1. 词向量上的操作(Operations on word vectors)

因为词嵌入的训练是非常耗资源的，所以ML从业者通常 都是 选择加载训练好 的 词嵌入(Embedding)数据集。(不用自己训练啦~~~)

任务：

导入 预训练词向量，使用余弦相似性(cosine similarity)计算相似度

使用词嵌入来解决 “Man is to Woman as King is to __.” 之类的 词语类比问题

修改词嵌入 来减少它们的性别歧视

import numpy as np

from w2v_utils import *

导入词向量，这个任务中，使用 50维的GloVe向量 来表示单词，导入 load the word_to_vec_map.

words, word_to_vec_map = read_glove_vecs('data/glove.6B.50d.txt') # Embedding vector已知

print(list(words)[:10])

print(word_to_vec_map['mauzac'])

['1945gmt', 'mauzac', 'kambojas', '4-b', 'wakan', 'lorikeet', 'paratroops', 'wittkower', 'messageries', 'oliver']

[ 0.049225 -0.36274 -0.31555 -0.2424 -0.58761 0.27733

0.059622 -0.37908 -0.59505 0.78046 0.3348 -0.90401

0.7552 -0.30247 0.21053 0.03027 0.22069 0.40635

0.11387 -0.79478 -0.57738 0.14817 0.054704 0.973

-0.22502 1.3677 0.14288 0.83708 -0.31258 0.25514

-1.2681 -0.41173 0.0058966 -0.64135 0.32456 -0.84562

-0.68853 -0.39517 -0.17035 -0.54659 0.014695 0.073697

0.1433 -0.38125 0.22585 -0.70205 0.9841 0.19452

-0.21459 0.65096 ]

导入的数据：

words: 词汇表中单词集.

word_to_vec_map: dictionary 映射单词到它们的 GloVe vector 表示.

Embedding vectors vs one-hot vectors

one-hot向量不能很好捕捉单词之间的相似度水平(每一个one-hot向量与任何其他one-hot向量有相同的欧几里得距离(Euclidean distance))

Embedding vector，如Glove vector提供了许多关于 单个单词含义 的有用信息

下面介绍如何使用 GloVe向量 来度量两个单词之间的 相似性

1.1 余弦相似度(Cosine similarity)

为了测量两个单词之间的相似性, 我们需要一个方法来测量两个单词的两个embedding vectors的相似性程度。 给定两个向量 \(u\) 和 \(v\), cosine similarity 定义如下：

\[\text{CosineSimilarity(u, v)} = \frac {u \cdot v} {||u||_2 ||v||_2} = cos(\theta) \tag{1}

\]

\(u \cdot v\) 是两个向量的点积(内积)

\(||u||_2\) 向量 \(u\) 的范数(长度)

\(\theta\) 是 \(u\) 与 \(v\) 之间的夹角角度

余弦相似性 依赖于 \(u\) and \(v\) 的角度.

如果 \(u\) 和 \(v\) 很相似， 那么 \(cos(\theta)\) 越接近1.

如果 \(u\) 和 \(v\) 不相似， 那么 \(cos(\theta)\) 得到一个很小的值.

**Figure 1**: The cosine of the angle between two vectors is a measure their similarity

Exercise： 实现函数 cosine_similarity() 来计算两个词向量之间的 相似性.

Reminder： \(u\) 的范式定义为 \(||u||_2 = \sqrt{\sum_{i=1}^{n} u_i^2}\)

提示： 使用 np.dot, np.sum, or np.sqrt 很有用.

# GRADED FUNCTION: cosine_similarity

def cosine_similarity(u, v):

"""

Cosine similarity reflects the degree of similarity between u and v

Arguments:

u -- a word vector of shape (n,)

v -- a word vector of shape (n,)

Returns:

cosine_similarity -- the cosine similarity between u and v defined by the formula above.

"""

distance = 0.0

### START CODE HERE ###

# Compute the dot product between u and v (≈1 line)

dot = np.sum(u * v)

# Compute the L2 norm of u (≈1 line)

norm_u = np.sqrt(np.sum(np.square(u)))

# Compute the L2 norm of v (≈1 line)

norm_v = np.sqrt(np.sum(np.square(v)))

# Compute the cosine similarity defined by formula (1) (≈1 line)

cosine_similarity = dot / (norm_u * norm_v)

### END CODE HERE ###

return cosine_similarity

测试：

father = word_to_vec_map["father"]

mother = word_to_vec_map["mother"]

ball = word_to_vec_map["ball"]

crocodile = word_to_vec_map["crocodile"]

france = word_to_vec_map["france"]

italy = word_to_vec_map["italy"]

paris = word_to_vec_map["paris"]

rome = word_to_vec_map["rome"]

print("cosine_similarity(father, mother) = ", cosine_similarity(father, mother))

print("cosine_similarity(ball, crocodile) = ",cosine_similarity(ball, crocodile))

print("cosine_similarity(france - paris, rome - italy) = ",cosine_similarity(france - paris, rome - italy)) # (国家-首都, 首都-国家)-->接近-1

print("cosine_similarity(france - paris, italy - rome) = ",cosine_similarity(france - paris, italy - rome))

cosine_similarity(father, mother) = 0.890903844289

cosine_similarity(ball, crocodile) = 0.274392462614

cosine_similarity(france - paris, rome - italy) = -0.675147930817

cosine_similarity(france - paris, italy - rome) = 0.675147930817

随意的修改单词，查看他们相似性。

1.2 词类比工作(Word analogy task)

在词类比工作(word analogy task)中，我们完成句子：

"a is to b as c is to ____".

举例：

'man is to woman as king is to queen'.

我们尝试找到一个单词 d，使得相关的单词向量 \(e_a, e_b, e_c, e_d\) 以下列方式关联:

\(e_b - e_a \approx e_d - e_c\)

我们将使用cosine similarity测量 \(e_b - e_a\) 和 \(e_d - e_c\) 的相似性.

Exercise：完成函数complete_analogy 实现 word analogies.

# GRADED FUNCTION: complete_analogy

def complete_analogy(word_a, word_b, word_c, word_to_vec_map):

"""

Performs the word analogy task as explained above: a is to b as c is to ____.

Arguments:

word_a -- a word, string

word_b -- a word, string

word_c -- a word, string

word_to_vec_map -- dictionary that maps words to their corresponding vectors.

Returns:

best_word -- the word such that v_b - v_a is close to v_best_word - v_c, as measured by cosine similarity

"""

# convert words to lower case

word_a, word_b, word_c = word_a.lower(), word_b.lower(), word_c.lower()

### START CODE HERE ###

# Get the word embeddings v_a, v_b and v_c (≈1-3 lines)

e_a, e_b, e_c = word_to_vec_map[word_a],word_to_vec_map[word_b],word_to_vec_map[word_c]

### END CODE HERE ###

words = word_to_vec_map.keys()

max_cosine_sim = -100 # Initialize max_cosine_sim to a large negative number

best_word = None # Initialize best_word with None, it will help keep track of the word to output

# loop over the whole word vector set

for w in words:

# to avoid best_word being one of the input words, pass on them.

if w in [word_a, word_b, word_c] :

continue

### START CODE HERE ###

# Compute cosine similarity between the vector (e_b - e_a) and the vector ((w's vector representation) - e_c) (≈1 line)

cosine_sim = cosine_similarity(e_b - e_a,word_to_vec_map[w] - e_c)

# If the cosine_sim is more than the max_cosine_sim seen so far,

# then: set the new max_cosine_sim to the current cosine_sim and the best_word to the current word (≈3 lines)

if cosine_sim > max_cosine_sim:

max_cosine_sim = cosine_sim

best_word = w

### END CODE HERE ###

return best_word

测试：

triads_to_try = [('italy', 'italian', 'spain'), ('india', 'delhi', 'japan'), ('man', 'woman', 'boy'), ('small', 'smaller', 'large')]

for triad in triads_to_try:

print ('{} -> {} :: {} -> {}'.format( *triad, complete_analogy(*triad,word_to_vec_map)))

italy -> italian :: spain -> spanish

india -> delhi :: japan -> tokyo

man -> woman :: boy -> girl

small -> smaller :: large -> larger

也存在一些单词，算法不能给出正确答案：

triad = ['small', 'smaller', 'big']

print ('{} -> {} :: {} -> {}'.format( *triad, complete_analogy(*triad, word_to_vec_map)))

small -> smaller :: big -> competitors

1.3 总结

Cosine similarity 求两个词向量的相似度不错

对于NLP应用，通常使用预训练好的词向量数据集

2. 除偏词向量(Debiasing word vectors)

在这一部分，我们将研究反映在词嵌入中的性别偏差，并试着去去除这一些偏差.

首先看一下 GloVe词嵌入如何关联性别的，你将计算一个向量 \(g = e_{woman}-e_{man}\)，\(e_{woman}\) 代表 woman 的词向量，\(e_{man}\)代表man的词向量，得到的结果 \(g\) 粗略的包含性别这一概念，计算 \(g_1 = e_{mother}-e_{father}\)，\(g_2 = e_{girl}-e_{boy}\) 的平均值可能会更准确点，现在使用 \(e_{woman}-e_{man}\) 足够了。

g = word_to_vec_map['woman'] - word_to_vec_map['man'] # 计算w与g的余弦相似度时，为正更接近女人，为负更接近男人

print(g)

[-0.087144 0.2182 -0.40986 -0.03922 -0.1032 0.94165

-0.06042 0.32988 0.46144 -0.35962 0.31102 -0.86824

0.96006 0.01073 0.24337 0.08193 -1.02722 -0.21122

0.695044 -0.00222 0.29106 0.5053 -0.099454 0.40445

0.30181 0.1355 -0.0606 -0.07131 -0.19245 -0.06115

-0.3204 0.07165 -0.13337 -0.25068714 -0.14293 -0.224957

-0.149 0.048882 0.12191 -0.27362 -0.165476 -0.20426

0.54376 -0.271425 -0.10245 -0.32108 0.2516 -0.33455

-0.04371 0.01258 ]

考虑不同单词与 \(g\) 的余弦相似度， 考虑一下正相似值 与 负余弦相似值的关系。

print ('List of names and their similarities with constructed vector:')

# girls and boys name

name_list = ['john', 'marie', 'sophie', 'ronaldo', 'priya', 'rahul', 'danielle', 'reza', 'katy', 'yasmin']

for w in name_list:

print (w, cosine_similarity(word_to_vec_map[w], g))

List of names and their similarities with constructed vector:

john -0.23163356146

marie 0.315597935396

sophie 0.318687898594

ronaldo -0.312447968503

priya 0.17632041839

rahul -0.169154710392

danielle 0.243932992163

reza -0.079304296722

katy 0.283106865957

yasmin 0.233138577679

可以看到，女性的名字与 \(g\) 的余弦相似度为正，男性的名字与 \(g\) 的余弦相似度为负。

尝试其他

print('Other words and their similarities:')

word_list = ['lipstick', 'guns', 'science', 'arts', 'literature', 'warrior','doctor', 'tree', 'receptionist',

'technology', 'fashion', 'teacher', 'engineer', 'pilot', 'computer', 'singer']

for w in word_list:

print (w, cosine_similarity(word_to_vec_map[w], g))

Other words and their similarities:

lipstick 0.276919162564

guns -0.18884855679

science -0.0608290654093

arts 0.00818931238588

literature 0.0647250443346

warrior -0.209201646411

doctor 0.118952894109

tree -0.0708939917548

receptionist 0.330779417506

technology -0.131937324476

fashion 0.0356389462577

teacher 0.179209234318

engineer -0.0803928049452

pilot 0.00107644989919

computer -0.103303588739

singer 0.185005181365

可以发现，比如“computer”就接近于“man”，“literature ”接近于“woman”，但是这些都是不对的一些观念，那么我们该如何减少这些偏差呢？

对于一些特殊的词汇而言，比如“男演员(actor)”与“女演员(actress)”或者“祖母(grandmother)”与“祖父(grandfather)”之间应该是具有性别差异的，而其他词汇比如“接待员(receptionist)”与“技术(technology )”是不应该有性别差异的，当我们处理这些词汇的时候应该区别对待。

2.1 中和非性别特定词汇的偏见(Neutralize bias for non-gender specific words)

下面的一张图表示了消除偏差之后的效果。如果我们使用的是50维的词嵌入，那么50维的空间可以分为两个部分： 偏置方向 \(g\)，和 剩下的49维 \(g_{\perp}\)。 在线性代数中，将 49维的 \(g_{\perp}\) 与 \(g\) 垂直(perpendicular)或正交("orthogonal")，即 \(g_{\perp}\) 与 \(g\) 成90°角 。在中和步骤中，取一个向量，如 \(e_{receptionist}\)向量， 将 \(g\) 方向的组成 归零，得到 \(e_{receptionist}^{debiased}\) 向量。

即使 \(g_{\perp}\) 是 49维， 鉴于只能在2D屏幕上绘制图像的局限性，我们使用下面的一维坐标轴来说明它。

**Figure 2**: The word vector for "receptionist" represented before and after applying the neutralize operation.

Exercise: 实现 neutralize() 来消除词汇 的偏见，如 "receptionist" 和 "scientist"。 给定一个词嵌入输入\(e\)，可以使用下面的公式计算 \(e^{debiased}\):

\[e^{bias\_component} = \frac{e \cdot g}{||g||_2^2} * g\tag{2}

\]

\[e^{debiased} = e - e^{bias\_component}\tag{3}

\]

\(e^{bias\_component}\) 是 \(e\) 在 \(g\) 方向的投影。

def neutralize(word, g, word_to_vec_map):

"""

Removes the bias of "word" by projecting it on the space orthogonal to the bias axis.

This function ensures that gender neutral words are zero in the gender subspace.

Arguments:

word -- string indicating the word to debias

g -- numpy-array of shape (50,), corresponding to the bias axis (such as gender)

word_to_vec_map -- dictionary mapping words to their corresponding vectors.

Returns:

e_debiased -- neutralized word vector representation of the input "word"

"""

### START CODE HERE ###

# Select word vector representation of "word". Use word_to_vec_map. (≈ 1 line)

e = word_to_vec_map[word]

# Compute e_biascomponent using the formula give above. (≈ 1 line)

e_biascomponent = np.divide(np.sum(e * g), np.square(np.linalg.norm(g))) * g

# Neutralize e by substracting e_biascomponent from it

# e_debiased should be equal to its orthogonal projection. (≈ 1 line)

e_debiased = e - e_biascomponent

### END CODE HERE ###

return e_debiased

测试：

e = "receptionist"

print("cosine similarity between " + e + " and g, before neutralizing: ", cosine_similarity(word_to_vec_map["receptionist"], g))

e_debiased = neutralize("receptionist", g, word_to_vec_map)

print("cosine similarity between " + e + " and g, after neutralizing: ", cosine_similarity(e_debiased, g))

cosine similarity between receptionist and g, before neutralizing: 0.330779417506

cosine similarity between receptionist and g, after neutralizing: -5.60374039375e-17

2.2 性别词的均衡算法(Equalization algorithm for gender-specific words)

接下来我们来看看在关于有特定性别词组中，如何将它们进行均衡，比如“男演员”与“女演员”中，与“保姆”一词更接近的是“女演员”，我们可以消去“保姆”的性别偏差，但是这并不能保证“保姆”一词与“男演员”与“女演员”之间的距离相等，我们要学的均衡算法将解决这个问题。

均衡(equalization)关键思想：确保 一对特定的单词 与 49维 \(g_\perp\) 距离相等。均衡步骤还确保 两个被均衡的step 现在与 \(e_{receptionist}^{debiased}\) 或 任何已被中和的其他工作的距离相同。下面展示equalization如何工作：

线性代数的推导过程很复杂(See Bolukbasi et al., 2016 for details.) 关键的公式：

\[\mu = \frac{e_{w1} + e_{w2}}{2}\tag{4}

\]

\[ \mu_{B} = \frac {\mu \cdot \text{bias_axis}}{||\text{bias_axis}||_2^2} *\text{bias_axis}

\tag{5}\]

\[\mu_{\perp} = \mu - \mu_{B} \tag{6}

\]

\[ e_{w1B} = \frac {e_{w1} \cdot \text{bias_axis}}{||\text{bias_axis}||_2^2} *\text{bias_axis}

\tag{7}\]

\[ e_{w2B} = \frac {e_{w2} \cdot \text{bias_axis}}{||\text{bias_axis}||_2^2} *\text{bias_axis}

\tag{8}\]

\[e_{w1B}^{corrected} = \sqrt{ |{1 - ||\mu_{\perp} ||^2_2} |} * \frac{e_{\text{w1B}} - \mu_B} {|(e_{w1} - \mu_{\perp}) - \mu_B|} \tag{9}

\]

\[e_{w2B}^{corrected} = \sqrt{ |{1 - ||\mu_{\perp} ||^2_2} |} * \frac{e_{\text{w2B}} - \mu_B} {|(e_{w2} - \mu_{\perp}) - \mu_B|} \tag{10}

\]

\[e_1 = e_{w1B}^{corrected} + \mu_{\perp} \tag{11}

\]

\[e_2 = e_{w2B}^{corrected} + \mu_{\perp} \tag{12}

\]

Exercise: 使用这些上述公式 得到这对单词的最终均衡版本.

def equalize(pair, bias_axis, word_to_vec_map):

"""

Debias gender specific words by following the equalize method described in the figure above.

Arguments:

pair -- pair of strings of gender specific words to debias, e.g. ("actress", "actor")

bias_axis -- numpy-array of shape (50,), vector corresponding to the bias axis, e.g. gender

word_to_vec_map -- dictionary mapping words to their corresponding vectors

Returns

e_1 -- word vector corresponding to the first word

e_2 -- word vector corresponding to the second word

"""

### START CODE HERE ###

# Step 1: Select word vector representation of "word". Use word_to_vec_map. (≈ 2 lines)

w1, w2 = pair

e_w1, e_w2 = word_to_vec_map[w1],word_to_vec_map[w2]

# Step 2: Compute the mean of e_w1 and e_w2 (≈ 1 line)

mu = (e_w1 + e_w2) / 2

# Step 3: Compute the projections of mu over the bias axis and the orthogonal axis (≈ 2 lines)

mu_B = np.divide(np.dot(mu, bias_axis) * bias_axis, np.sum(bias_axis**2))

mu_orth = mu - mu_B

# Step 4: Use equations (7) and (8) to compute e_w1B and e_w2B (≈2 lines)

e_w1B = (np.dot(e_w1,bias_axis) * bias_axis) / (np.sum(bias_axis**2))

e_w2B = (np.dot(e_w2,bias_axis) * bias_axis) / (np.sum(bias_axis**2))

# Step 5: Adjust the Bias part of e_w1B and e_w2B using the formulas (9) and (10) given above (≈2 lines)

corrected_e_w1B = (np.sqrt(np.abs(1 - (np.sum(mu_orth**2))))) * np.divide(e_w1B - mu_B, np.abs((e_w1 - mu_orth) - mu_B))

corrected_e_w2B = (np.sqrt(np.abs(1 - (np.sum(mu_orth**2))))) * np.divide(e_w2B - mu_B, np.abs((e_w2 - mu_orth) - mu_B))

# Step 6: Debias by equalizing e1 and e2 to the sum of their corrected projections (≈2 lines)

e1 = corrected_e_w1B + mu_orth

e2 = corrected_e_w2B + mu_orth

### END CODE HERE ###

return e1, e2

测试：

print("cosine similarities before equalizing:")

print("cosine_similarity(word_to_vec_map[\"man\"], gender) = ", cosine_similarity(word_to_vec_map["man"], g))

print("cosine_similarity(word_to_vec_map[\"woman\"], gender) = ", cosine_similarity(word_to_vec_map["woman"], g))

print()

e1, e2 = equalize(("man", "woman"), g, word_to_vec_map)

print("cosine similarities after equalizing:")

print("cosine_similarity(e1, gender) = ", cosine_similarity(e1, g))

print("cosine_similarity(e2, gender) = ", cosine_similarity(e2, g))

cosine similarities before equalizing:

cosine_similarity(word_to_vec_map["man"], gender) = -0.117110957653

cosine_similarity(word_to_vec_map["woman"], gender) = 0.356666188463

cosine similarities after equalizing:

cosine_similarity(e1, gender) = -0.716572752584

cosine_similarity(e2, gender) = 0.739659647493

Debiasing algorithms 减少偏见很有用，但并不完美，不能消除所有的偏置痕迹

如：这种实现的一个缺点： \(g\) 的偏差方向 只使用 单词对 woman 和 man 来定义。如果通过计算 \(g_1 = e_{woman} - e_{man}\)，\(g_2 = e_{mother} - e_{father}\)，\(g_3 = e_{girl} - e_{boy}\)，然后使用 \(g = avg(g_1, g_2, g_3)\) 来计算他它们平均，你将更好的估计50维词嵌入空间中的 "gender" 维度。