吴恩达深度学习笔记——第一课第四周
2024-04-17 14:31:09
深层神经网络
- 内容概述
- 深层神经网络概述
- 前向传播和反向传播（Forward and backward propagation）
- 前向传播
- 反向传播
- 搭建神经网络块
- 超参数
- 代码作业——helper function
- 初始化
- 对于一个2层的神经网络
- 初始化一个L层的神经网络
- 向前传播模块
- 线性Forward
- 带有激活函数的线性Forward
- L层的Forward部分
- Cost函数
- Backward部分
- 线性backward
- Linear-Activation backward
- L 层的backward过程
- 参数更新
- 代码作业——实现神经网络
- 数据集加载
- 神经网络结构
- 二层网络（单隐层）
- L层神经网络
- 实现两层的神经网络
- 实现L层神经网络
内容概述
这一次课的内容主要是把之前的内容进行了一个综合，从而可以实现一个神经网络。
在作业部分，第一份作业是helper function，完成了第一份，第二份就顺水推舟了，注意一下参数选取即可。
相关练习可以在我的资源中免费下载，我会尽快上传。
深层神经网络概述
之前介绍的神经网络只有一个隐藏层，下面是一个四层神经网络的示意图
第一层（第一个隐藏层）有5个神经元，第二层也是5个，第三层是3个
用符号表示， n [ 0 ] = 3 , n [ 1 ] = 5 , n [ 2 ] = 5 , n [ 3 ] = 3 , n [ 4 ] = 1 n^{[0]}=3,n^{[1]}=5,n^{[2]}=5,n^{[3]}=3,n^{[4]}=1 n[0]=3,n[1]=5,n[2]=5,n[3]=3,n[4]=1
对于第i层，用 a [ i ] a^{[i]} a[i]来表示这一层激活后的结果。forward部分会算出 a [ i ] a^{[i]} a[i]
前向传播和反向传播（Forward and backward propagation）
前向传播
前向传播需要用X来初始化。
反向传播
搭建神经网络块
超参数
learning rate α \alpha α（学习率）、iterations(梯度下降法循环的数量)、L（隐藏层数目）、 n [ l ] n^{[l]} n[l]（隐藏层单元数目）、choice of activation function（激活函数的选择）…这些参数控制了最后的W和b
代码作业——helper function
首先还是要引入相关的包
import numpy as np
import h5py
import matplotlib.pyplot as plt
from testCases_v2 import *
from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward%matplotlib inline
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'%load_ext autoreload
%autoreload 2np.random.seed(1)
在搭建一个完整的深层神经网络之前，需要先实现一些“helper function”
初始化
对于一个2层的神经网络
def initialize_parameters(n_x, n_h, n_y):"""Argument:n_x -- 输入层大小size of the input layern_h -- 隐藏层大小size of the hidden layern_y -- 输出层大小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)# 注意这里的shape应该是Tuple，括号层数要注意！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
初始化一个L层的神经网络
对于一个多层的神经网络，初始化的时候，层数一定要匹配上
def 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
向前传播模块
线性Forward
Z [ l ] = W [ l ] A [ l − 1 ] + b [ l ] Z^{[l]}=W^{[l]}A^{[l-1]+b^{[l]}} Z[l]=W[l]A[l−1]+b[l]，其中 Z [ 0 ] = X Z^{[0]}=X Z[0]=X
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
带有激活函数的线性Forward
def 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
L层的Forward部分
def 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)],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
Cost函数
这里的成本函数依然使用了之前用过的公式
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(np.log(AL)*Y+(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
Backward部分
线性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 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]print(A_prev.shape)print(dZ.shape)### 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 = 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
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 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
L 层的backward过程
def 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, 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_tempgrads["dW" + str(l + 1)] = dW_tempgrads["db" + str(l + 1)] = db_temp### END CODE HERE ###return grads
参数更新
def 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 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
代码作业——实现神经网络
这一次练习的目的是用神经网络进行监督学习
还是先引入包
import time
import numpy as np
import h5py
import matplotlib.pyplot as plt
import scipy
from PIL import Image
from scipy import ndimage
from dnn_app_utils_v2 import *%matplotlib inline
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'%load_ext autoreload
%autoreload 2np.random.seed(1)
数据集加载
train_x_orig, train_y, test_x_orig, test_y, classes = load_data()
下面的代码可以查看数据集中的图像，以及查看样本数等
# Example of a picture
index = 6
plt.imshow(train_x_orig[index])
print ("y = " + str(train_y[0,index]) + ". It's a " + classes[train_y[0,index]].decode("utf-8") + " picture.")# Explore your dataset
m_train = train_x_orig.shape[0]
num_px = train_x_orig.shape[1]
m_test = test_x_orig.shape[0]print ("Number of training examples: " + str(m_train))
print ("Number of testing examples: " + str(m_test))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_x_orig shape: " + str(train_x_orig.shape))
print ("train_y shape: " + str(train_y.shape))
print ("test_x_orig shape: " + str(test_x_orig.shape))
print ("test_y shape: " + str(test_y.shape))
在把样本输入神经网络之前，要reshape一下，获得一个feature向量
# 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))
神经网络结构
二层网络（单隐层）
feature向量在输入之后，经过一个线性的forward和ReLU激活函数后，再经历一次线性的forward和sigmoid激活函数，最后输出。
L层神经网络
把二层神经网络中的linear-activation forward重复L-1次，最后一次还是linear-sigmoid forward
实现两层的神经网络
n_x = 12288 # num_px * num_px * 3
n_h = 7
n_y = 1
layers_dims = (n_x, n_h, n_y)def 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, activation="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, activation="sigmoid")dA0, dW1, db1 = linear_activation_backward(dA1, cache1, activation="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=0.0075)### 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
调用如下
parameters = two_layer_model(train_x, train_y, layers_dims = (n_x, n_h, n_y), num_iterations = 2500, print_cost=True)
二层网络的准确率为72%，和逻辑回归的准确率差不多，不过，增加网络层数，或许能提升准确率。
实现L层神经网络
在开始搭建L层神经网络之前，先列出可能用到的help function
这里就先搭一个5层的神经网络
layers_dims = [12288, 20, 7, 5, 1] # 5-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=0.0075)### 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
遇到的问题
上面这个图像明显不太对劲，事实证明，其准确率是74%，并没有比逻辑回归高那么多。
之后把学习率learning_rate调成0.0015,测试集上的准确率上升到了80%
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