神经网络的梯度下降公式推导及代码实现

神经网络的梯度下降公式推导及代码实现

1. 神经网络结构

以 2-Layers-Neural Network 为例,其结构如下。
该神经网络有两层,仅有一层为隐藏层。输入相应的数据 X = { X 1 , X 2 , ⋯   , X f } X=\{X_1,X_2,\cdots,X_f\} X={X1,X2,,Xf},输出为对应标签 Y Y Y

神经网络的梯度下降公式推导及代码实现_第1张图片

神经网络的梯度下降(Gradient descent)更新步骤基本为:

Step 1 Step 2 Step 3 Step 4
Forward Propagation 计算Cost Function Back Propagation 更新参数W,b

表格描述的是一个epoch的步骤,需要不停的重复Step 1~4, 需经过多个epochs直到Cost Function 收敛


2. Step 1: Forward Propagation

其第一层(隐藏层)由 $ h $ 个神经元(隐藏单元)构成, 每个神经元的都要经过两部计算:

  1. Z 1 [ 1 ] = W 1 [ 1 ] T X + b [ 1 ] ,   X = { X 1 , X 2 , ⋯   , X f } T Z_1^{[1]} = W_1^{[1]T}X+b^{[1]} , \ X = \{X_1,X_2,\cdots,X_f\}^T Z1[1]=W1[1]TX+b[1], X={X1,X2,,Xf}T
  2. A 1 [ 1 ] = σ ( Z 1 [ 1 ] ) = 1 / ( 1 + e x p ( − Z 1 [ 1 ] ) ) A_1^{[1]} = \sigma(Z_1^{[1]}) = 1/(1+exp(-Z_1^{[1]})) A1[1]=σ(Z1[1])=1/(1+exp(Z1[1]))

同理, 只需要改变一下对应的下角标, Z 2 [ 1 ] ∼ Z h [ 1 ] , A 2 [ 1 ] ∼ A h [ 1 ] Z_2^{[1]} \sim Z_h^{[1]},A_2^{[1]} \sim A_h^{[1]} Z2[1]Zh[1],A2[1]Ah[1] 也能被计算出来。
此时,第一层神经元的输出为 A [ 1 ] = { A 1 [ 1 ] , A 2 [ 1 ] , ⋯   , A h [ 1 ] } A^{[1]} = \{A_1^{[1]},A_2^{[1]},\cdots, A_h^{[1]}\} A[1]={A1[1],A2[1],,Ah[1]}
接下来进行下一层的计算:

  1. Z 1 [ 2 ] = W 1 [ 2 ] T X + b [ 2 ] Z_1^{[2]} = W_1^{[2]T}X+b^{[2]} Z1[2]=W1[2]TX+b[2]
  2. A 1 [ 2 ] = σ ( Z 1 [ 2 ] ) = 1 / ( 1 + e x p ( − Z 1 [ 2 ] ) ) A_1^{[2]} = \sigma (Z_1^{[2]}) = 1/(1+exp(-Z_1^{[2]})) A1[2]=σ(Z1[2])=1/(1+exp(Z1[2]))

此时, A 1 [ 2 ] A_1^{[2]} A1[2] 为模型生成的一个0~1之间的数字,需要利用一定的阈值使得其能变为标签值 Y ∈ { 0 , 1 } Y \in \{0,1\} Y{0,1}

if {A2 < 0.5}:
    Y = 0 
else:
    Y = 1 

由于第二层只有一个神经元,因此本文中 $Z_1^{[2]} $ 和 $Z^{[2]} $ 代表相同含义

  • 简单的描述一下数据集的结构:
    数据集为 T = { ( ( X 1 ( i ) , X 2 ( i ) , ⋯   , X f ( i ) ) , Y ( i ) ) } ,   i = { 1 , 2 , ⋯   , m } T=\{((X_1^{(i)},X_2^{(i)},\cdots,X_f^{(i)}),Y^{(i)})\}, \ i =\{1,2,\cdots,m\} T={((X1(i),X2(i),,Xf(i)),Y(i))}, i={1,2,,m}
    其中, f f f 为 feature个数, m m m 为训练集的sample数, $Y \in {0,1} $. 下图举了两个samples的例子.
    神经网络的梯度下降公式推导及代码实现_第2张图片
    因此, Forward Propagation 可以简化为:

[ Z [ 1 ] = W [ 1 ] ⋅ X + b [ 1 ] A [ 1 ] = σ ( Z [ 1 ] ) Z [ 2 ] = W [ 2 ] A [ 1 ] + b [ 2 ] A [ 2 ] = σ ( Z [ 2 ] ) \left[ \begin{array}{rl} Z^{[1]} & =W^{[1]} \cdot X+b^{[1]} \\ A^{[1]} & =\sigma\left(Z^{[1]}\right) \\ Z^{[2]} & =W^{[2]} A^{[1]}+b^{[2]} \\ A^{[2]} & =\sigma\left(Z^{[2]}\right) \end{array}\right. Z[1]A[1]Z[2]A[2]=W[1]X+b[1]=σ(Z[1])=W[2]A[1]+b[2]=σ(Z[2])


3. Step 2: 计算 Cost Function

Cost Function是对于每一个样本的 Loss 函数的平均值,计算如下:
C o s t   F u n c t i o n : J ( w , b ) = 1 m ∑ i = 1 m L ( y ^ ( i ) , y ( i ) ) = − 1 m ∑ i = 1 m [ ( y ( i ) log ⁡ ( y ^ ( i ) ) + ( 1 − y ( i ) ) log ⁡ ( 1 − y ^ ( i ) ) ] \begin{aligned} Cost \ Function: J(w, b) &=\frac{1}{m} \sum_{i=1}^{m} L\left(\hat{y}^{(i)}, y^{(i)}\right)\\ &=-\frac{1}{m} \sum_{i=1}^{m}\left[\left(y^{(i)} \log \left(\hat{y}^{(i)}\right)+\left(1-y^{(i)}\right) \log \left(1-\hat{y}^{(i)}\right)\right]\right. \end{aligned} Cost FunctionJ(w,b)=m1i=1mL(y^(i),y(i))=m1i=1m[(y(i)log(y^(i))+(1y(i))log(1y^(i))]
其中, y ^ ( i ) = A [ 2 ] ( i ) \hat{y}^{(i)} = A^{[2](i)} y^(i)=A[2](i),及第二层第 i i i 个 sample.

4. Step 3: Back Propagation

Back Propagation 是基于Chain Theory的梯度计算方法。其输出为W,b参数的梯度,以实现参数更新。

梯度下降的参数更新为:
W ← W − α ∂ J ( W , b ) ∂ W ,    b ← b − α ∂ J ( W , b ) ∂ b W \leftarrow W-\alpha\frac{\partial J(W,b)}{\partial W}, \ \ b \leftarrow b-\alpha\frac{\partial J(W,b)}{\partial b} WWαWJ(W,b),  bbαbJ(W,b)
α : l e a r n i n g   r a t e \alpha: learning \ rate α:learning rate 学习率小更新慢,反之则快。
因此,如果要更新 W 和 b,则需要计算对应的梯度。

由Forward Propagation知道,我么们需要计算的梯度由:
∂ J ( W , b ) ∂ W [ 2 ] ,   ∂ J ( W , b ) ∂ W [ 1 ] ,   ∂ J ( W , b ) ∂ b [ 2 ] ,   ∂ J ( W , b ) ∂ b [ 1 ]   \frac{\partial J(W,b)}{\partial W^{[2]}}, \ \frac{\partial J(W,b)}{\partial W^{[1]}}, \ \frac{\partial J(W,b)}{\partial b^{[2]}}, \ \frac{\partial J(W,b)}{\partial b^{[1]}} \ W[2]J(W,b), W[1]J(W,b), b[2]J(W,b), b[1]J(W,b) 
则其计算如下:

4.1 梯度 ∂ J ( W , b ) ∂ W [ 2 ] \frac{\partial J(W,b)}{\partial W^{[2]}} W[2]J(W,b) 的计算

∂ J ( W , b ) ∂ W [ 2 ] = d J d Z [ 2 ] d Z [ 2 ] d W [ 2 ] \frac{\partial J(W,b)}{\partial W^{[2]}} =\frac{dJ}{dZ^{[2]}}\frac{dZ^{[2]}}{d W^{[2]}} W[2]J(W,b)=dZ[2]dJdW[2]dZ[2]
其中后面一项可以直接求解:
因为 Z [ 2 ] = W [ 2 ] A [ 1 ] + b [ 2 ] Z^{[2]} =W^{[2]} A^{[1]}+b^{[2]} Z[2]=W[2]A[1]+b[2]
所以 d Z [ 2 ] d W [ 2 ] = 1 m A [ 1 ] T \frac{dZ^{[2]}}{d W^{[2]}} = \frac{1}{m}A^{[1]T} dW[2]dZ[2]=m1A[1]T
因此只需要求 d J d Z [ 2 ] \frac{dJ}{dZ^{[2]}} dZ[2]dJ.利用 Chain Theory 可得:
d J d Z [ 2 ] = d J d A [ 2 ] d A [ 2 ] d Z [ 2 ] = A [ 2 ] − Y \frac{dJ}{dZ^{[2]}} = \frac{dJ}{dA^{[2]}}\frac{dA^{[2]}}{dZ^{[2]}} = A^{[2]} - Y dZ[2]dJ=dA[2]dJdZ[2]dA[2]=A[2]Y
∂ J ( W , b ) ∂ W [ 2 ] = ( A [ 2 ] − Y ) × 1 m A [ 1 ] T \frac{\partial J(W,b)}{\partial W^{[2]}} = (A^{[2]}-Y)\times \frac{1}{m}A^{[1]T} W[2]J(W,b)=(A[2]Y)×m1A[1]T

推导1: 求证下面等式 d Z [ 2 ] d W [ 2 ] = 1 m A [ 1 ] T \frac{dZ^{[2]}}{d W^{[2]}} = \frac{1}{m}A^{[1]T} dW[2]dZ[2]=m1A[1]T

  • 如何理解 d Z [ 2 ] d W [ 2 ] \frac{dZ^{[2]}}{d W^{[2]}} dW[2]dZ[2] 的维度问题?
    Z [ 2 ] = W [ 2 ] A [ 1 ] + b [ 2 ] Z^{[2]} =W^{[2]} A^{[1]}+b^{[2]} Z[2]=W[2]A[1]+b[2]
    其中 A [ 1 ] ∈ R h × m A^{[1]} \in \mathbb{R}^{h \times m} A[1]Rh×m, Z [ 2 ] ∈ R 1 × m Z^{[2]} \in \mathbb{R}^{1 \times m} Z[2]R1×m
    , W [ 2 ] ∈ R 1 × h W^{[2]} \in \mathbb{R}^{1 \times h} W[2]R1×h, b [ 2 ] ∈ R 1 × m b^{[2]} \in \mathbb{R}^{1 \times m} b[2]R1×m.
    拿出 Z [ 2 ] Z^{[2]} Z[2] W [ 2 ] W^{[2]} W[2] 具体分析:
    Z [ 2 ] = { Z [ 2 ] ( 1 ) , Z [ 2 ] ( 2 ) , Z [ 2 ] ( 3 ) , ⋯   , Z [ 2 ] ( m ) } 1 × m Z^{[2]} = \{Z^{[2](1)},Z^{[2](2)},Z^{[2](3)},\cdots,Z^{[2](m)}\}_{1\times m} Z[2]={Z[2](1),Z[2](2),Z[2](3),,Z[2](m)}1×m
    A [ 1 ] = [ ∣ ∣ ∣ ∣ a [ 1 ] ( 1 ) a [ 1 ] ( 2 ) ⋯ a [ 1 ] ( m ) ∣ ∣ ∣ ∣ ] h × m A^{[1]}=\left[ \begin{array}{cccc} \mid & \mid &\mid & \mid \\ a^{[1](1)} & a^{[1](2)} & \cdots & a^{[1](m)} \\ \mid & \mid & \mid & \mid \end{array}\right]_{h\times m} A[1]=a[1](1)a[1](2)a[1](m)h×m
    W [ 2 ] = { W 1 [ 2 ] , W 2 [ 2 ] , W 3 [ 2 ] , ⋯   , W h [ 2 ] } 1 × h W^{[2]} = \{ W_1^{[2]},W_2^{[2]},W_3^{[2]},\cdots,W_h^{[2]}\}_{1\times h} W[2]={W1[2],W2[2],W3[2],,Wh[2]}1×h
    Z [ 2 ] = { Z [ 2 ] ( 1 ) , Z [ 2 ] ( 2 ) , Z [ 2 ] ( 3 ) , ⋯   , Z [ 2 ] ( m ) } 1 × m = { W 1 [ 2 ] , W 2 [ 2 ] , W 3 [ 2 ] , ⋯   , W h [ 2 ] } × [ ∣ ∣ ∣ ∣ a [ 1 ] ( 1 ) a [ 1 ] ( 2 ) ⋯ a [ 1 ] ( m ) ∣ ∣ ∣ ∣ ] h × m \begin{aligned} Z^{[2]} &=\{Z^{[2](1)},Z^{[2](2)},Z^{[2](3)},\cdots,Z^{[2](m)}\}_{1\times m} \\ & = \{ W_1^{[2]},W_2^{[2]},W_3^{[2]},\cdots,W_h^{[2]}\}\times \left[ \begin{array}{cccc} \mid & \mid &\mid & \mid \\ a^{[1](1)} & a^{[1](2)} & \cdots & a^{[1](m)} \\ \mid & \mid & \mid & \mid \end{array}\right]_{h\times m} \end{aligned} Z[2]={Z[2](1),Z[2](2),Z[2](3),,Z[2](m)}1×m={W1[2],W2[2],W3[2],,Wh[2]}×a[1](1)a[1](2)a[1](m)h×m
    因为: d Z [ 2 ] d W [ 2 ] \frac{dZ^{[2]}}{d W^{[2]}} dW[2]dZ[2]则表示:
    { d Z [ 2 ] } m × 1 T = d Z [ 2 ] d W [ 2 ] { d W [ 2 ] } h × 1 T \{dZ^{[2]}\}^T_{m \times 1} = \frac{dZ^{[2]}}{d W^{[2]}} \{d W^{[2]}\}^T_{h \times 1} {dZ[2]}m×1T=dW[2]dZ[2]{dW[2]}h×1T
    d Z [ 2 ] d W [ 2 ] \frac{dZ^{[2]}}{d W^{[2]}} dW[2]dZ[2] 维度为 (m,h)
    [ d Z [ 2 ] ( 1 ) d Z [ 2 ] ( 2 ) d Z [ 2 ] ( 3 ) ⋮ d Z [ 2 ] ( m ) ] = 1 m [ d Z [ 2 ] ( 1 ) d W 1 [ 2 ] d Z [ 2 ] ( 1 ) d W 2 [ 2 ] ⋯ d Z [ 2 ] ( 1 ) d W h [ 2 ] ⋮ ⋮ ⋮ ⋮ d Z [ 2 ] ( m ) d W 1 [ 2 ] d Z [ 2 ] ( m ) d W 2 [ 2 ] ⋯ d Z [ 2 ] ( m ) d W h [ 2 ] ] m × h × [ d W 1 [ 2 ] d W 2 [ 2 ] d W 3 [ 2 ] ⋮ d W h [ 2 ] ] = 1 m A [ 1 ] T × [ d W 1 [ 2 ] d W 2 [ 2 ] d W 3 [ 2 ] ⋮ d W h [ 2 ] ] \left[\begin{array}{c} dZ^{[2](1)} \\dZ^{[2](2)}\\dZ^{[2](3)} \\ \vdots \\ dZ^{[2](m)} \end{array}\right] = \frac{1}{m} \left[ \begin{array}{cccc} \frac{dZ^{[2](1)}}{d W_{1}^{[2]}} & \frac{dZ^{[2](1)}}{d W_{2}^{[2]}} & \cdots & \frac{dZ^{[2](1)}}{d W_{h}^{[2]}} \\ \vdots & \vdots &\vdots & \vdots \\ \frac{dZ^{[2](m)}}{d W_{1}^{[2]}} & \frac{dZ^{[2](m)}}{d W_{2}^{[2]}} & \cdots& \frac{dZ^{[2](m)}}{d W_{h}^{[2]}} \end{array} \right]_{m \times h} \times \left[\begin{array}{c} dW_1^{[2]} \\dW_2^{[2]} \\dW_3^{[2]} \\ \vdots \\ dW_h^{[2]} \end{array}\right] = \frac{1}{m} A^{[1]T} \times \left[\begin{array}{c} dW_1^{[2]} \\dW_2^{[2]} \\dW_3^{[2]} \\ \vdots \\ dW_h^{[2]} \end{array}\right] dZ[2](1)dZ[2](2)dZ[2](3)dZ[2](m)=m1dW1[2]dZ[2](1)dW1[2]dZ[2](m)dW2[2]dZ[2](1)dW2[2]dZ[2](m)dWh[2]dZ[2](1)dWh[2]dZ[2](m)m×h×dW1[2]dW2[2]dW3[2]dWh[2]=m1A[1]T×dW1[2]dW2[2]dW3[2]dWh[2]
    因此,可以推导出 d Z [ 2 ] d W [ 2 ] = 1 m A [ 1 ] T \frac{dZ^{[2]}}{d W^{[2]}} = \frac{1}{m}A^{[1]T} dW[2]dZ[2]=m1A[1]T

4.1 梯度 ∂ J ( W , b ) ∂ b [ 2 ] \frac{\partial J(W,b)}{\partial b^{[2]}} b[2]J(W,b) 的计算

∂ J ( W , b ) ∂ b [ 2 ] = d J d Z [ 2 ] d Z [ 2 ] d b [ 2 ] = ( A [ 2 ] − Y ) 1 × m × { 1 } m × 1 × 1 m \frac{\partial J(W,b)}{\partial b^{[2]}} = \frac{dJ}{dZ^{[2]}}\frac{dZ^{[2]}}{db^{[2]}} = (A^{[2]}-Y)_{1 \times m}\times\{1\}_{m \times 1 } \times \frac{1}{m} b[2]J(W,b)=dZ[2]dJdb[2]dZ[2]=(A[2]Y)1×m×{1}m×1×m1
这相当于是对 ( A [ 2 ] − Y ) 1 × m (A^{[2]}-Y)_{1 \times m} (A[2]Y)1×m求平均值。

import numpy as np

dZ2 = A2 - Y
db2 = 1.0 / m * np.sum(dZ2, axis=1, keepdims=True)

4.2 梯度 ∂ J ( W , b ) ∂ b [ 1 ] \frac{\partial J(W,b)}{\partial b^{[1]}} b[1]J(W,b) ∂ J ( W , b ) ∂ W [ 1 ] \frac{\partial J(W,b)}{\partial W^{[1]}} W[1]J(W,b) 的计算

其他梯度也可以通过类似的方式求解
∂ J ( W , b ) ∂ W [ 1 ] = d J d Z [ 1 ] X T ∂ J ( W , b ) ∂ b [ 1 ] = d J d Z [ 1 ] d J d Z [ 1 ] = W [ 2 ] d J d Z [ 2 ] ∗ σ [ 1 ] ′ ( Z [ 1 ] ) \frac{\partial J(W,b)}{\partial W^{[1]}} = \frac{dJ}{dZ^{[1]}}X^T \\ \frac{\partial J(W,b)}{\partial b^{[1]}} = \frac{dJ}{dZ^{[1]}} \\ \frac{dJ}{dZ^{[1]}} = W^{[2]} \frac{dJ}{dZ^{[2]}}*\sigma^{[1]'}(Z^{[1]}) W[1]J(W,b)=dZ[1]dJXTb[1]J(W,b)=dZ[1]dJdZ[1]dJ=W[2]dZ[2]dJσ[1](Z[1])

5. 参数更新

梯度下降的参数更新为:
W [ 1 ] ← W [ 1 ] − α ∂ J ( [ 1 ] , b [ 1 ] ) ∂ [ 1 ] ,    b [ 1 ] ← b [ 1 ] − α ∂ J ( W [ 1 ] , b [ 1 ] ) ∂ b [ 1 ] W [ 2 ] ← W [ 2 ] − α ∂ J ( [ 2 ] , b [ 2 ] ) ∂ [ 2 ] ,    b [ 2 ] ← b [ 2 ] − α ∂ J ( W [ 2 ] , b [ 2 ] ) ∂ b [ 2 ] W^{[1]} \leftarrow W^{[1]}-\alpha\frac{\partial J(^{[1]},b^{[1]})}{\partial ^{[1]}}, \ \ b^{[1]} \leftarrow b^{[1]}-\alpha\frac{\partial J(W^{[1]},b^{[1]})}{\partial b^{[1]}} \\ W^{[2]} \leftarrow W^{[2]}-\alpha\frac{\partial J(^{[2]},b^{[2]})}{\partial ^{[2]}}, \ \ b^{[2]} \leftarrow b^{[2]}-\alpha\frac{\partial J(W^{[2]},b^{[2]})}{\partial b^{[2]}} \\ W[1]W[1]α[1]J([1],b[1]),  b[1]b[1]αb[1]J(W[1],b[1])W[2]W[2]α[2]J([2],b[2]),  b[2]b[2]αb[2]J(W[2],b[2])
α : l e a r n i n g   r a t e \alpha: learning \ rate α:learning rate 学习率小更新慢,反之则快。

6. 部分函数以及代码实现

详情可见 Coursera Deep Learning.

def forward_propagation(X, parameters):
    """
    Argument:
    X -- input data of size (n_x, m)
    parameters -- python dictionary containing your parameters (output of initialization function)

    Returns:
    A2 -- The sigmoid output of the second activation
    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
    """
    # Retrieve each parameter from the dictionary "parameters"

    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]

    # Implement Forward Propagation to calculate A2 (probabilities)

    Z1 = np.dot(W1, X) + b1
    A1 = np.tanh(Z1)
    Z2 = np.dot(W2, A1) + b2
    A2 = sigmoid(Z2)

    assert (A2.shape == (1, X.shape[1]))

    cache = {"Z1": Z1,
             "A1": A1,
             "Z2": Z2,
             "A2": A2}

    return A2, cache


def compute_cost(A2, Y):
    """
    Computes the cross-entropy cost given in equation (13)

    Arguments:
    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
    Y -- "true" labels vector of shape (1, number of examples)
    parameters -- python dictionary containing your parameters W1, b1, W2 and b2

    Returns:
    cost -- cross-entropy cost given equation (13)
    """

    m = Y.shape[1]  # number of example

    # Compute the cross-entropy cost

    logprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1 - A2), (1 - Y))
    cost = -(1.0 / m) * np.sum(logprobs)

    cost = np.squeeze(cost)  # makes sure cost is the dimension we expect. # E.g., turns [[17]] into 17

    assert (isinstance(cost, float))

    return cost

    
def backward_propagation(parameters, cache, X, Y):
    """
    Implement the backward propagation using the instructions above.

    Arguments:
    parameters -- python dictionary containing our parameters
    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
    X -- input data of shape (2, number of examples)
    Y -- "true" labels vector of shape (1, number of examples)

    Returns:
    grads -- python dictionary containing your gradients with respect to different parameters
    """
    m = X.shape[1]  # sample size

    # First, retrieve W1 and W2 from the dictionary "parameters".

    W1 = parameters["W1"]
    W2 = parameters["W2"]

    # Retrieve also A1 and A2 from dictionary "cache".

    A1 = cache["A1"]
    A2 = cache["A2"]

    # Backward propagation: calculate dW1, db1, dW2, db2.

    dZ2 = A2 - Y
    dW2 = 1.0 / m * np.dot(dZ2, A1.T)
    db2 = 1.0 / m * np.sum(dZ2, axis=1, keepdims=True)
    dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2))
    dW1 = 1.0 / m * np.dot(dZ1, X.T)
    db1 = 1.0 / m * np.sum(dZ1, axis=1, keepdims=True)

    grads = {"dW1": dW1,
             "db1": db1,
             "dW2": dW2,
             "db2": db2}

    return grads
def update_parameters(parameters, grads, learning_rate=1.2):
    """
    Updates parameters using the gradient descent update rule given above

    Arguments:
    parameters -- python dictionary containing your parameters
    grads -- python dictionary containing your gradients

    Returns:
    parameters -- python dictionary containing your updated parameters
    """
    # Retrieve each parameter from the dictionary "parameters"

    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]

    # Retrieve each gradient from the dictionary "grads"

    dW1 = grads["dW1"]
    db1 = grads["db1"]
    dW2 = grads["dW2"]
    db2 = grads["db2"]

    # Update rule for each parameter

    W1 = W1 - learning_rate * dW1
    b1 = b1 - learning_rate * db1
    W2 = W2 - learning_rate * dW2
    b2 = b2 - learning_rate * db2

    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}

    return parameters


def nn_model(X, Y, n_h, num_iterations=10000, print_cost=False):
    """
    Arguments:
    X -- dataset of shape (2, number of examples)
    Y -- labels of shape (1, number of examples)
    n_h -- size of the hidden layer
    num_iterations -- Number of iterations in gradient descent loop
    print_cost -- if True, print the cost every 1000 iterations

    Returns:
    parameters -- parameters learnt by the model. They can then be used to predict.
    """

    np.random.seed(3)
    n_x = layer_sizes(X, Y)[0]
    n_y = layer_sizes(X, Y)[2]

    # Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2,
    # parameters".

    parameters = initialize_parameters(n_x, n_h, n_y)
    # W1 = parameters["W1"]
    # b1 = parameters["b1"]
    # W2 = parameters["W2"]
    # b2 = parameters["b2"]

    # Loop (gradient descent)
    Cost_plot = []
    for i in range(0, num_iterations):

        # Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
        A2, cache = forward_propagation(X, parameters)

        # Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
        cost = compute_cost(A2, Y)

        # Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
        grads = backward_propagation(parameters, cache, X, Y)

        # Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
        parameters = update_parameters(parameters, grads, learning_rate=1.2)

        # Stack all the cost for plot

        Cost_plot = np.append(Cost_plot, cost)
        # Print the cost every 1000 iterations
        if print_cost and i % 10 == 0:
            print("Cost after iteration %i: %f" % (i, cost))

    return parameters, Cost_plot

def predict(parameters, X):
    """
    Using the learned parameters, predicts a class for each example in X

    Arguments:
    parameters -- python dictionary containing your parameters
    X -- input data of size (n_x, m)

    Returns
    predictions -- vector of predictions of our model (red: 0 / blue: 1)
    """

    # Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.

    A2, cache = forward_propagation(X, parameters)
    predictions = (A2 > 0.5)

    return predictions

  • Main Process
# Package imports
import numpy as np
import matplotlib.pyplot as plt
import sklearn.linear_model

from NN_Compute_Function import nn_model, predict
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets

np.random.seed(1)  # set a seed so that the results are consistent

'''
    Import the data from Datasets
'''
# Visualize the data:
X, Y = load_planar_dataset()

'''
    Build a model with a n_h-dimensional hidden layer
'''
parameters, cost_for_plot = nn_model(X, Y, n_h=9, num_iterations=10000, print_cost=True)

plt.plot(cost_for_plot, label='Cost function')
plt.legend()
plt.show()

# Print accuracy of 2-Layers-NN
predictions = predict(parameters, X)
print('Accuracy of 2-Layers-NN: %d' % float(
    (np.dot(Y, predictions.T) + np.dot(1 - Y, 1 - predictions.T)) / float(Y.size) * 100) + '%'
      + " with hidden layer size of " + str(4))

if __name__ == "__main__":
    print('=====[This is implementation of two layers neural network on classification]=====')

你可能感兴趣的:(深度学习,统计学,神经网络,深度学习,机器学习)