[人工智能]python深度学习之深层神经网络基础与图像分类

前言

本文参考吴恩达老师的机器学习课程作业，结合自己的理解所作笔记

1、热身：创建并初始化二层神经网络

模型的结构为：LINEAR -> RELU -> LINEAR -> SIGMOID

矩阵的形状要搞清楚，以这个五层神经网络为例（和本题无关），W矩阵的行数一定等于本层神经元的个数n[l]，由Z = Wa + b 可知 W的列和a的行相等，也就是等于前一层神经元的个数，最终得到的乘积要和b相加等于一个（n[l],1）的结果Z,再输入激活函数a = g(Z),把a作为下一层神经网络的输入，如此反复。牢记下面画红圈的部分

随机初始化W1，W2。随意处理b1，b2 (一般赋值为零)
parameters是存放梯度下降所需要的参数字典

# GRADED FUNCTION: initialize_parameters

def initialize_parameters(n_x, n_h, n_y):
    """
    Argument:
    n_x -- size of the input layer
    n_h -- size of the hidden layer
    n_y -- size of the output layer
    
    Returns:
    parameters -- python dictionary containing your parameters:
                    W1 -- weight matrix of shape (n_h, n_x)
                    b1 -- bias vector of shape (n_h, 1)
                    W2 -- weight matrix of shape (n_y, n_h)
                    b2 -- bias vector of shape (n_y, 1)
    """
    
    np.random.seed(1)
    
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = np.random.randn(n_h, n_x) * 0.01
    b1 = np.zeros((n_h, 1))
    W2 = np.random.randn(n_y, n_h) * 0.01
    b2 = np.zeros((n_y, 1))
    ### END CODE HERE ###
    
    assert(W1.shape == (n_h, n_x))
    assert(b1.shape == (n_h, 1))
    assert(W2.shape == (n_y, n_h))
    assert(b2.shape == (n_y, 1))
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters    
    

2、进阶：L层神经网络

模型的结构为 [LINEAR -> RELU] (L-1) -> LINEAR -> SIGMOID。也就是说，L-1层使用ReLU作为激活函数，最后一层采用sigmoid激活函数输出。

函数调用过程如下图所示：
我们要实现的四个主要函数

第一部分——正向传播

函数解释汇总

initialize_parameters_deep（layer_dims） 初始化神经网络的W和b,需要指定每层神经元的个数，返回parameter字典包含网络的参数W,b

linear_forward（A, W, b）计算Z = WA + b，返回Z和包含A,W,b的cache，便于后续反向传播更新参数

linear_activation_forward(A_prev, W, b, activation) 首先调用线性传播，获得Z和linear_cache，再计算A =g(Z),函数g可以是sigmoid或者relu，返回为A和包含Z的activitation_cache,最后把两个cache组合成为一个cache返回，“cache = (linear_cache, activation_cache)”，总结一下，其实最后返回的cache中包含的只不过四个参数W,b,Z,A,

L_model_forward(X, parameters)是正向传播的总函数，先用for循环进行L-1次[Linear->Relu]传播,调用linear_activation_forward()，参数activitation选择relu,然后获取每一层的cache,把它加到caches中备用（反向传播要用到），然后调用一次linear_activitation_forward()参数activitation 选择sigmoid,获取最后一层的cache和AL(也就是y_hat)

(1)初始化参数initialize_parameters_deep

initialize_parameters_deep（layer_dims） 初始化神经网络的W和b,需要指定每层神经元的个数，返回parameter字典包含网络的参数W,b

# GRADED FUNCTION: initialize_parameters_deep

def initialize_parameters_deep(layer_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the dimensions of each layer in our network
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
                    bl -- bias vector of shape (layer_dims[l], 1)
    """
    
    np.random.seed(3)
    parameters = {}
    L = len(layer_dims)            # number of layers in the network

    for l in range(1, L):
        ### START CODE HERE ### (≈ 2 lines of code)
        parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l-1])
        parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))
        ### END CODE HERE ###
        
        assert(parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[l-1]))
        assert(parameters['b' + str(l)].shape == (layer_dims[l], 1))

        
    return parameters

(2)线性传播linear_forward

linear_forward（A, W, b）计算Z = WA + b，返回Z和包含A,W,b的cache，便于后续反向传播更新参数

# GRADED FUNCTION: linear_forward

def linear_forward(A, W, b):
    """
    Implement the linear part of a layer's forward propagation.

    Arguments:
    A -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)

    Returns:
    Z -- the input of the activation function, also called pre-activation parameter 
    cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
    """
    
    ### START CODE HERE ### (≈ 1 line of code)
    Z = np.dot(W, A) + b
    ### END CODE HERE ###
    
    assert(Z.shape == (W.shape[0], A.shape[1]))
    cache = (A, W, b)
    
    return Z, cache

(3)正向线性激活linear_activation_forward

linear_activation_forward(A_prev, W, b, activation) 首先调用线性传播，获得Z和linear_cache，再计算A =g(Z),函数g可以是sigmoid或者relu，返回为A和包含Z的activitation_cache,最后把两个cache组合成为一个cache返回，“cache = (linear_cache, activation_cache)”，总结一下，其实最后返回的cache中包含的只不过四个参数W,b,Z,A,

# GRADED FUNCTION: linear_activation_forward

def linear_activation_forward(A_prev, W, b, activation):
    """
    Implement the forward propagation for the LINEAR->ACTIVATION layer

    Arguments:
    A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

    Returns:
    A -- the output of the activation function, also called the post-activation value 
    cache -- a python dictionary containing "linear_cache" and "activation_cache";
             stored for computing the backward pass efficiently
    """
    
    if activation == "sigmoid":
        # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
        ### START CODE HERE ### (≈ 2 lines of code)
        Z, linear_cache = linear_forward(A_prev,W,b)
        A, activation_cache = sigmoid(Z)
        ### END CODE HERE ###
    
    elif activation == "relu":
        # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
        ### START CODE HERE ### (≈ 2 lines of code)
        Z, linear_cache = linear_forward(A_prev,W,b)
        A, activation_cache = relu(Z)
        ### END CODE HERE ###
    
    assert (A.shape == (W.shape[0], A_prev.shape[1]))
    cache = (linear_cache, activation_cache)

    return A, cache

(4)正向传播L_model_forward(X, parameters)

L_model_forward(X, parameters)是正向传播的总函数，先用for循环进行L-1次[Linear->Relu]传播,调用linear_activation_forward()，参数activitation选择relu,然后获取每一层的cache,把它加到caches中备用（反向传播要用到），然后调用一次linear_activitation_forward()参数activitation 选择sigmoid,获取最后一层的cache和AL(也就是y_hat)

# GRADED FUNCTION: L_model_forward

def L_model_forward(X, parameters):
    """
    Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation
    
    Arguments:
    X -- data, numpy array of shape (input size, number of examples)
    parameters -- output of initialize_parameters_deep()
    
    Returns:
    AL -- last post-activation value
    caches -- list of caches containing:
                every cache of linear_relu_forward() (there are L-1 of them, indexed from 0 to L-2)
                the cache of linear_sigmoid_forward() (there is one, indexed L-1)
    """

    caches = []
    A = X
    L = len(parameters) // 2                  # number of layers in the neural network
    
    # Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
    for l in range(1, L):
        A_prev = A 
        ### START CODE HERE ### (≈ 2 lines of code)
        A, cache = linear_activation_forward(A_prev,parameters['W' + str(l)],parameters['b' + str(l)],activation = "relu")
        caches.append(cache)
        ### END CODE HERE ###
    
    # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
    ### START CODE HERE ### (≈ 2 lines of code)
    AL, cache = linear_activation_forward(A,parameters['W' + str(L)],parameters['b' + str(L)],activation = "sigmoid")
    caches.append(cache)
    
    ### END CODE HERE ###
    
    assert(AL.shape == (1,X.shape[1]))
            
    return AL, caches

第二部分——计算交叉熵损失

计算交叉熵损失compute_cost(AL, Y)

compute_cost(AL, Y)计算交叉熵损失，两个大小相同（1, n_x）的矩阵逐个元素（elementwise）相乘再相加，然后求和返回

# GRADED FUNCTION: compute_cost

def compute_cost(AL, Y):
    """
    Implement the cost function defined by equation (7).

    Arguments:
    AL -- probability vector corresponding to your label predictions, shape (1, number of examples)
    Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)

    Returns:
    cost -- cross-entropy cost
    """
    
    m = Y.shape[1]

    # Compute loss from aL and y.
    ### START CODE HERE ### (≈ 1 lines of code)
    cost = -1 / m * np.sum(Y * np.log(AL) + (1-Y) * np.log(1-AL),axis=1,keepdims=True)
    ### END CODE HERE ###
    
    cost = np.squeeze(cost)      # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).
    assert(cost.shape == ())
    
    return cost

第三部分——反向传播（梯度下降）

这一部分多了一些数学内容，函数基本和正向传播相似，不过计算是求梯度
推导如下
（C1W3L010 slides）


最终结论

总体思路

从最后一层开始往前传播，先求解最后一层，如下：
先求dAL(损失函数对倒数第一个神经元的激活输出y_hat求导）公式如下：
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
有了dAL，我们可以调用linear_activation_backward（第一次先选择sigmoid,其他L-1次选择relu）求出dZ，内部调用linear_backward求出dAL_prev，dWL，dbL，再把dAL_prev入作为新的输入到linear_activation_backward如此传播（for循环）直到达到第一层

(1)线性反向linear_backward(dZ, cache)

求dW,db

# GRADED FUNCTION: linear_backward

def linear_backward(dZ, cache):
    """
    Implement the linear portion of backward propagation for a single layer (layer l)

    Arguments:
    dZ -- Gradient of the cost with respect to the linear output (of current layer l)
    cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer

    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    A_prev, W, b = cache
    m = A_prev.shape[1]

    ### START CODE HERE ### (≈ 3 lines of code)
    dW = 1 / m * np.dot(dZ ,A_prev.T)
    db = 1 / m * np.sum(dZ,axis = 1 ,keepdims=True)
    dA_prev = np.dot(W.T,dZ) 
    ### END CODE HERE ###
    
    assert (dA_prev.shape == A_prev.shape)
    assert (dW.shape == W.shape)
    assert (db.shape == b.shape)
    
    return dA_prev, dW, db

(2)线性激活反向linear_activation_backward(dA, cache, activation)

# GRADED FUNCTION: linear_activation_backward

def linear_activation_backward(dA, cache, activation):
    """
    Implement the backward propagation for the LINEAR->ACTIVATION layer.
    
    Arguments:
    dA -- post-activation gradient for current layer l 
    cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
    
    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    linear_cache, activation_cache = cache
    
    if activation == "relu":
        ### START CODE HERE ### (≈ 2 lines of code)
        dZ = relu_backward(dA, activation_cache)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
        ### END CODE HERE ###
        
    elif activation == "sigmoid":
        ### START CODE HERE ### (≈ 2 lines of code)
        dZ = sigmoid_backward(dA, activation_cache)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
        ### END CODE HERE ###
    
    return dA_prev, dW, db

(3)反向传播L_model_backward(AL, Y, caches)

# GRADED FUNCTION: L_model_backward

def L_model_backward(AL, Y, caches):
    """
    Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group
    
    Arguments:
    AL -- probability vector, output of the forward propagation (L_model_forward())
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat)
    caches -- list of caches containing:
                every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)
                the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1])
    
    Returns:
    grads -- A dictionary with the gradients
             grads["dA" + str(l)] = ... 
             grads["dW" + str(l)] = ...
             grads["db" + str(l)] = ... 
    """
    grads = {}
    L = len(caches) # the number of layers
    m = AL.shape[1]
    Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL
    
    # Initializing the backpropagation
    ### START CODE HERE ### (1 line of code)
    dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
    ### END CODE HERE ###
    
    # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]
    ### START CODE HERE ### (approx. 2 lines)
    current_cache = caches[L-1]
    grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, current_cache, activation = "sigmoid")
    ### END CODE HERE ###
    
    for l in reversed(range(L-1)):
        # lth layer: (RELU -> LINEAR) gradients.
        # Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)] 
        ### START CODE HERE ### (approx. 5 lines)
        current_cache = caches[l]
        dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l+2)], current_cache, activation = "relu")
        grads["dA" + str(l + 1)] = dA_prev_temp
        grads["dW" + str(l + 1)] = dW_temp
        grads["db" + str(l + 1)] = db_temp
        ### END CODE HERE ###

    return grads

(4)更新参数update_parameters(parameters, grads, learning_rate)

# GRADED FUNCTION: update_parameters

def update_parameters(parameters, grads, learning_rate):
    """
    Update parameters using gradient descent
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    grads -- python dictionary containing your gradients, output of L_model_backward
    
    Returns:
    parameters -- python dictionary containing your updated parameters 
                  parameters["W" + str(l)] = ... 
                  parameters["b" + str(l)] = ...
    """
    
    L = len(parameters) // 2 # number of layers in the neural network

    # Update rule for each parameter. Use a for loop.
    ### START CODE HERE ### (≈ 3 lines of code)
    for l in range(L):
        parameters["W" + str(l+1)] =  parameters["W" + str(l+1)] - learning_rate * grads["dW" + str(l + 1)]
        parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * grads["db" + str(l + 1)]
    ### END CODE HERE ###
        
    return parameters

3、深层神经网络的应用——图像分类

(1)函数调用

# GRADED FUNCTION: L_layer_model

def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):#lr was 0.009
    """
    Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.
    
    Arguments:
    X -- data, numpy array of shape (number of examples, num_px * num_px * 3)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
    layers_dims -- list containing the input size and each layer size, of length (number of layers + 1).
    learning_rate -- learning rate of the gradient descent update rule
    num_iterations -- number of iterations of the optimization loop
    print_cost -- if True, it prints the cost every 100 steps
    
    Returns:
    parameters -- parameters learnt by the model. They can then be used to predict.
    """

    np.random.seed(1)
    costs = []                         # keep track of cost
    
    # Parameters initialization.
    ### START CODE HERE ###
    parameters = initialize_parameters_deep(layers_dims)
    ### END CODE HERE ###
    
    # Loop (gradient descent)
    for i in range(0, num_iterations):

        # Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.
        ### START CODE HERE ### (≈ 1 line of code)
        AL, caches = L_model_forward(X, parameters)
        ### END CODE HERE ###
        
        # Compute cost.
        ### START CODE HERE ### (≈ 1 line of code)
        cost = compute_cost(AL, Y)
        ### END CODE HERE ###
    
        # Backward propagation.
        ### START CODE HERE ### (≈ 1 line of code)
        grads = L_model_backward(AL, Y, caches)
        ### END CODE HERE ###
 
        # Update parameters.
        ### START CODE HERE ### (≈ 1 line of code)
        parameters = update_parameters(parameters, grads, learning_rate)
        ### END CODE HERE ###
                
        # Print the cost every 100 training example
        if print_cost and i % 100 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))
        if print_cost and i % 100 == 0:
            costs.append(cost)
            
    # plot the cost
    plt.plot(np.squeeze(costs))
    plt.ylabel('cost')
    plt.xlabel('iterations (per tens)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()
    
    return parameters

采用五层神经网络训练，结果如下

预测准确率80%

(2)结果分析

首先，让我们看一下L层模型标记错误的一些图像。 这将显示一些分类错误的图像

该模型在表现效果较差的的图像包括：

猫身处于异常位置
图片背景与猫颜色类似
猫的种类和颜色稀有
相机角度
图片的亮度
比例变化（猫的图像很大或很小）

4、reference

deeplearning.ai by Andrew Ng on couresa