本文節選自吳恩達老師《深度學習專項課程》編程作業，在此表示感謝。

課程鏈接：https://www.deeplearning.ai/deep-learning-specialization

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目錄

1 - Building basic functions with numpy

1.1 - np.exp(), sigmoid function

1.2 - Sigmoid gradient

1.3 - Reshaping arrays（常用）

1.4 - Normalizing rows（常用）

1.5 - Broadcasting and the softmax function（理解）

2 - Vectorization

2.1 - Implement the L1 and L2 loss functions

3 - 推薦閱讀：

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1 - Building basic functions with numpy

1.1 - np.exp(), sigmoid function

import numpy as npx = np.array([1, 2, 3]) print(np.exp(x))x = np.array([1, 2, 3]) print (x + 3)

Exercise: Implement the sigmoid function using numpy.

Instructions: x could now be either a real number, a vector, or a matrix. The data structures we use in numpy to represent these shapes (vectors, matrices...) are called numpy arrays. You don't need to know more for now.

提示：x可能代表一個實數、向量或者矩陣。在numpy中使用數組來表示x的數據類型。

import numpy as np def sigmoid(x):s = 1 / (1 + np.exp(-x)) return s

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1.2 - Sigmoid gradient

Exercise: Implement the function sigmoid_grad() to compute the gradient of the sigmoid function with respect to its input x. The formula is:

實現sigmoid_grad()函數來計算sigmoid函數的梯度，sigmoid函數求導公式如下：

You often code this function in two steps:

Set s to be the sigmoid of x. You might find your sigmoid(x) function useful.

Compute sigmoid gradient

def sigmoid_derivative(x):s = 1 / (1 + np.exp(-x))ds = s *(1-s)return dsx = np.array([1, 2, 3]) print ("sigmoid_derivative(x) = " + str(sigmoid_derivative(x)))

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1.3 - Reshaping arrays（常用）

Two common numpy functions used in deep learning are?np.shape?and?np.reshape().

• X.shape is used to get the shape (dimension) of a matrix/vector X
• X.reshape(...) is used to reshape X into some other dimension

For example, in computer science, an image is represented by a 3D array of shape?(length,height,depth=3). However, when you read an image as the input of an algorithm you convert it to a vector of shape (length?height?3,1). In other words, you "unroll", or reshape, the 3D array into a 1D vector.

Exercise: Implement?image2vector()?that takes an input of shape (length, height, 3) and returns a vector of shape (length*height*3, 1). For example, if you would like to reshape an array v of shape (a, b, c) into a vector of shape (a*b,c) you would do:

def image2vector(image):v = image.reshape((image.shape[0] * image.shape[1] * image.shape[2], 1))return v

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1.4 - Normalizing rows（常用）

Another common technique we use in Machine Learning and Deep Learning is to normalize our data. It often leads to a better performance because gradient descent converges faster after normalization. Here, by normalization we mean changing x to (dividing each row vector of x by its norm).

Exercise: Implement normalizeRows() to normalize the rows of a matrix. After applying this function to an input matrix x, each row of x should be a vector of unit length (meaning length 1).

def normalizeRows(x):"""Argument:x -- A numpy matrix of shape (n, m)Returns:x -- The normalized (by row) numpy matrix. """# Compute x_norm as the norm 2 of x. Use np.linalg.norm(..., ord = 2, axis = ..., keepdims = True)x_norm = np.linalg.norm(x, ord=2, axis=1, keepdims=True) # Divide x by its norm.x = x / x_normreturn x

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1.5 - Broadcasting and the softmax function（理解）

A very important concept to understand in numpy is "broadcasting". It is very useful for performing mathematical operations between arrays of different shapes. For the full details on broadcasting, you can read the official?broadcasting documentation.

Exercise: Implement a softmax function using numpy. You can think of softmax as a normalizing function used when your algorithm needs to classify two or more classes. You will learn more about softmax in the second course of this specialization.

Instructions:

def softmax(x):"""Argument:x -- A numpy matrix of shape (n,m)Returns:s -- A numpy matrix equal to the softmax of x, of shape (n,m)"""x_exp = np.exp(x)# Create a vector x_sum that sums each row of x_exp. Use np.sum(..., axis = 1, keepdims = True).x_sum = np.sum(x_exp, axis=1, keepdims=True)s = x_exp / x_sumreturn sx = np.array([[9, 2, 5, 0, 0],[7, 5, 0, 0 ,0]]) print("softmax(x) = " + str(softmax(x)))

**What you need to remember:** - np.exp(x) works for any np.array x and applies the exponential function to every coordinate - the sigmoid function and its gradient - image2vector is commonly used in deep learning - np.reshape is widely used. In the future, you'll see that keeping your matrix/vector dimensions straight will go toward eliminating a lot of bugs. - numpy has efficient built-in functions - broadcasting is extremely useful.

你需要記住的：

• np.exp(x)對任意數組都有效并且對每個元素都進行指數計算；
• sigmoid和其梯度函數以及image2vector函數以及np.reshape函數是深度學習中最常使用的函數；
• numpy的廣播機制是最常使用到的。

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2 - Vectorization

2.1 - Implement the L1 and L2 loss functions

Exercise: Implement the numpy vectorized version of the L1 loss. You may find the function abs(x) (absolute value of x) useful.

Reminder:

• The loss is used to evaluate the performance of your model. The bigger your loss is, the more different your predictions ?are from the true values . In deep learning, you use optimization algorithms like Gradient Descent to train your model and to minimize the cost.
• L1 loss is defined as:

def L1(yhat, y):loss = np.sum(abs(yhat-y))return lossyhat = np.array([.9, 0.2, 0.1, .4, .9]) y = np.array([1, 0, 0, 1, 1]) print("L1 = " + str(L1(yhat,y)))

L2 loss is defined as:

def L2(yhat, y):loss = np.dot((y-yhat),(y-yhat)) return lossyhat = np.array([.9, 0.2, 0.1, .4, .9]) y = np.array([1, 0, 0, 1, 1]) print("L2 = " + str(L2(yhat,y)))

**What to remember:** - Vectorization is very important in deep learning. It provides computational efficiency and clarity. - You have reviewed the L1 and L2 loss. - You are familiar with many numpy functions such as np.sum, np.dot, np.multiply, np.maximum, etc.

需要記住的：

• 向量化是深度學習中最重要的，使得計算更加高效和清晰；
• L1和 L2損失函數公式；
• numpy中其它有用的函數：np.sum，np.dot，np.multiply，np.maximum。

3 - 推薦閱讀：

• Python之Numpy入門實戰教程（1）：基礎篇
• Python之Numpy入門實戰教程（2）：進階篇之線性代數
• numpy手冊

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