深度学习作业L1W4：深层神经网络

本周的作业分为两部分，第一部分是构建一个深层神经网络的模板，第二部分是进行一个识别猫图片的实践。
我先进行我的经验总结，再贴出实验代码。

构建一个神经网络模型的步骤：
1.进行数据处理，如将图片拍扁
2.进行参数初始化（注意w应该进行随机初始化）
3.构建前向传播：利用向量化公式，同时注意保留cache，以方便反向传播的计算。
4.计算损失函数：方便我们直观的判断损失函数的下降过程，以调整超参数
5.构建反向传播：利用前向传播提供的cache求解
6.更新参数：梯度下降法
7.统计测试集训练集的准确率

上代码！

part1

两层简单神经网络参数初始化，利用np.random.randn

# 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: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)*0.01b1 = 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

对于多层参数初始化，利用for循环

# GRADED FUNCTION: initialize_parameters_deepdef initialize_parameters_deep(layer_dims):"""Arguments:layer_dims -- python array (list) containing the dimensions of each layer in our networkReturns: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 networkfor 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])*0.01parameters['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

计算z值并存储cache

# GRADED FUNCTION: linear_forwarddef 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 = W.dot(A)+b### END CODE HERE ###assert(Z.shape == (W.shape[0], A.shape[1]))cache = (A, W, b)return Z, cache

计算A值并记录cache

# GRADED FUNCTION: linear_activation_forwarddef linear_activation_forward(A_prev, W, b, activation):"""Implement the forward propagation for the LINEAR->ACTIVATION layerArguments: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

这里的cache分成了两部分，这是为了方便后续反向传播的调用

构建训练模型

# GRADED FUNCTION: L_model_forwarddef L_model_forward(X, parameters):"""Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computationArguments:X -- data, numpy array of shape (input size, number of examples)parameters -- output of initialize_parameters_deep()Returns:AL -- last post-activation valuecaches -- 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 = XL = 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)], "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)], "sigmoid")caches.append(cache)### END CODE HERE ###assert(AL.shape == (1,X.shape[1]))return AL, caches

计算损失函数

# GRADED FUNCTION: compute_costdef 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 = -np.sum(np.log(AL)*Y+np.log(1-AL)*(1-Y))/m### 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

计算反向传播

# GRADED FUNCTION: linear_backwarddef 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 layerReturns:dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prevdW -- Gradient of the cost with respect to W (current layer l), same shape as Wdb -- Gradient of the cost with respect to b (current layer l), same shape as b"""A_prev, W, b = cachem = A_prev.shape[1]### START CODE HERE ### (≈ 3 lines of code)dW = dZ.dot(A_prev.T)/mdb = np.sum(dZ, axis=1, keepdims=True)/mdA_prev = W.T.dot(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

# GRADED FUNCTION: linear_activation_backwarddef 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 efficientlyactivation -- 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_prevdW -- Gradient of the cost with respect to W (current layer l), same shape as Wdb -- Gradient of the cost with respect to b (current layer l), same shape as b"""linear_cache, activation_cache = cacheif 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

# GRADED FUNCTION: L_model_backwarddef L_model_backward(AL, Y, caches):"""Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID groupArguments: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 gradientsgrads["dA" + str(l)] = ... grads["dW" + str(l)] = ...grads["db" + str(l)] = ... """grads = {}L = len(caches) # the number of layersm = 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, "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, "relu")grads["dA" + str(l + 1)] = dA_prev_tempgrads["dW" + str(l + 1)] = dW_tempgrads["db" + str(l + 1)] = db_temp### END CODE HERE ###return grads

更新参数

# GRADED FUNCTION: update_parametersdef update_parameters(parameters, grads, learning_rate):"""Update parameters using gradient descentArguments:parameters -- python dictionary containing your parameters grads -- python dictionary containing your gradients, output of L_model_backwardReturns: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 i in range(L):parameters["W"+str(i+1)]=parameters["W"+str(i+1)]-grads["dW"+str(i+1)]*learning_rateparameters["b"+str(i+1)]=parameters["b"+str(i+1)]-grads["db"+str(i+1)]*learning_rate### END CODE HERE ###return parameters

part2

这部分还是找猫，我们观察一下有什么提升

数据处理（压扁）

# Reshape the training and test examples
train_x_flatten = train_x_orig.reshape(train_x_orig.shape[0], -1).T   # The "-1" makes reshape flatten the remaining dimensions
test_x_flatten = test_x_orig.reshape(test_x_orig.shape[0], -1).T# Standardize data to have feature values between 0 and 1.
train_x = train_x_flatten/255.
test_x = test_x_flatten/255.print ("train_x's shape: " + str(train_x.shape))
print ("test_x's shape: " + str(test_x.shape))

构建一个两层的神经网络

### CONSTANTS DEFINING THE MODEL ####
n_x = 12288     # num_px * num_px * 3
n_h = 7
n_y = 1
layers_dims = (n_x, n_h, n_y)
# GRADED FUNCTION: two_layer_modeldef two_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):"""Implements a two-layer neural network: LINEAR->RELU->LINEAR->SIGMOID.Arguments:X -- input data, of shape (n_x, number of examples)Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)layers_dims -- dimensions of the layers (n_x, n_h, n_y)num_iterations -- number of iterations of the optimization looplearning_rate -- learning rate of the gradient descent update ruleprint_cost -- If set to True, this will print the cost every 100 iterations Returns:parameters -- a dictionary containing W1, W2, b1, and b2"""np.random.seed(1)grads = {}costs = []                              # to keep track of the costm = X.shape[1]                           # number of examples(n_x, n_h, n_y) = layers_dims# Initialize parameters dictionary, by calling one of the functions you'd previously implemented### START CODE HERE ### (≈ 1 line of code)parameters = initialize_parameters(n_x, n_h, n_y)### END CODE HERE #### Get W1, b1, W2 and b2 from the dictionary parameters.W1 = parameters["W1"]b1 = parameters["b1"]W2 = parameters["W2"]b2 = parameters["b2"]# Loop (gradient descent)for i in range(0, num_iterations):# Forward propagation: LINEAR -> RELU -> LINEAR -> SIGMOID. Inputs: "X, W1, b1". Output: "A1, cache1, A2, cache2".### START CODE HERE ### (≈ 2 lines of code)A1, cache1 = linear_activation_forward(X, W1, b1, "relu")A2, cache2 = linear_activation_forward(A1, W2, b2, activation="sigmoid")### END CODE HERE #### Compute cost### START CODE HERE ### (≈ 1 line of code)cost = compute_cost(A2, Y)### END CODE HERE #### Initializing backward propagationdA2 = - (np.divide(Y, A2) - np.divide(1 - Y, 1 - A2))# Backward propagation. Inputs: "dA2, cache2, cache1". Outputs: "dA1, dW2, db2; also dA0 (not used), dW1, db1".### START CODE HERE ### (≈ 2 lines of code)dA1, dW2, db2 = linear_activation_backward(dA2, cache2, "sigmoid")dA0, dW1, db1 = linear_activation_backward(dA1, cache1, "relu")### END CODE HERE #### Set grads['dWl'] to dW1, grads['db1'] to db1, grads['dW2'] to dW2, grads['db2'] to db2grads['dW1'] = dW1grads['db1'] = db1grads['dW2'] = dW2grads['db2'] = db2# Update parameters.### START CODE HERE ### (approx. 1 line of code)parameters = update_parameters(parameters, grads, learning_rate)### END CODE HERE #### Retrieve W1, b1, W2, b2 from parametersW1 = parameters["W1"]b1 = parameters["b1"]W2 = parameters["W2"]b2 = parameters["b2"]# Print the cost every 100 training exampleif print_cost and i % 100 == 0:print("Cost after iteration {}: {}".format(i, np.squeeze(cost)))if print_cost and i % 100 == 0:costs.append(cost)# plot the costplt.plot(np.squeeze(costs))plt.ylabel('cost')plt.xlabel('iterations (per tens)')plt.title("Learning rate =" + str(learning_rate))plt.show()return parameters

这个模型的测试集准确率为100%（在我的电脑上莫名其妙0.9999999999999998），测试集准确率0.72，比logistic回归好一点点。

接下来是L层的神经网络

### CONSTANTS ###
layers_dims = [12288, 20, 7, 5, 1] #  5-layer model
# GRADED FUNCTION: L_layer_modeldef 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 rulenum_iterations -- number of iterations of the optimization loopprint_cost -- if True, it prints the cost every 100 stepsReturns: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 exampleif 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 costplt.plot(np.squeeze(costs))plt.ylabel('cost')plt.xlabel('iterations (per tens)')plt.title("Learning rate =" + str(learning_rate))plt.show()return parameters

这次的训练集准确率有所下降，但在测试集上表现非常好，准确率为0.8
可见在本问题中，深层的神经网络更胜一筹。