deeplearning.ai-lecture1-building deep neural network steps

 该实验主要是实现一些“Helper function”,为下一步实现两层神经网络和L层神经网络做准备，实现一个两层网络或深层网络的步骤如下：

Step 1.分别初始化一个两层神经网络和L层神经网络的参数

Step 2: 前向传播的实现：

1.完成一个网络的前向传播的线性部分（linear part）,即计算出 Z [l]     

2.实现relu和 sigmoid激活函数

3.联合前两步，实现网络前向传播的一个【linear->activation】 层 函数

4.实现前向传播的前L-1层【linear->relu】最后一层的【linear->sigmoid】函数

Step 3:计算损失函数

Step 4:反向传播的实现：

1.计算神经网络线性部分（linear part）的反向传播 

2.求出relu和sigmoid函数的梯度函数（relu_backward/relu_backward）

3.联合前两步，实现一个新的【linear->Activation】反向函数

4.整合，实现最后一层的【linear->sigmoid】和前L-1层的【linear->relu】的反向函数

Step 5:更新参数

下面开始实现神经网络的函数

Step 1:

1.   2层神经网络参数初始化

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)        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))        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. L层神经网络参数初始化

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)    for l in range(1,L):        parameters['W'+str(l)]=np.random.randn(layer_dims[l],layer_dims[l-1])*0.01        parameters['b'+str(l)]=np.zeros((layer_dims[l],1))        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

Step 2:

1.网络的前向传播的线性部分

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        """    Z=np.dot(W,A)+b    assert (Z.shape==(W.shape[0],A.shape[1]))    cache=(A,W,b)    return Z,cache

2.实现relu和 sigmoid激活函数

def sigmoid(Z):    """    Implements the sigmoid activation in numpy        Arguments:    Z -- numpy array of any shape        Returns:    A -- output of sigmoid(z), same shape as Z    cache -- returns Z as well, useful during backpropagation    """        A = 1/(1+np.exp(-Z))    cache = Z        return A, cachedef relu(Z):    """    Implement the RELU function.    Arguments:    Z -- Output of the linear layer, of any shape    Returns:    A -- Post-activation parameter, of the same shape as Z    cache -- a python dictionary containing "A" ; stored for computing the backward pass efficiently    """        A = np.maximum(0,Z)        assert(A.shape == Z.shape)        cache = Z     return A, cache

3.联合前两步，实现网络前向传播的一个【linear->activation】 层 函数

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".        Z, linear_cache = linear_forward(A_prev, W, b)        A, activation_cache = sigmoid(Z)        elif activation == "relu":        # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".        Z, linear_cache = linear_forward(A_prev, W, b)        A, activation_cache = relu(Z)        assert (A.shape == (W.shape[0], A_prev.shape[1]))    cache = (linear_cache, activation_cache)    return A, cache

4.实现前向传播的前L-1层【linear->relu】最后一层的【linear->sigmoid】函数

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         A, cache = linear_activation_forward(A_prev, parameters['W' + str(l)], parameters['b' + str(l)], activation = "relu")        caches.append(cache)        # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.    AL, cache = linear_activation_forward(A, parameters['W' + str(L)], parameters['b' + str(L)], activation = "sigmoid")    caches.append(cache)        assert(AL.shape == (1,X.shape[1]))                return AL, caches

Step 3:计算损失函数

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.    cost = (1./m) * (-np.dot(Y,np.log(AL).T) - np.dot(1-Y, np.log(1-AL).T))        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

Step 4:反向传播的实现：

1.计算神经网络线性部分（linear part）的反向传播 （假设你已经知道dZ[l]，计算dW[l],db[l],dA[l-1]）

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]    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)        assert (dA_prev.shape == A_prev.shape)    assert (dW.shape == W.shape)    assert (db.shape == b.shape)        return dA_prev, dW, db

2.求出relu和sigmoid函数的梯度函数（relu_backward/relu_backward）

假设dA已经，

def relu_backward(dA, cache):    """    Implement the backward propagation for a single RELU unit.    Arguments:    dA -- post-activation gradient, of any shape    cache -- 'Z' where we store for computing backward propagation efficiently    Returns:    dZ -- Gradient of the cost with respect to Z    """        Z = cache    dZ = np.array(dA, copy=True) # just converting dz to a correct object.        # When z <= 0, you should set dz to 0 as well.     dZ[Z <= 0] = 0        assert (dZ.shape == Z.shape)        return dZdef sigmoid_backward(dA, cache):    """    Implement the backward propagation for a single SIGMOID unit.    Arguments:    dA -- post-activation gradient, of any shape    cache -- 'Z' where we store for computing backward propagation efficiently    Returns:    dZ -- Gradient of the cost with respect to Z    """        Z = cache        s = 1/(1+np.exp(-Z))    dZ = dA * s * (1-s)        assert (dZ.shape == Z.shape)        return dZ

3.联合前两步，实现一个新的【linear->Activation】反向函数

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":        dZ = relu_backward(dA, activation_cache)        dA_prev, dW, db = linear_backward(dZ, linear_cache)            elif activation == "sigmoid":        dZ = sigmoid_backward(dA, activation_cache)        dA_prev, dW, db = linear_backward(dZ, linear_cache)        return dA_prev, dW, db

4.整合，实现最后一层的【linear->sigmoid】和前L-1层的【linear->relu】的反向函数

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" (there are (L-1) or them, indexes from 0 to L-2)                the cache of linear_activation_forward() with "sigmoid" (there is one, index 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    dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))        # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]    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")        for l in reversed(range(L-1)):        # lth layer: (RELU -> LINEAR) gradients.        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    return grads

Step 5:更新参数

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.    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)]            return parameters

6.实现一个预测函数，来预测测试集的正确率

def predict(X, y, parameters):    """    This function is used to predict the results of a  L-layer neural network.        Arguments:    X -- data set of examples you would like to label    parameters -- parameters of the trained model        Returns:    p -- predictions for the given dataset X    """        m = X.shape[1]    n = len(parameters) // 2 # number of layers in the neural network    p = np.zeros((1,m))        # Forward propagation    probas, caches = L_model_forward(X, parameters)        # convert probas to 0/1 predictions    for i in range(0, probas.shape[1]):        if probas[0,i] > 0.5:            p[0,i] = 1        else:            p[0,i] = 0    print("p="+str(p))    #print results    #print ("predictions: " + str(p))    #print ("true labels: " + str(y))    print("Accuracy: "  + str(np.sum((p == y)/m)))            return p