浅层神经网络

文章目录

一、两层神经网络

下面是逻辑回归和两层神经网络的对比图：可以看出两层神经网络（激活函数为逻辑回归）就是做了两大次逻辑回归。

上述隐藏层的计算过程如下：每一个圈圈代表一个隐藏层单元，对应一个a值（做一次逻辑回归）。

神经网络中两层的计算过程（矢量化计算）如下：

二、激活函数

（一）激活函数的种类

这里提出了一种新的激活函数------tanh()，它是sigmois函数平移之后得到的，y坐标的值在-1和1之间。sigmoid函数和tanh函数有一个共同的缺点：在z无穷大时，函数的斜率接近0，会导致梯度下降变得非常缓慢，所以后面提出了relu激活函数（修正线性单元），现在作为默认的激活函数选择。relu没有当斜率为0时减慢梯度下降的效应。尽管函数当z为负数时，斜率为0，但是实际应用中，有足够多的隐藏单元令z大于0。

（二）神经网络为什么要使用非线性激活函数

在之前的机器学习课程中我们也提到过，如何使用线性激活函数，我们的神经网络输出结果跟直接使用线性回归没有什么区别。不管你的神经网络有多少层，最终做的只是计算线性激活函数。同样，如果你在所有隐藏层都使用线性回归，最终的输出层使用逻辑回归，这与你直接使用逻辑回归的结果是一样的。

（三）参数的初始化

当我们将所有的参数w和b全部初始化为0时，同一层的神经单元具有对称性，做的是相同的梯度下降，导致梯度下降过后的的各个神经单元的参数一模一样，导致多个隐藏单元没有意义。

所以我们需要随机初始化参数：

为什么生成符合正态分布的随机数之后还要×一个0.01，当我们的激活函数是sigmoid时，如果w太大，z就会越大，导致数据落在了激活函数较为平缓的区域，大大减慢了梯度下降的速度。0.01是一个比较合理的数值，当然如果不是sigmoid函数，我们也可以多多尝试一下其他的数值，后续会再讲这个数值的选择。

（四）构建具有一个隐藏层的分类神经网络

（1）数据集

（2）使用逻辑回归直接拟合数据：准确率只有47%

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# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y[0, :].T);

（3）根据数据构建神经网络模型

Logistic 回归在"花卉数据集"上效果不佳。下面将训练具有单个隐藏层的神经网络。

计算成本的函数如下：

提醒： 构建神经网络的一般方法是：1. 定义神经网络结构（# 输入单元、# 隐藏单元等）。2. 初始化模型的参数 3.循环： - 实现前向传播 - 计算损失 - 实现向后传播以获得梯度 - 更新参数（梯度下降）。

（4）初始化模型的参数

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# 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:
    params -- 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(2) # we set up a seed so that your output matches ours although the initialization is random.
    
    ### 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

（5）前向传播（矢量化）

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# GRADED FUNCTION: forward_propagation

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"
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    ### END CODE HERE ###
    
    # Implement Forward Propagation to calculate A2 (probabilities)
    ### START CODE HERE ### (≈ 4 lines of code)
    Z1 = np.dot(W1,X)+b1
    A1 = np.tanh(Z1)
    Z2 = np.dot(W2,A1)+b2
    A2 = np.tanh(Z2)
    ### END CODE HERE ###

    assert(A2.shape == (1, X.shape[1]))
    
    cache = {"Z1": Z1,
             "A1": A1,
             "Z2": Z2,
             "A2": A2}
    
    return A2, cache

（6）计算成本函数

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# GRADED FUNCTION: compute_cost

def compute_cost(A2, Y, parameters):
    """
    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
    ### START CODE HERE ### (≈ 2 lines of code)
    logprobs = np.multiply(np.log(A2),Y)+np.multiply(np.log(1-A2),1-Y)
    cost = np.sum(logprobs)/-m
                    # no need to use a for loop!

    ### END CODE HERE ###
    
    cost = np.squeeze(cost)     # makes sure cost is the dimension we expect. 
                                # E.g., turns [[17]] into 17 
    assert(isinstance(cost, float))
    
    return cost

（7）根据导数计算公式进行向后传播

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# GRADED FUNCTION: backward_propagation

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]
    
    # First, retrieve W1 and W2 from the dictionary "parameters".
    ### START CODE HERE ### (≈ 2 lines of code)
    W1 = parameters["W1"]
    W2 = parameters["W2"] 
    print(W1.shape)
    print(W2.shape)
    ### END CODE HERE ###
        
    # Retrieve also A1 and A2 from dictionary "cache".
    ### START CODE HERE ### (≈ 2 lines of code)
    A1 = cache["A1"]
    A2 = cache["A2"] 
    print(A1.shape)
    print(A2.shape)
    ### END CODE HERE ###
    
    # Backward propagation: calculate dW1, db1, dW2, db2. 
    ### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)
    dZ2 = A2-Y
    dW2 = np.dot(dZ2,A1.T)/m
    db2 = np.sum(dZ2,axis = 1,keepdims=True)/m
    dZ1 = np.dot(W2.T,dZ2)*(1-np.power(A1,2))
    dW1 = np.dot(dZ1,X.T)/m
    db1 = np.sum(dZ1,axis = 1,keepdims=True)/m
    ### END CODE HERE ###
    
    grads = {"dW1": dW1,
             "db1": db1,
             "dW2": dW2,
             "db2": db2}
    
    return grads

其中的cache是在进行向前传播时，将计算过程中关于A1和A2的变量缓存保存了下来，提供给计算梯度使用。

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# GRADED FUNCTION: update_parameters

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"
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    ### END CODE HERE ###
    
    # Retrieve each gradient from the dictionary "grads"
    ### START CODE HERE ### (≈ 4 lines of code)
    dW1 = grads["dW1"]
    db1 = grads["db1"]
    dW2 = grads["dW2"]
    db2 = grads["db2"]
    ## END CODE HERE ###
    
    # Update rule for each parameter
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = W1 - dW1*learning_rate
    b1 = b1 - db1*learning_rate
    W2 = W2 - dW2*learning_rate
    b2 = b2 - db2*learning_rate
    
    ### END CODE HERE ###
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters

（8）集成上述辅助函数

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# GRADED FUNCTION: nn_model

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".
    ### START CODE HERE ### (≈ 5 lines of code)
    parameters = initialize_parameters(n_x, n_h, n_y)
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    ### END CODE HERE ###
    
    # Loop (gradient descent)

    for i in range(0, num_iterations):
         
        ### START CODE HERE ### (≈ 4 lines of code)
        # 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, parameters)
        # 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)
        ### END CODE HERE ###
        
        # Print the cost every 1000 iterations
        if print_cost and i % 1000 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))

    return parameters

（9）使用模型进行预测

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# GRADED FUNCTION: predict

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.
    ### START CODE HERE ### (≈ 2 lines of code)
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    Z1 = np.dot(W1,X)+b1
    A1 = np.tanh(Z1)
    Z2 = np.dot(W2,A1)+b2
    predictions = np.tanh(Z2)
    
    ### END CODE HERE ###
    
    return predictions

使用神经网络模型进行训练，准确率达到90%。同时我们可以设置不同的隐藏层中的隐藏单元个数，看看模型的准确率的变化情况，从而选择最佳的隐藏单元个数。