Distributed Representations of Words and Phrases
and their Compositionality
Tomas Mikolov
Google Inc.
Mountain View
mikolov@google.com
Ilya Sutskever
Google Inc.
Mountain View
ilyasu@google.com
Kai Chen
Google Inc.
Mountain View
kai@google.com
Greg Corrado
Google Inc.
Mountain View
gcorrado@google.com
Jeffrey Dean
Google Inc.
Mountain View
jeff@google.com
Abstract
The recently introduced continuous Skip-gram model is an efﬁcient method for
learning high-quality distributed vector representations that capture a large number
of precise syntactic and semantic word relationships. In this paper we present
several extensions that improve both the quality of the vectors and the training
speed. By subsampling of the frequent words we obtain signiﬁcant speedup and
also learn more regular word representations. We also describe a simple alternative
to the hierarchical softmax called negative sampling.
An inherent limitation of word representations is their indifference to word order
and their inability to represent idiomatic phrases. For example, the meanings of
“Canada” and “Air” cannot be easily combined to obtain “Air Canada”. Motivated
by this example, we present a simple method for ﬁnding phrases in text, and show
that learning good vector representations for millions of phrases is possible.
1 Introduction
Distributed representations of words in a vector space help learning algorithms to achieve better
performance in natural language processing tasks by grouping similar words. One of the earliest use
of word representations dates back to 1986 due to Rumelhart, Hinton, and Williams [13]. This idea
has since been applied to statistical language modeling with considerable success [1]. The follow
up work includes applications to automatic speech recognition and machine translation [14, 7], and
a wide range of NLP tasks [2, 20, 15, 3, 18, 19, 9].
Recently, Mikolov et al. [8] introduced the Skip-gram model, an efﬁcient method for learning highquality
vector representations of words from large amounts of unstructured text data. Unlike most
of the previously used neural network architectures for learning word vectors, training of the Skipgram
model (see Figure 1) does not involve dense matrix multiplications. This makes the training
extremely efﬁcient: an optimized single-machine implementation can train on more than 100 billion
words in one day.
The word representations computed using neural networks are very interesting because the learned
vectors explicitly encode many linguistic regularities and patterns. Somewhat surprisingly, many of
these patterns can be represented as linear translations. For example, the result of a vector calculation
vec(“Madrid”) - vec(“Spain”) + vec(“France”) is closer to vec(“Paris”) than to any other word
vector [9, 8].
1
Figure 1: The Skip-gram model architecture. The training objective is to learn word vector representations
that are good at predicting the nearby words.
In this paper we present several extensions of the original Skip-gram model. We show that subsampling
of frequent words during training results in a signiﬁcant speedup (around 2x - 10x), and
improves accuracy of the representations of less frequent words. In addition, we present a simpliﬁed
variant of Noise Contrastive Estimation (NCE) [4] for training the Skip-gram model that results
in faster training and better vector representations for frequent words, compared to more complex
hierarchical softmax that was used in the prior work [8].
Word representations are limited by their inability to represent idiomatic phrases that are not compositions
of the individual words. For example, “Boston Globe” is a newspaper, and so it is not a
natural combination of the meanings of “Boston” and “Globe”. Therefore, using vectors to represent
the whole phrases makes the Skip-gram model considerably more expressive. Other techniques
that aim to represent meaning of sentences by composing the word vectors, such as the recursive
autoencoders [15], would also beneﬁt from using phrase vectors instead of the word vectors.
The extension from word based to phrase based models is relatively simple. First we identify a large
number of phrases using a data-driven approach, and then we treat the phrases as individual tokens
during the training. To evaluate the quality of the phrase vectors, we developed a test set of analogical
reasoning tasks that contains both words and phrases. A typical analogy pair from our test set is
“Montreal”:“Montreal Canadiens”::“Toronto”:“Toronto Maple Leafs”. It is considered to have been
answered correctly if the nearest representation to vec(“Montreal Canadiens”) - vec(“Montreal”) +
vec(“Toronto”) is vec(“Toronto Maple Leafs”).
Finally, we describe another interesting property of the Skip-gram model. We found that simple
vector addition can often produce meaningful results. For example, vec(“Russia”) + vec(“river”) is
close to vec(“Volga River”), and vec(“Germany”) + vec(“capital”) is close to vec(“Berlin”). This
compositionality suggests that a non-obvious degree of language understanding can be obtained by
using basic mathematical operations on the word vector representations.
2 The Skip-gram Model
The training objective of the Skip-gram model is to ﬁnd word representations that are useful for
predicting the surrounding words in a sentence or a document. More formally, given a sequence of
training words w1, w2, w3, . . . , wT , the objective of the Skip-gram model is to maximize the average
log probability
1
T
T
t=1 −c≤j≤c,j=0
log p(wt+j|wt) (1)
where c is the size of the training context (which can be a function of the center word wt). Larger
c results in more training examples and thus can lead to a higher accuracy, at the expense of the
2
training time. The basic Skip-gram formulation deﬁnes p(wt+j|wt) using the softmax function:
p(wO|wI) =
exp v′
wO
⊤
vwI
W
w=1 exp v′
w
⊤
vwI
(2)
where vw and v′
w are the “input” and “output” vector representations of w, and W is the number
of words in the vocabulary. This formulation is impractical because the cost of computing
∇ log p(wO|wI ) is proportional to W, which is often large (105
–107
terms).
2.1 Hierarchical Softmax
A computationally efﬁcient approximation of the full softmax is the hierarchical softmax. In the
context of neural network language models, it was ﬁrst introduced by Morin and Bengio [12]. The
main advantage is that instead of evaluating W output nodes in the neural network to obtain the
probability distribution, it is needed to evaluate only about log2(W) nodes.
The hierarchical softmax uses a binary tree representation of the output layer with the W words as
its leaves and, for each node, explicitly represents the relative probabilities of its child nodes. These
deﬁne a random walk that assigns probabilities to words.
More precisely, each word w can be reached by an appropriate path from the root of the tree. Let
n(w, j) be the j-th node on the path from the root to w, and let L(w) be the length of this path, so
n(w, 1) = root and n(w, L(w)) = w. In addition, for any inner node n, let ch(n) be an arbitrary
ﬁxed child of n and let [[x]] be 1 if x is true and -1 otherwise. Then the hierarchical softmax deﬁnes
p(wO|wI ) as follows:
p(w|wI ) =
L(w)−1
j=1
σ [[n(w, j + 1) = ch(n(w, j))]] · v′
n(w,j)
⊤
vwI (3)
where σ(x) = 1/(1 + exp(−x)). It can be veriﬁed that
W
w=1 p(w|wI ) = 1. This implies that the
cost of computing log p(wO|wI ) and ∇ log p(wO|wI ) is proportional to L(wO), which on average
is no greater than log W. Also, unlike the standard softmax formulation of the Skip-gram which
assigns two representations vw and v′
w to each word w, the hierarchical softmax formulation has
one representation vw for each word w and one representation v′
n for every inner node n of the
binary tree.
The structure of the tree used by the hierarchical softmax has a considerable effect on the performance.
Mnih and Hinton explored a number of methods for constructing the tree structure and the
effect on both the training time and the resulting model accuracy [10]. In our work we use a binary
Huffman tree, as it assigns short codes to the frequent words which results in fast training. It has
been observed before that grouping words together by their frequency works well as a very simple
speedup technique for the neural network based language models [5, 8].
2.2 Negative Sampling
An alternative to the hierarchical softmax is Noise Contrastive Estimation (NCE), which was introduced
by Gutmann and Hyvarinen [4] and applied to language modeling by Mnih and Teh [11].
NCE posits that a good model should be able to differentiate data from noise by means of logistic
regression. This is similar to hinge loss used by Collobert and Weston [2] who trained the models
by ranking the data above noise.
While NCE can be shown to approximately maximize the log probability of the softmax, the Skipgram
model is only concerned with learning high-quality vector representations, so we are free to
simplify NCE as long as the vector representations retain their quality. We deﬁne Negative sampling
(NEG) by the objective
log σ(v′
wO
⊤
vwI ) +
k
i=1
Ewi∼Pn(w) log σ(−v′
wi
⊤
vwI ) (4)
3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Country and Capital Vectors Projected by PCA
China
Japan
France
Russia
Germany
Italy
Spain
Greece
Turkey
Beijing
Paris
Tokyo
Poland
Moscow
Portugal
Berlin
Rome
Athens
Madrid
Ankara
Warsaw
Lisbon
Figure 2: Two-dimensional PCA projection of the 1000-dimensional Skip-gram vectors of countries and their
capital cities. The ﬁgure illustrates ability of the model to automatically organize concepts and learn implicitly
the relationships between them, as during the training we did not provide any supervised information about
what a capital city means.
which is used to replace every log P(wO|wI ) term in the Skip-gram objective. Thus the task is to
distinguish the target word wO from draws from the noise distribution Pn(w) using logistic regression,
where there are k negative samples for each data sample. Our experiments indicate that values
of k in the range 5–20 are useful for small training datasets, while for large datasets the k can be as
small as 2–5. The main difference between the Negative sampling and NCE is that NCE needs both
samples and the numerical probabilities of the noise distribution, while Negative sampling uses only
samples. And while NCE approximately maximizes the log probability of the softmax, this property
is not important for our application.
Both NCE and NEG have the noise distribution Pn(w) as a free parameter. We investigated a number
of choices for Pn(w) and found that the unigram distribution U(w) raised to the 3/4rd power (i.e.,
U(w)3/4
/Z) outperformed signiﬁcantly the unigram and the uniform distributions, for both NCE
and NEG on every task we tried including language modeling (not reported here).
2.3 Subsampling of Frequent Words
In very large corpora, the most frequent words can easily occur hundreds of millions of times (e.g.,
“in”, “the”, and “a”). Such words usually provide less information value than the rare words. For
example, while the Skip-gram model beneﬁts from observing the co-occurrences of “France” and
“Paris”, it beneﬁts much less from observing the frequent co-occurrences of “France” and “the”, as
nearly every word co-occurs frequently within a sentence with “the”. This idea can also be applied
in the opposite direction; the vector representations of frequent words do not change signiﬁcantly
after training on several million examples.
To counter the imbalance between the rare and frequent words, we used a simple subsampling approach:
each word wi in the training set is discarded with probability computed by the formula
P(wi) = 1 −
t
f(wi)
(5)
4
Method Time [min] Syntactic [%] Semantic [%] Total accuracy [%]
NEG-5 38 63 54 59
NEG-15 97 63 58 61
HS-Huffman 41 53 40 47
NCE-5 38 60 45 53
The following results use 10−5
subsampling
NEG-5 14 61 58 60
NEG-15 36 61 61 61
HS-Huffman 21 52 59 55
Table 1: Accuracy of various Skip-gram 300-dimensional models on the analogical reasoning task
as deﬁned in [8]. NEG-k stands for Negative Sampling with k negative samples for each positive
sample; NCE stands for Noise Contrastive Estimation and HS-Huffman stands for the Hierarchical
Softmax with the frequency-based Huffman codes.
where f(wi) is the frequency of word wi and t is a chosen threshold, typically around 10−5
.
We chose this subsampling formula because it aggressively subsamples words whose frequency
is greater than t while preserving the ranking of the frequencies. Although this subsampling formula
was chosen heuristically, we found it to work well in practice. It accelerates learning and even
signiﬁcantly improves the accuracy of the learned vectors of the rare words, as will be shown in the
following sections.
3 Empirical Results
In this section we evaluate the Hierarchical Softmax (HS), Noise Contrastive Estimation, Negative
Sampling, and subsampling of the training words. We used the analogical reasoning task1
introduced
by Mikolov et al. [8]. The task consists of analogies such as “Germany” : “Berlin” :: “France” : ?,
which are solved by ﬁnding a vector x such that vec(x) is closest to vec(“Berlin”) - vec(“Germany”)
+ vec(“France”) according to the cosine distance (we discard the input words from the search). This
speciﬁc example is considered to have been answered correctly if x is “Paris”. The task has two
broad categories: the syntactic analogies (such as “quick” : “quickly” :: “slow” : “slowly”) and the
semantic analogies, such as the country to capital city relationship.
For training the Skip-gram models, we have used a large dataset consisting of various news articles
(an internal Google dataset with one billion words). We discarded from the vocabulary all words
that occurred less than 5 times in the training data, which resulted in a vocabulary of size 692K.
The performance of various Skip-gram models on the word analogy test set is reported in Table 1.
The table shows that Negative Sampling outperforms the Hierarchical Softmax on the analogical
reasoning task, and has even slightly better performance than the Noise Contrastive Estimation. The
subsampling of the frequent words improves the training speed several times and makes the word
representations signiﬁcantly more accurate.
It can be argued that the linearity of the skip-gram model makes its vectors more suitable for such
linear analogical reasoning, but the results of Mikolov et al. [8] also show that the vectors learned
by the standard sigmoidal recurrent neural networks (which are highly non-linear) improve on this
task signiﬁcantly as the amount of the training data increases, suggesting that non-linear models also
have a preference for a linear structure of the word representations.
4 Learning Phrases
As discussed earlier, many phrases have a meaning that is not a simple composition of the meanings
of its individual words. To learn vector representation for phrases, we ﬁrst ﬁnd words that
appear frequently together, and infrequently in other contexts. For example, “New York Times” and
“Toronto Maple Leafs” are replaced by unique tokens in the training data, while a bigram “this is”
will remain unchanged.
1
code.google.com/p/word2vec/source/browse/trunk/questions-words.txt
5
Newspapers
New York New York Times Baltimore Baltimore Sun
San Jose San Jose Mercury News Cincinnati Cincinnati Enquirer
NHL Teams
Boston Boston Bruins Montreal Montreal Canadiens
Phoenix Phoenix Coyotes Nashville Nashville Predators
NBA Teams
Detroit Detroit Pistons Toronto Toronto Raptors
Oakland Golden State Warriors Memphis Memphis Grizzlies
Airlines
Austria Austrian Airlines Spain Spainair
Belgium Brussels Airlines Greece Aegean Airlines
Company executives
Steve Ballmer Microsoft Larry Page Google
Samuel J. Palmisano IBM Werner Vogels Amazon
Table 2: Examples of the analogical reasoning task for phrases (the full test set has 3218 examples).
The goal is to compute the fourth phrase using the ﬁrst three. Our best model achieved an accuracy
of 72% on this dataset.
This way, we can form many reasonable phrases without greatly increasing the size of the vocabulary;
in theory, we can train the Skip-gram model using all n-grams, but that would be too memory
intensive. Many techniques have been previously developed to identify phrases in the text; however,
it is out of scope of our work to compare them. We decided to use a simple data-driven approach,
where phrases are formed based on the unigram and bigram counts, using
score(wi, wj) =
count(wiwj) − δ
count(wi) × count(wj)
. (6)
The δ is used as a discounting coefﬁcient and prevents too many phrases consisting of very infrequent
words to be formed. The bigrams with score above the chosen threshold are then used as
phrases. Typically, we run 2-4 passes over the training data with decreasing threshold value, allowing
longer phrases that consists of several words to be formed. We evaluate the quality of the phrase
representations using a new analogical reasoning task that involves phrases. Table 2 shows examples
of the ﬁve categories of analogies used in this task. This dataset is publicly available on the web2
.
4.1 Phrase Skip-Gram Results
Starting with the same news data as in the previous experiments, we ﬁrst constructed the phrase
based training corpus and then we trained several Skip-gram models using different hyperparameters.
As before, we used vector dimensionality 300 and context size 5. This setting already
achieves good performance on the phrase dataset, and allowed us to quickly compare the Negative
Sampling and the Hierarchical Softmax, both with and without subsampling of the frequent tokens.
The results are summarized in Table 3.
The results show that while Negative Sampling achieves a respectable accuracy even with k = 5,
using k = 15 achieves considerably better performance. Surprisingly, while we found the Hierarchical
Softmax to achieve lower performance when trained without subsampling, it became the best
performing method when we downsampled the frequent words. This shows that the subsampling
can result in faster training and can also improve accuracy, at least in some cases.
2
code.google.com/p/word2vec/source/browse/trunk/questions-phrases.txt
Method Dimensionality No subsampling [%] 10−5
subsampling [%]
NEG-5 300 24 27
NEG-15 300 27 42
HS-Huffman 300 19 47
Table 3: Accuracies of the Skip-gram models on the phrase analogy dataset. The models were
trained on approximately one billion words from the news dataset.
6
NEG-15 with 10−5
subsampling HS with 10−5
subsampling
Vasco de Gama Lingsugur Italian explorer
Lake Baikal Great Rift Valley Aral Sea
Alan Bean Rebbeca Naomi moonwalker
Ionian Sea Ruegen Ionian Islands
chess master chess grandmaster Garry Kasparov
Table 4: Examples of the closest entities to the given short phrases, using two different models.
Czech + currency Vietnam + capital German + airlines Russian + river French + actress
koruna Hanoi airline Lufthansa Moscow Juliette Binoche
Check crown Ho Chi Minh City carrier Lufthansa Volga River Vanessa Paradis
Polish zolty Viet Nam ﬂag carrier Lufthansa upriver Charlotte Gainsbourg
CTK Vietnamese Lufthansa Russia Cecile De
Table 5: Vector compositionality using element-wise addition. Four closest tokens to the sum of two
vectors are shown, using the best Skip-gram model.
To maximize the accuracy on the phrase analogy task, we increased the amount of the training data
by using a dataset with about 33 billion words. We used the hierarchical softmax, dimensionality
of 1000, and the entire sentence for the context. This resulted in a model that reached an accuracy
of 72%. We achieved lower accuracy 66% when we reduced the size of the training dataset to 6B
words, which suggests that the large amount of the training data is crucial.
To gain further insight into how different the representations learned by different models are, we did
inspect manually the nearest neighbours of infrequent phrases using various models. In Table 4, we
show a sample of such comparison. Consistently with the previous results, it seems that the best
representations of phrases are learned by a model with the hierarchical softmax and subsampling.
5 Additive Compositionality
We demonstrated that the word and phrase representations learned by the Skip-gram model exhibit
a linear structure that makes it possible to perform precise analogical reasoning using simple vector
arithmetics. Interestingly, we found that the Skip-gram representations exhibit another kind of linear
structure that makes it possible to meaningfully combine words by an element-wise addition of their
vector representations. This phenomenon is illustrated in Table 5.
The additive property of the vectors can be explained by inspecting the training objective. The word
vectors are in a linear relationship with the inputs to the softmax nonlinearity. As the word vectors
are trained to predict the surrounding words in the sentence, the vectors can be seen as representing
the distribution of the context in which a word appears. These values are related logarithmically
to the probabilities computed by the output layer, so the sum of two word vectors is related to the
product of the two context distributions. The product works here as the AND function: words that
are assigned high probabilities by both word vectors will have high probability, and the other words
will have low probability. Thus, if “Volga River” appears frequently in the same sentence together
with the words “Russian” and “river”, the sum of these two word vectors will result in such a feature
vector that is close to the vector of “Volga River”.
6 Comparison to Published Word Representations
Many authors who previously worked on the neural network based representations of words have
published their resulting models for further use and comparison: amongst the most well known authors
are Collobert and Weston [2], Turian et al. [17], and Mnih and Hinton [10]. We downloaded
their word vectors from the web3
. Mikolov et al. [8] have already evaluated these word representations
on the word analogy task, where the Skip-gram models achieved the best performance with a
huge margin.
3
http://metaoptimize.com/projects/wordreprs/
7
Model Redmond Havel ninjutsu grafﬁti capitulate
(training time)
Collobert (50d) conyers plauen reiki cheesecake abdicate
(2 months) lubbock dzerzhinsky kohona gossip accede
keene osterreich karate dioramas rearm
Turian (200d) McCarthy Jewell - gunﬁre (few
weeks) Alston Arzu - emotion Cousins
Ovitz - impunity Mnih
(100d) Podhurst Pontiff - anaesthetics Mavericks
(7 days) Harlang Pinochet - monkeys planning
Agarwal Rodionov - Jews hesitated
Skip-Phrase Redmond Wash. Vaclav Havel ninja spray paint capitulation
(1000d, 1 day) Redmond Washington president Vaclav Havel martial arts graﬁtti capitulated
Microsoft Velvet Revolution swordsmanship taggers capitulating
Table 6: Examples of the closest tokens given various well known models and the Skip-gram model
trained on phrases using over 30 billion training words. An empty cell means that the word was not
in the vocabulary.
To give more insight into the difference of the quality of the learned vectors, we provide empirical
comparison by showing the nearest neighbours of infrequent words in Table 6. These examples show
that the big Skip-gram model trained on a large corpus visibly outperforms all the other models in
the quality of the learned representations. This can be attributed in part to the fact that this model
has been trained on about 30 billion words, which is about two to three orders of magnitude more
data than the typical size used in the prior work. Interestingly, although the training set is much
larger, the training time of the Skip-gram model is just a fraction of the time complexity required by
the previous model architectures.
7 Conclusion
This work has several key contributions. We show how to train distributed representations of words
and phrases with the Skip-gram model and demonstrate that these representations exhibit linear
structure that makes precise analogical reasoning possible. The techniques introduced in this paper
can be used also for training the continuous bag-of-words model introduced in [8].
We successfully trained models on several orders of magnitude more data than the previously published
models, thanks to the computationally efﬁcient model architecture. This results in a great
improvement in the quality of the learned word and phrase representations, especially for the rare
entities. We also found that the subsampling of the frequent words results in both faster training
and signiﬁcantly better representations of uncommon words. Another contribution of our paper is
the Negative sampling algorithm, which is an extremely simple training method that learns accurate
representations especially for frequent words.
The choice of the training algorithm and the hyper-parameter selection is a task speciﬁc decision,
as we found that different problems have different optimal hyperparameter conﬁgurations. In our
experiments, the most crucial decisions that affect the performance are the choice of the model
architecture, the size of the vectors, the subsampling rate, and the size of the training window.
A very interesting result of this work is that the word vectors can be somewhat meaningfully combined
using just simple vector addition. Another approach for learning representations of phrases
presented in this paper is to simply represent the phrases with a single token. Combination of these
two approaches gives a powerful yet simple way how to represent longer pieces of text, while having
minimal computational complexity. Our work can thus be seen as complementary to the existing
approach that attempts to represent phrases using recursive matrix-vector operations [16].
We made the code for training the word and phrase vectors based on the techniques described in this
paper available as an open-source project4
.
4
code.google.com/p/word2vec
8
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