前兩個(gè)筆記筆者集中探討了卷積神經(jīng)網(wǎng)絡(luò)中的卷積原理，對于二維卷積和三維卷積的原理進(jìn)行了深入的剖析，對 CNN 的卷積、池化、全連接、濾波器、感受野等關(guān)鍵概念進(jìn)行了充分的理解。本節(jié)內(nèi)容將繼續(xù)秉承之前 DNN 的學(xué)習(xí)路線，在利用 Tensorflow 搭建神經(jīng)網(wǎng)絡(luò)之前，先嘗試?yán)?numpy 手動(dòng)搭建卷積神經(jīng)網(wǎng)絡(luò)，以期對卷積神經(jīng)網(wǎng)絡(luò)的卷積機(jī)制、前向傳播和反向傳播的原理和過程有更深刻的理解。

單步卷積過程

在正式搭建 CNN 之前，我們先依據(jù)前面筆記提到的卷積機(jī)制的線性計(jì)算的理解，利用 numpy 定義一個(gè)單步卷積過程。代碼如下：

def conv_single_step(a_slice_prev, W, b):
  s = a_slice_prev * W  # Sum over all entries of the volume s.
  Z = np.sum(s)  # Add bias b to Z. Cast b to a float() so that Z results in a scalar value.
  Z = float(Z + b)  
  return Z

在上述的單步卷積定義中，我們傳入了一個(gè)前一層輸入的要進(jìn)行卷積的區(qū)域，即感受野 a_slice_prev ，濾波器 W，即卷積層的權(quán)重參數(shù)，偏差 b，對其執(zhí)行 Z=Wx+b 的線性計(jì)算即可實(shí)現(xiàn)一個(gè)單步的卷積過程。

CNN前向傳播過程：卷積

正如 DNN 中一樣，CNN 即使多了卷積和池化過程，模型仍然是前向傳播和反向傳播的訓(xùn)練過程。CNN 的前向傳播包括卷積和池化兩個(gè)過程，我們先來看如何利用 numpy 基于上面定義的單步卷積實(shí)現(xiàn)完整的卷積過程。卷積計(jì)算并不難，我們在單步卷積中就已經(jīng)實(shí)現(xiàn)了，難點(diǎn)在于如何實(shí)現(xiàn)濾波器在輸入圖像矩陣上的的掃描和移動(dòng)過程。

這其中我們需要搞清楚一些變量和參數(shù)，以及每一個(gè)輸入輸出的 shape，這對于我們執(zhí)行卷積和矩陣相乘至關(guān)重要。首先我們的輸入是原始圖像矩陣，也可以是前一層經(jīng)過激活后的圖像輸出矩陣，這里以前一層的激活輸出為準(zhǔn)，輸入像素的 shape 我們必須明確，然后是濾波器矩陣和偏差，還需要考慮步幅和填充，在此基礎(chǔ)上我們基于濾波器移動(dòng)和單步卷積搭建定義如下前向卷積過程：

def conv_forward(A_prev, W, b, hparameters):  
  """
  Arguments:
  A_prev -- output activations of the previous layer, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
  W -- Weights, numpy array of shape (f, f, n_C_prev, n_C)
  b -- Biases, numpy array of shape (1, 1, 1, n_C)
  hparameters -- python dictionary containing "stride" and "pad"

  Returns:
  Z -- conv output, numpy array of shape (m, n_H, n_W, n_C)
  cache -- cache of values needed for the conv_backward() function
  """
  # 前一層輸入的shape

  (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape  
  
  # 濾波器權(quán)重的shape
  (f, f, n_C_prev, n_C) = W.shape  
  # 步幅參數(shù)
  stride = hparameters['stride']  
  # 填充參數(shù)
  pad = hparameters['pad']  
  # 計(jì)算輸出圖像的高寬
  n_H = int((n_H_prev + 2 * pad - f) / stride + 1)
  n_W = int((n_W_prev + 2 * pad - f) / stride + 1)  
  # 初始化輸出
  Z = np.zeros((m, n_H, n_W, n_C))  
  # 對輸入執(zhí)行邊緣填充
  A_prev_pad = zero_pad(A_prev, pad)  
  for i in range(m):               
    a_prev_pad = A_prev_pad[i, :, :, :]         
    for h in range(n_H):             
      for w in range(n_W):           
        for c in range(n_C):          

          # 濾波器在輸入圖像上掃描
          vert_start = h * stride
          vert_end = vert_start + f
          horiz_start = w * stride
          horiz_end = horiz_start + f          
          # 定義感受野
          a_slice_prev = a_prev_pad[vert_start : vert_end, horiz_start : horiz_end, :]          # 對感受野執(zhí)行單步卷積
          Z[i, h, w, c] = conv_single_step(a_slice_prev, W[:,:,:,c], b[:,:,:,c])  
  assert(Z.shape == (m, n_H, n_W, n_C))
  cache = (A_prev, W, b, hparameters)  
  return Z, cache

這樣，卷積神經(jīng)網(wǎng)絡(luò)前向傳播中一個(gè)完整的卷積計(jì)算過程就被我們定義好了。通常而言，我們也會對卷積后輸出加一個(gè) relu 激活操作，正如前面的圖2所示，這里我們就省略不加了。

CNN前向傳播過程：池化

池化簡單而言就是取局部區(qū)域最大值，池化的前向傳播跟卷積過程類似，但相對簡單一點(diǎn)，無需執(zhí)行單步卷積那樣的乘積運(yùn)算。同樣需要注意的是各參數(shù)和輸入輸出的 shape，因此我們定義如下前向傳播池化過程：

def pool_forward(A_prev, hparameters, mode = "max"):  
  """
  Arguments:
  A_prev -- Input data, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
  hparameters -- python dictionary containing "f" and "stride"
  mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

  Returns:
  A -- output of the pool layer, a numpy array of shape (m, n_H, n_W, n_C)
  cache -- cache used in the backward pass of the pooling layer, contains the input and hparameters 
  """

  # 前一層輸入的shape
  (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape  
  # 步幅和權(quán)重參數(shù)
  f = hparameters["f"]
  stride = hparameters["stride"]  
  # 計(jì)算輸出圖像的高寬
  n_H = int(1 + (n_H_prev - f) / stride)
  n_W = int(1 + (n_W_prev - f) / stride)
  n_C = n_C_prev  
  # 初始化輸出
  A = np.zeros((m, n_H, n_W, n_C))       

  for i in range(m):            
    for h in range(n_H):          
      for w in range(n_W):         
        for c in range (n_C):     

          # 樹池在輸入圖像上掃描
          vert_start = h * stride
          vert_end = vert_start + f
          horiz_start = w * stride
          horiz_end = horiz_start + f         
          # 定義池化區(qū)域
          a_prev_slice = A_prev[i, vert_start:vert_end, horiz_start:horiz_end, c]          
          # 選擇池化類型
          if mode == "max":
            A[i, h, w, c] = np.max(a_prev_slice)          
          elif mode == "average":
            A[i, h, w, c] = np.mean(a_prev_slice)

  cache = (A_prev, hparameters)  
  assert(A.shape == (m, n_H, n_W, n_C))  
  return A, cache

由上述代碼結(jié)構(gòu)可以看出，前向傳播的池化過程的代碼結(jié)構(gòu)和卷積過程非常類似。

CNN反向傳播過程：卷積

定義好前向傳播之后，難點(diǎn)和關(guān)鍵點(diǎn)就在于如何給卷積和池化過程定義反向傳播過程。卷積層的反向傳播向來是個(gè)復(fù)雜的過程，Tensorflow 中我們只要定義好前向傳播過程，反向傳播會自動(dòng)進(jìn)行計(jì)算。但利用 numpy 搭建 CNN 反向傳播就還得我們自己定義了。其關(guān)鍵還是在于準(zhǔn)確的定義損失函數(shù)對于各個(gè)變量的梯度：

由上述梯度計(jì)算公式和卷積的前向傳播過程，我們定義如下卷積的反向傳播函數(shù)：

def conv_backward(dZ, cache):  """
  Arguments:
  dZ -- gradient of the cost with respect to the output of the conv layer (Z), numpy array of shape (m, n_H, n_W, n_C)
  cache -- cache of values needed for the conv_backward(), output of conv_forward()

  Returns:
  dA_prev -- gradient of the cost with respect to the input of the conv layer (A_prev),
        numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
  dW -- gradient of the cost with respect to the weights of the conv layer (W)
     numpy array of shape (f, f, n_C_prev, n_C)
  db -- gradient of the cost with respect to the biases of the conv layer (b)
     numpy array of shape (1, 1, 1, n_C)
  """
  # 獲取前向傳播中存儲的cache
  (A_prev, W, b, hparameters) = cache  
  # 前一層輸入的shape
  (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape  
  # 濾波器的 shape
  (f, f, n_C_prev, n_C) = W.shape  
  # 步幅和權(quán)重參數(shù)
  stride = hparameters['stride']
  pad = hparameters['pad']  
  # dZ 的shape
  (m, n_H, n_W, n_C) = dZ.shape  
  # 初始化 dA_prev, dW, db 
  dA_prev = np.zeros((m, n_H_prev, n_W_prev, n_C_prev))              
  dW = np.zeros((f, f, n_C_prev, n_C))
  db = np.zeros((1, 1, 1, n_C))  
  # 對A_prev 和 dA_prev 執(zhí)行零填充
  A_prev_pad = zero_pad(A_prev, pad)
  dA_prev_pad = zero_pad(dA_prev, pad)  
  for i in range(m):        
    # select ith training example from A_prev_pad and dA_prev_pad
    a_prev_pad = A_prev_pad[i,:,:,:]
    da_prev_pad = dA_prev_pad[i,:,:,:]    
    for h in range(n_H):         
      for w in range(n_W):       
        for c in range(n_C):     

          # 獲取當(dāng)前感受野
          vert_start = h * stride
          vert_end = vert_start + f
          horiz_start = w * stride
          horiz_end = horiz_start + f          
          # 獲取當(dāng)前濾波器矩陣
          a_slice = a_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :]          
          # 梯度更新
          da_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :] += W[:,:,:,c] * dZ[i, h, w, c]
          dW[:,:,:,c] += a_slice * dZ[i, h, w, c]
          db[:,:,:,c] += dZ[i, h, w, c]

          dA_prev[i, :, :, :] = da_prev_pad[pad:-pad, pad:-pad, :]  
  assert(dA_prev.shape == (m, n_H_prev, n_W_prev, n_C_prev))  
  return dA_prev, dW, db

CNN反向傳播過程：池化

反向傳播中的池化操作跟卷積也是類似的。再此之前，我們需要根據(jù)濾波器為最大池化和平均池化分別創(chuàng)建一個(gè) mask 和一個(gè) distribute_value :

def create_mask_from_window(x):  
  """
  Creates a mask from an input matrix x, to identify the max entry of x.

  Arguments:
  x -- Array of shape (f, f)

  Returns:
  mask -- Array of the same shape as window, contains a True at the position corresponding to the max entry of x.
  """
  mask = (x == np.max(x))  
  return mask

def distribute_value(dz, shape):  
  """
  Distributes the input value in the matrix of dimension shape

  Arguments:
  dz -- input scalar
  shape -- the shape (n_H, n_W) of the output matrix for which we want to distribute the value of dz

  Returns:
  a -- Array of size (n_H, n_W) for which we distributed the value of dz
  """

  (n_H, n_W) = shape  
  # Compute the value to distribute on the matrix 
  average = dz / (n_H * n_W)  
  # Create a matrix where every entry is the "average" value
  a = np.full(shape, average)  
  return a

然后整合封裝最大池化的反向傳播過程：

def pool_backward(dA, cache, mode = "max"):  
  """
  Arguments:
  dA -- gradient of cost with respect to the output of the pooling layer, same shape as A
  cache -- cache output from the forward pass of the pooling layer, contains the layer's input and hparameters 
  mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

  Returns:
  dA_prev -- gradient of cost with respect to the input of the pooling layer, same shape as A_prev
  """
  # Retrieve information from cache 
  (A_prev, hparameters) = cache  
  # Retrieve hyperparameters from "hparameters" 
  stride = hparameters['stride']
  f = hparameters['f']  
  # Retrieve dimensions from A_prev's shape and dA's shape 
  m, n_H_prev, n_W_prev, n_C_prev = A_prev.shape
  m, n_H, n_W, n_C = dA.shape  
  # Initialize dA_prev with zeros
  dA_prev = np.zeros((m, n_H_prev, n_W_prev, n_C_prev))  
  for i in range(m):           
    # select training example from A_prev
    a_prev = A_prev[i,:,:,:]    
    for h in range(n_H):         
      for w in range(n_W):       
        for c in range(n_C):     

          # Find the corners of the current "slice" 
          vert_start = h * stride
          vert_end = vert_start + f
          horiz_start = w * stride
          horiz_end = horiz_start + f          
          # Compute the backward propagation in both modes.
          if mode == "max":
            a_prev_slice = a_prev[vert_start:vert_end, horiz_start:horiz_end, c]
            mask = create_mask_from_window(a_prev_slice)
            dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += np.multiply(mask, dA[i,h,w,c])          elif mode == "average":           # Get the value a from dA
            da = dA[i,h,w,c]            
            # Define the shape of the filter as fxf 
            shape = (f,f)            
            # Distribute it to get the correct slice of dA_prev. i.e. Add the distributed value of da. 
            dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += distribute_value(da, shape)  
  # Making sure your output shape is correct
  assert(dA_prev.shape == A_prev.shape)  
  return dA_prev

這樣卷積神經(jīng)網(wǎng)絡(luò)的整個(gè)前向傳播和反向傳播過程我們就搭建好了。可以說是非常費(fèi)力的操作了，但我相信，經(jīng)過這樣一步步的根據(jù)原理的手寫，你一定會對卷積神經(jīng)網(wǎng)絡(luò)的原理理解更加深刻了。

本文由《自興動(dòng)腦人工智能》項(xiàng)目部 凱文 投稿。