浅层神经网络

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%

# 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

# GRADED FUNCTION: initialize_parametersdef initialize_parameters(n_x, n_h, n_y):"""Argument:n_x -- size of the input layern_h -- size of the hidden layern_y -- size of the output layerReturns: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)*0.01b1 = np.zeros((n_h,1))W2 = np.random.randn(n_y,n_h)*0.01b2 = 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）

# GRADED FUNCTION: forward_propagationdef 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 activationcache -- 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)+b1A1 = np.tanh(Z1)Z2 = np.dot(W2,A1)+b2A2 = 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

# GRADED FUNCTION: compute_costdef 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 b2Returns: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

# GRADED FUNCTION: backward_propagationdef 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-YdW2 = np.dot(dZ2,A1.T)/mdb2 = np.sum(dZ2,axis = 1,keepdims=True)/mdZ1 = np.dot(W2.T,dZ2)*(1-np.power(A1,2))dW1 = np.dot(dZ1,X.T)/mdb1 = 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。

# GRADED FUNCTION: update_parametersdef update_parameters(parameters, grads, learning_rate = 1.2):"""Updates parameters using the gradient descent update rule given aboveArguments: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_rateb1 = b1 - db1*learning_rateW2 = W2 - dW2*learning_rateb2 = 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

# GRADED FUNCTION: nn_modeldef 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 layernum_iterations -- Number of iterations in gradient descent loopprint_cost -- if True, print the cost every 1000 iterationsReturns: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 iterationsif 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

# GRADED FUNCTION: predictdef predict(parameters, X):"""Using the learned parameters, predicts a class for each example in XArguments:parameters -- python dictionary containing your parameters X -- input data of size (n_x, m)Returnspredictions -- 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)+b1A1 = np.tanh(Z1)Z2 = np.dot(W2,A1)+b2predictions = 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。