简单起见，我们考虑一个无偏差项的循环神经网络 ϕ ( x ) = x phi(x)=x ϕ ( x ) = x t t t x t ∈ R d boldsymbol{x}_t in mathbb{R}^d x t ∈ R d y t y_t y t h t ∈ R h boldsymbol{h}_t in mathbb{R}^h h t ∈ R h
h t = W h x x t + W h h h t − 1 , boldsymbol{h}_t = boldsymbol{W}_{hx} boldsymbol{x}_t + boldsymbol{W}_{hh} boldsymbol{h}_{t-1}, h t = W h x x t + W h h h t − 1 ,
其中W h x ∈ R h × d boldsymbol{W}_{hx} in mathbb{R}^{h times d} W h x ∈ R h × d W h h ∈ R h × h boldsymbol{W}_{hh} in mathbb{R}^{h times h} W h h ∈ R h × h W q h ∈ R q × h boldsymbol{W}_{qh} in mathbb{R}^{q times h} W q h ∈ R q × h t t t o t ∈ R q boldsymbol{o}_t in mathbb{R}^q o t ∈ R q
o t = W q h h t . boldsymbol{o}_t = boldsymbol{W}_{qh} boldsymbol{h}_{t}. o t = W q h h t .
设时间步t t t ℓ ( o t , y t ) ell(boldsymbol{o}_t, y_t) ℓ ( o t , y t ) T T T L L L
L = 1 T ∑ t = 1 T ℓ ( o t , y t ) . L = frac{1}{T} sum_{t=1}^T ell (boldsymbol{o}_t, y_t). L = T 1 t = 1 ∑ T ℓ ( o t , y t ) .
我们将L L L
为了可视化循环神经网络中模型变量和参数在计算中的依赖关系，我们可以绘制模型计算图，如图所示。例如，时间步3的隐藏状态h 3 boldsymbol{h}_3 h 3 W h x boldsymbol{W}_{hx} W h x W h h boldsymbol{W}_{hh} W h h h 2 boldsymbol{h}_2 h 2 x 3 boldsymbol{x}_3 x 3
刚刚提到，图6.3中的模型的参数是 W h x boldsymbol{W}_{hx} W h x W h h boldsymbol{W}_{hh} W h h W q h boldsymbol{W}_{qh} W q h ∂ L ∂ W h x frac{partial L}{partial boldsymbol{W}_{hx}} ∂ W h x ∂ L ∂ L ∂ W h h frac{partial L}{partial boldsymbol{W}_{hh}} ∂ W h h ∂ L ∂ L ∂ W q h frac{partial L}{partial boldsymbol{W}_{qh}} ∂ W q h ∂ L
首先，目标函数有关各时间步输出层变量的梯度∂ L ∂ o t ∈ R q frac{partial L}{partial boldsymbol{o}_t} in mathbb{R}^q ∂ o t ∂ L ∈ R q
∂ L ∂ o t = ∂ ℓ ( o t , y t ) T ⋅ ∂ o t . frac{partial L}{partial boldsymbol{o}_t} = frac{partial ell (boldsymbol{o}_t, y_t)}{T cdot partial boldsymbol{o}_t}. ∂ o t ∂ L = T ⋅ ∂ o t ∂ ℓ ( o t , y t ) .
下面，我们可以计算目标函数有关模型参数W q h boldsymbol{W}_{qh} W q h ∂ L ∂ W q h ∈ R q × h frac{partial L}{partial boldsymbol{W}_{qh}} in mathbb{R}^{q times h} ∂ W q h ∂ L ∈ R q × h L L L o 1 , … , o T boldsymbol{o}_1, ldots, boldsymbol{o}_T o 1 , … , o T W q h boldsymbol{W}_{qh} W q h
∂ L ∂ W q h = ∑ t = 1 T prod ( ∂ L ∂ o t , ∂ o t ∂ W q h ) = ∑ t = 1 T ∂ L ∂ o t h t ⊤ . frac{partial L}{partial boldsymbol{W}{qh}} = sum_{t=1}^T text{prod}left(frac{partial L}{partial boldsymbol{o}_t}, frac{partial boldsymbol{o}_t}{partial boldsymbol{W}_{qh}}right) = sum_{t=1}^T frac{partial L}{partial boldsymbol{o}_t} boldsymbol{h}_t^top. ∂ W q h ∂ L = t = 1 ∑ T prod ( ∂ o t ∂ L , ∂ W q h ∂ o t ) = t = 1 ∑ T ∂ o t ∂ L h t ⊤ .
其次，我们注意到隐藏状态之间也存在依赖关系。 在计算图中，L L L o T boldsymbol{o}_T o T T T T h T boldsymbol{h}_T h T ∂ L ∂ h T ∈ R h frac{partial L}{partial boldsymbol{h}_T} in mathbb{R}^h ∂ h T ∂ L ∈ R h
∂ L ∂ h T = prod ( ∂ L ∂ o T , ∂ o T ∂ h T ) = W q h ⊤ ∂ L ∂ o T . frac{partial L}{partial boldsymbol{h}_T} = text{prod}left(frac{partial L}{partial boldsymbol{o}_T}, frac{partial boldsymbol{o}_T}{partial boldsymbol{h}_T} right) = boldsymbol{W}_{qh}^top frac{partial L}{partial boldsymbol{o}_T}. ∂ h T ∂ L = prod ( ∂ o T ∂ L , ∂ h T ∂ o T ) = W q h ⊤ ∂ o T ∂ L .
接下来对于时间步t < T t < T t < T L L L h t + 1 boldsymbol{h}_{t+1} h t + 1 o t boldsymbol{o}_t o t h t boldsymbol{h}_t h t t < T t < T t < T ∂ L ∂ h T ∈ R h frac{partial L}{partial boldsymbol{h}_T} in mathbb{R}^h ∂ h T ∂ L ∈ R h ∂ L ∂ h t = prod ( ∂ L ∂ h t + 1 , ∂ h t + 1 ∂ h t ) + prod ( ∂ L ∂ o t , ∂ o t ∂ h t ) = W h h ⊤ ∂ L ∂ h t + 1 + W q h ⊤ ∂ L ∂ o t frac{partial L}{partial boldsymbol{h}_t} = text{prod} (frac{partial L}{partial boldsymbol{h}_{t+1}}, frac{partial boldsymbol{h}_{t+1}}{partial boldsymbol{h}_t}) + text{prod} (frac{partial L}{partial boldsymbol{o}_t}, frac{partial boldsymbol{o}_t}{partial boldsymbol{h}_t} ) = boldsymbol{W}_{hh}^top frac{partial L}{partial boldsymbol{h}_{t+1}} + boldsymbol{W}_{qh}^top frac{partial L}{partial boldsymbol{o}_t} ∂ h t ∂ L = prod ( ∂ h t + 1 ∂ L , ∂ h t ∂ h t + 1 ) + prod ( ∂ o t ∂ L , ∂ h t ∂ o t ) = W h h ⊤ ∂ h t + 1 ∂ L + W q h ⊤ ∂ o t ∂ L
将上面的递归公式展开，对任意时间步1 ≤ t ≤ T 1 leq t leq T 1 ≤ t ≤ T
∂ L ∂ h t = ∑ i = t T ( W h h ⊤ ) T − i W q h ⊤ ∂ L ∂ o T + t − i . frac{partial L}{partial boldsymbol{h}_t} = sum_{i=t}^T {left(boldsymbol{W}_{hh}^topright)}^{T-i} boldsymbol{W}_{qh}^top frac{partial L}{partial boldsymbol{o}_{T+t-i}}. ∂ h t ∂ L = i = t ∑ T ( W h h ⊤ ) T − i W q h ⊤ ∂ o T + t − i ∂ L .
由上式中的指数项可见，当时间步数 T T T t t t ∂ L ∂ h T frac{partial L}{partial boldsymbol{h}_T} ∂ h T ∂ L ∂ L ∂ W h x ∈ R h × d frac{partial L}{partial boldsymbol{W}_{hx}} in mathbb{R}^{h times d} ∂ W h x ∂ L ∈ R h × d ∂ L ∂ W h h ∈ R h × h frac{partial L } {partial boldsymbol{W}_{hh}} in mathbb{R}^{h times h} ∂ W h h ∂ L ∈ R h × h L L L h 1 , … , h T boldsymbol{h}_1, ldots, boldsymbol{h}_T h 1 , … , h T
∂ L ∂ W h x = ∑ t = 1 T prod ( ∂ L ∂ h t , ∂ h t ∂ W h x ) = ∑ t = 1 T ∂ L ∂ h t x t ⊤ frac{partial L}{partial boldsymbol{W}_{hx}} = sum_{t=1}^T text{prod}left(frac{partial L}{partial boldsymbol{h}_t}, frac{partial boldsymbol{h}_t}{partial boldsymbol{W}_{hx}}right) = sum_{t=1}^T frac{partial L}{partial boldsymbol{h}_t} boldsymbol{x}t^top ∂ W h x ∂ L = t = 1 ∑ T prod ( ∂ h t ∂ L , ∂ W h x ∂ h t ) = t = 1 ∑ T ∂ h t ∂ L x t ⊤ ∂ L ∂ W h h = ∑ t = 1 T prod ( ∂ L ∂ h t , ∂ h t ∂ W h h ) = ∑ t = 1 T ∂ L ∂ h t h t − 1 ⊤ frac{partial L}{partial boldsymbol{W}_{hh}} = sum_{t=1}^T text{prod}left(frac{partial L}{partial boldsymbol{h}_t}, frac{partial boldsymbol{h}_t}{partial boldsymbol{W}_{hh}}right) = sum_{t=1}^T frac{partial L}{partial boldsymbol{h}_t} boldsymbol{h}_{t-1}^top ∂ W h h ∂ L = t = 1 ∑ T prod ( ∂ h t ∂ L , ∂ W h h ∂ h t ) = t = 1 ∑ T ∂ h t ∂ L h t − 1 ⊤
每次迭代中，我们在依次计算完以上各个梯度后，会将它们存储起来，从而避免重复计算。例如，由于隐藏状态梯度∂ L ∂ h t frac{partial L}{partial boldsymbol{h}_t} ∂ h t ∂ L ∂ L ∂ W h x frac{partial L}{partial boldsymbol{W}_{hx}} ∂ W h x ∂ L ∂ L ∂ W h h frac{partial L}{partial boldsymbol{W}_{hh}} ∂ W h h ∂ L ∂ L ∂ h t frac{partial L}{partial boldsymbol{h}_t} ∂ h t ∂ L ∂ L ∂ W h h frac{partial L}{partial boldsymbol{W}_{hh}} ∂ W h h ∂ L t = 0 , … , T − 1 t = 0, ldots, T-1 t = 0 , … , T − 1 h t boldsymbol{h}_t h t h 0 boldsymbol{h}_0 h 0
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