Building your Deep Neural Network Step by Step

Posted by LudoArt on February 17, 2019

1 导入所需的包

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'

np.random.seed(1)

包介绍

  • numpy 用Python进行科学计算的基本软件包。
  • matplotlib 是一个用于在Python中绘制图表的库。
  • dnn_utils 提供一些必要的函数。
  • testCases 提供一些测试用例来评估函数的正确性。

2 概要

主要步骤如下:

  • 为一个2层神经网络和一个L层神经网络初始化参数
  • 建立前向传播模型
    • 完成一层中前向传播的线性部分(得到
    • 激活函数 relu/sigmoid
    • 将前两步结合到一个新的 LINEAR->ACTIVATION 前向函数中
    • 在新建的 L_model_forward 函数中,将 LINEAR->RELU 前向函数执行L-1 次(1~L-1层),然后再执行一次 LINEAR->SIGMOID 前向函数(最后的L层)
  • 计算损失
  • 建立反向传播模型
    • 完成一层中反向传播的线性部分
    • 激活函数的导数 relu_backward/sigmoid_backward
    • 将前两步结合到一个新的 LINEAR->ACTIVATION 反向函数中
    • 在新建的 L_model_backward 函数中,将 LINEAR->SIGMOID 反向函数执行1次,然后再执行L-1次 LINEAR->RELU 反向函数
  • 更新参数

final outline

3 初始化参数

3.1 2层神经网络

该模型的结构是: LINEAR -> RELU -> LINEAR -> SIGMOID

# 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:
    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

3.2 L层神经网络

该模型的结构是: [LINEAR -> RELU] × (L-1) -> LINEAR -> SIGMOID

# GRADED FUNCTION: initialize_parameters_deep
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)  # number of layers in the network

    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

4 前向传播模型

4.1 线性部分(Linear)

线性部分要计算的公式如下:

此处

# GRADED FUNCTION: linear_forward
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

4.2 线性激活部分(Linear-Activation)

为了更方便,将把两个函数( LINEARACTIVATION )放入一个函数( LINEAR-> ACTIVATION )当中。

因此,该函数将首先执行LINEAR步骤,然后执行ACTIVATION步骤。

# GRADED FUNCTION: linear_activation_forward
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":
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = sigmoid(Z)

    elif activation == "relu":
        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

4.3 L层前向模型

该模型需执行使用RELU的 linear_activation_forward 函数L-1次,然后执行使用SIGMOID的 linear_activation_forward 函数一次

model_architecture_kiank

# GRADED FUNCTION: L_model_forward
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)], "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)], "sigmoid")
    caches.append(cache)

    assert (AL.shape == (1, X.shape[1]))

    return AL, caches

5 计算成本

计算成本的公式如下:

# GRADED FUNCTION: compute_cost
def compute_cost(AL, Y):
    """
    Implement the cost function

    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 = -np.sum(np.multiply(Y, np.log(AL)) + np.multiply(1 - Y, np.log(1 - AL))) / m

    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

6 反向传播模型

backprop_kiank

6.1 线性部分(Linear)

假设已经计算了导数

现在需要求得参数

linearback_kiank

可使用以下公式:

# GRADED FUNCTION: linear_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 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 = np.dot(dZ, A_prev.T) / m
    db = np.sum(dZ, axis=1, keepdims=True) / m
    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

6.2 线性激活部分(Linear-Activation)

为了更方便,将把两个函数(LINEARACTIVATION)放入一个函数(LINEAR-> ACTIVATION)当中。

因此,该函数将首先执行ACTIVATION步骤,然后执行LINEAR步骤。

其中 为激活函数的导数

# GRADED FUNCTION: 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 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

6.3 L层反向模型

mn_backward

首先计算

dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))

紧接着,便可以利用这个导数 dAL 继续进行反向计算

首先进行一次 LINEAR->SIGMOID 反向函数的计算,然后再循环计算 LINEAR->RELU 反向函数L-1次

应将每个 dAdWdb 存储在 grads 字典中,以便进行后续参数的更新

# GRADED FUNCTION: L_model_backward
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" (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 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,
                                                                                                  "sigmoid")

    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)]
        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_temp
        grads["dW" + str(l + 1)] = dW_temp
        grads["db" + str(l + 1)] = db_temp

    return grads

6.4 更新参数

使用梯度下降来更新参数:

此处的 是学习率

# GRADED FUNCTION: update_parameters
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