辑回归的损失函数
首先复习一下之前学过的逻辑回归的损失函数：
$J( heta) = - frac{1}{m} sum_{i=1}^m [ y^{(i)} log (h_ heta (x^{(i)})) + (1 - y^{(i)}) log (1 - h_ heta(x^{(i)}))] + frac{lambda}{2m}sum_{j=1}^n heta_j^2$
其中，$m$是样本总数。逻辑回归只有一个输出，取值是1或0。
最右边的加号，将公式分为两个部分。左侧是负log损失，可由最大似然估计得到。左侧是正则项，用以降低模型的复杂度，避免过拟合。
1.2、神经网络的损失函数
神经网络损失函数：
$J(Theta) = - frac{1}{m} sum_{i=1}^m sum_{k=1}^K left[y^{(i)}_k log ((h_Theta (x^{(i)}))_k) + (1 - y^{(i)}_k)log (1 - (h_Theta(x^{(i)}))_k) ight] + frac{lambda}{2m}sum_{l=1}^{L-1} sum_{i=1}^{s_l} sum_{j=1}^{s_{l+1}} ( Theta_{j,i}^{(l)})^2$
其中，$m$依然是样本总数，$K$是输出单元的个数。前面的逻辑回归只有一个输出，而神经网络可以用$K$输出，这$K$
另外，后边部分也是正则项，只是把各层的权重（每层都是个矩阵）加了起来，显得复杂。
2、计算梯度
反向传播算法就是通过训练集，使用梯度下降算法，调整各层的权重。
2.1、符号定义
$a_i^{(j)}$：第$j$层的第$i$个激活单元（activation units）
$Theta^{(j)}$：第$j$层到$j+1$层的权重矩阵
$g(z) = dfrac{1}{1 + e^{-z}}$：激活函数
图1
如上所示的神经网络，我们的目标是求出所有的权重矩阵$Theta^{(j)}$，使得损失函数$J( heta)$最小。这可以通过梯度下降求解，首先要计算偏导数：
$dfrac{partial}{partial Theta_{i,j}^{(l)}}J(Theta)$
只给一个训练样本$(x, y)$，根据前向传播算法（forward propagation）：
egin{align*}
& a^{(1)} = x \
& z^{(2)} = Theta^{(1)}a^{(1)} \
& a^{(2)} = g(z^{(2)}) quad ext(add a_{0}^{2}) \
& z^{(3)} = Theta^{(2)}a^{(2)} \
& a^{(3)} = g(z^{(3)}) quad ext(add a_{0}^{3}) \
& z^{(4)} = Theta^{(3)}a^{(3)} \
& a^{(4)} = h_{Theta}(x) = g(z^{(4)}) \
end{align*}
此时损失函数为：
$J(Theta) = - sum_{k=1}^K left[y_k log ((h_Theta (x))_k) + (1 - y_k)log (1 - (h_Theta(x))_k) ight] + frac{lambda}{2}sum_{l=1}^{L-1} sum_{i=1}^{s_l} sum_{j=1}^{s_{l+1}} ( Theta_{j,i}^{(l)})^2$
这个公式在博客园显示的有问题，${s_l}$、$s_{l+1}$中，$l$和$l+1$都是下标。再贴上一张正确显示的图片吧：
2.2、输出层权重的调整
输出层也就是$a^{(4)}$，这一步我们要调整的权重就是$Theta^{(3)}$。图1所示的神经网络，第3层5个节点，第4层4个节点，那么$Theta^{(3)}$就是一个$4 imes (5 + 1) $(其中，+1是因为要加上偏置的权重)的矩阵，并且有：
egin{align*}
& a_1^{(4)} = g(Theta_{10}^{(3)}a_0^{(3)} + Theta_{11}^{(3)}a_1^{(3)} + Theta_{12}^{(3)}a_2^{(3)} + Theta_{13}^{(3)}a_3^{(3)} + Theta_{14}^{(3)}a_4^{(3)} + Theta_{15}^{(3)}a_5^{(3)}) \
& a_2^{(4)} = g(Theta_{20}^{(3)}a_0^{(3)} + Theta_{21}^{(3)}a_1^{(3)} + Theta_{22}^{(3)}a_2^{(3)} + Theta_{23}^{(3)}a_3^{(3)} + Theta_{24}^{(3)}a_4^{(3)} + Theta_{25}^{(3)}a_5^{(3)}) \
& a_3^{(4)} = g(Theta_{30}^{(3)}a_0^{(3)} + Theta_{31}^{(3)}a_1^{(3)} + Theta_{32}^{(3)}a_2^{(3)} + Theta_{33}^{(3)}a_3^{(3)} + Theta_{34}^{(3)}a_4^{(3)} + Theta_{35}^{(3)}a_5^{(3)}) \
& a_4^{(4)} = g(Theta_{40}^{(3)}a_0^{(3)} + Theta_{41}^{(3)}a_1^{(3)} + Theta_{42}^{(3)}a_2^{(3)} + Theta_{43}^{(3)}a_3^{(3)} + Theta_{44}^{(3)}a_4^{(3)} + Theta_{45}^{(3)}a_5^{(3)}) \
end{align*}
$J(Theta)$关于$Theta_{ij}^{(3)}$的偏导数可以表示为：
$dfrac{partial}{partial Theta_{i,j}^{(3)}}J(Theta)$
对正则项求导比较简单，我们先不考虑正则项，将$J(Theta)$带入，
egin{align*}
dfrac{partial}{partial Theta_{i,j}^{(3)}}J(Theta) &= dfrac{partial}{partial Theta_{i,j}^{(3)}}- sum_{k=1}^K left[y_k log ((h_Theta (x))_k) + (1 - y_k)log (1 - (h_Theta(x))_k)
ight] \
&= dfrac{partial}{partial Theta_{i,j}^{(3)}}- sum_{k=1}^K left[y_k log (a_{k}^{(4)}) + (1 - y_k)log (1 - a_{k}^{(4)})
ight] \
end{align*}
在上式中，我们用到了$a_1^{(4)}$到$a_4^{(4)}$4个输出值。事实上，$Theta_{i,j}$只有在生成$a_i^{(4)}$时才会用到。所以有：
egin{align*}
dfrac{partial }{partial Theta_{i,j}^{(3)}}J(Theta) &= dfrac{partial}{partial Theta_{i,j}^{(3)}}- left[y_i log (a_{i}^{(4)}) + (1 - y_i)log (1 - a_{i}^{(4)})
ight] \
&= dfrac{partial left[-y_i log (a_{i}^{(4)}) - (1 - y_i)log (1 - a_{i}^{(4)})
ight]}{partial a_{i}^{(4)}} dfrac{partial a_{i}^{(4)}}{partial Theta_{i,j}^{(3)}} \
&= left[ - y_i frac{1}{a_{i}^{(4)}} + (1 - y_i) frac{1}{1 - a_{i}^{(4)}}
ight] dfrac{partial a_{i}^{(4)}}{partial Theta_{i,j}^{(3)}} \
&= frac{a_{i}^{(4)} - y_i}{a_{i}^{(4)}(1 - a_{i}^{(4)})} dfrac{partial a_{i}^{(4)}}{partial Theta_{i,j}^{(3)}} \
&= frac{a_{i}^{(4)} - y_i}{a_{i}^{(4)}(1 - a_{i}^{(4)})} dfrac{partial g(z_i^{(4)})}{partial Theta_{i,j}^{(3)}} \
&= frac{a_{i}^{(4)} - y_i}{a_{i}^{(4)}(1 - a_{i}^{(4)})} dfrac{partial g(z_i^{(4)})}{partial z_i^{(4)}} dfrac{partial z_i^{(4)}}{partial Theta_{i,j}^{(3)}} \
end{align*}
根据$g'(z^{(l)}) = a^{(l)} .* (1 - a^{(l)})$可得：
egin{align*}
dfrac{partial }{partial Theta_{i,j}^{(3)}}J(Theta) &= frac{a_{i}^{(4)} - y_i}{a_{i}^{(4)}(1 - a_{i}^{(4)})} a_{i}^{(4)}(1 - a_{i}^{(4)}) dfrac{partial z_i^{(4)}}{partial Theta_{i,j}^{(3)}} \
&= (a_{i}^{(4)} - y_i) dfrac{partial z_i^{(4)}}{partial Theta_{i,j}^{(3)}} \
end{align*}
根据$z_i^{(4)} = Theta_{i0}^{(3)}a_0^{(3)} + Theta_{i1}^{(1)}a_1^{(3)} + Theta_{i2}^{(1)}a_2^{(3)} + Theta_{i3}^{(1)}a_3^{(3)} + Theta_{i4}^{(1)}a_4^{(3)} + Theta_{i5}^{(1)}a_5^{(3)}$，$Theta_{i,j}^{(3)}$只与$a_j^{(3)}$有关，从而有：
egin{align*}
dfrac{partial }{partial Theta_{i,j}^{(3)}}J(Theta) = (a_{i}^{(4)} - y_i)a_j^{(3)}
end{align*}
2.3、隐藏层权重的调整
这一步我们来调整$Theta^{(2)}$。图1所示的神经网络，第2层5个节点，第3层5个节点，那么$Theta^{(2)}$就是一个$5 imes (5 + 1) $(其中，+1是因为要加上偏置的权重)的矩阵，并且有：
egin{align*}
& a_1^{(3)} = g(Theta_{10}^{(2)}a_0^{(2)} + Theta_{11}^{(2)}a_1^{(2)} + Theta_{12}^{(2)}a_2^{(2)} + Theta_{13}^{(2)}a_3^{(2)} + Theta_{14}^{(2)}a_4^{(2)} + Theta_{15}^{(2)}a_5^{(2)}) \
& a_2^{(3)} = g(Theta_{20}^{(2)}a_0^{(2)} + Theta_{21}^{(2)}a_1^{(2)} + Theta_{22}^{(2)}a_2^{(2)} + Theta_{23}^{(2)}a_3^{(2)} + Theta_{24}^{(2)}a_4^{(2)} + Theta_{25}^{(2)}a_5^{(2)}) \
& a_3^{(3)} = g(Theta_{30}^{(2)}a_0^{(2)} + Theta_{31}^{(2)}a_1^{(2)} + Theta_{32}^{(2)}a_2^{(2)} + Theta_{33}^{(2)}a_3^{(2)} + Theta_{34}^{(2)}a_4^{(2)} + Theta_{35}^{(2)}a_5^{(2)}) \
& a_4^{(3)} = g(Theta_{40}^{(2)}a_0^{(2)} + Theta_{41}^{(2)}a_1^{(2)} + Theta_{42}^{(2)}a_2^{(2)} + Theta_{43}^{(2)}a_3^{(2)} + Theta_{44}^{(2)}a_4^{(2)} + Theta_{45}^{(2)}a_5^{(2)}) \
& a_5^{(3)} = g(Theta_{50}^{(2)}a_0^{(2)} + Theta_{51}^{(2)}a_1^{(2)} + Theta_{52}^{(2)}a_2^{(2)} + Theta_{53}^{(2)}a_3^{(2)} + Theta_{54}^{(2)}a_4^{(2)} + Theta_{55}^{(2)}a_5^{(2)}) \
end{align*}
同样先不考虑正则项，
egin{align*}
dfrac{partial}{partial Theta_{i,j}^{(2)}}J(Theta) &= dfrac{partial}{partial Theta_{i,j}^{(2)}}- sum_{k=1}^K left[y_k log ((h_Theta (x))_k) + (1 - y_k)log (1 - (h_Theta(x))_k)
ight] \
&= dfrac{partial}{partial Theta_{i,j}^{(2)}}- sum_{k=1}^K left[y_k log (a_{k}^{(4)}) + (1 - y_k)log (1 - a_{k}^{(4)})
ight] \
&= sum_{k=1}^K dfrac{partial}{partial Theta_{i,j}^{(2)}}- left[y_k log (a_{k}^{(4)}) + (1 - y_k)log (1 - a_{k}^{(4)})
ight] \
&= sum_{k=1}^K dfrac{partial left[-y_k log (a_{k}^{(4)}) - (1 - y_k)log (1 - a_{k}^{(4)})
ight]}{partial a_{k}^{(4)}} dfrac{partial a_{k}^{(4)}}{partial Theta_{i,j}^{(2)}} \
&= sum_{k=1}^K frac{a_{k}^{(4)} - y_k}{a_{k}^{(4)}(1 - a_{k}^{(4)})} a_{k}^{(4)}(1 - a_{k}^{(4)}) dfrac{partial z_{k}^{(4)}}{partial Theta_{i,j}^{(2)}} \
&= sum_{k=1}^K (a_{k}^{(4)} - y_k) dfrac{partial z_{k}^{(4)}}{partial Theta_{i,j}^{(2)}} \
end{align*}
接着使用复合函数的链式求导法则：
egin{align*}
dfrac{partial}{partial Theta_{i,j}^{(2)}}J(Theta) &= sum_{k=1}^K (a_{k}^{(4)} - y_k) dfrac{partial z_{k}^{(4)}}{partial a_i^{(3)}} dfrac{partial a_i^{(3)}}{partial Theta_{i,j}^{(2)}} \
&= sum_{k=1}^K (a_{k}^{(4)} - y_k) Theta_{kj}^{(3)} dfrac{partial a_i^{(3)}}{partial Theta_{i,j}^{(2)}} \
&= dfrac{partial a_i^{(3)}}{partial Theta_{i,j}^{(2)}} sum_{k=1}^K (a_{k}^{(4)} - y_k) Theta_{kj}^{(3)} \
&= a_i^{(3)}(1 - a_i^{(3)}) dfrac{partial z_i^{(3)}}{partial Theta_{i,j}^{(2)}} sum_{k=1}^K (a_{k}^{(4)} - y_k) Theta_{kj}^{(3)} \
end{align*}
根据$z_i^{(3)} = Theta_{i0}^{(2)}a_0^{(2)} + Theta_{i1}^{(2)}a_1^{(2)} + Theta_{i2}^{(2)}a_2^{(2)} + Theta_{i3}^{(2)}a_3^{(2)} + Theta_{i4}^{(2)}a_4^{(2)} + Theta_{i5}^{(2)}a_5^{(2)}$，
egin{align*}
dfrac{partial}{partial Theta_{i,j}^{(2)}}J(Theta) &= a_i^{(3)}(1 - a_i^{(3)}) a_j^{(2)} sum_{k=1}^K (a_{k}^{(4)} - y_k) Theta_{kj}^{(3)} \
end{align*}
同样的，根据链式求导法则，可以计算$dfrac{partial}{partial Theta_{i,j}^{(1)}}J(Theta)$。
2.4、结论
对于$L$层的神经网络（图1我们认为有4层），记：
$delta^{(L)} = a^{(L)} - y^{(t)}$ (注意，这是向量)
$delta^{(l)} = ((Theta^{(l)})^T delta^{(l+1)}) .* a^{(l)} .* (1 - a^{(l)})$ 对于$0 < l < L$
则：$dfrac{partial}{partial Theta_{i,j}^{(l)}}J(Theta) = a_j^{(l)} delta_i^{(l+1)}$
参考文献：
Coursera：机器学习第五周课程
码农场hancks：反向传播神经网络极简入门