Transformers for Natural Language Processing

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By Denis Rothman
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  1. Getting Started with the Model Architecture of the Transformer

About this book

The transformer architecture has proved to be revolutionary in outperforming the classical RNN and CNN models in use today. With an apply-as-you-learn approach, Transformers for Natural Language Processing investigates in vast detail the deep learning for machine translations, speech-to-text, text-to-speech, language modeling, question answering, and many more NLP domains with transformers.

The book takes you through NLP with Python and examines various eminent models and datasets within the transformer architecture created by pioneers such as Google, Facebook, Microsoft, OpenAI, and Hugging Face.

The book trains you in three stages. The first stage introduces you to transformer architectures, starting with the original transformer, before moving on to RoBERTa, BERT, and DistilBERT models. You will discover training methods for smaller transformers that can outperform GPT-3 in some cases. In the second stage, you will apply transformers for Natural Language Understanding (NLU) and Natural Language Generation (NLG). Finally, the third stage will help you grasp advanced language understanding techniques such as optimizing social network datasets and fake news identification.

By the end of this NLP book, you will understand transformers from a cognitive science perspective and be proficient in applying pretrained transformer models by tech giants to various datasets.

Publication date:
January 2021


Getting Started with the Model Architecture of the Transformer

Language is the essence of human communication. Civilizations would never have been born without the word sequences that form language. We now mostly live in a world of digital representations of language. Our daily lives rely on Natural Language Processing (NLP) digitalized language functions: web search engines, emails, social networks, posts, tweets, smartphone texting, translations, web pages, speech-to-text on streaming sites for transcripts, text-to-speech on hotline services, and many more everyday functions.

In December 2017, the seminal Vaswani et al. Attention Is All You Need article, written by Google Brain members and Google Research, was published. The Transformer was born. The Transformer outperformed the existing state-of-the-art NLP models. The Transformer trained faster than previous architectures and obtained higher evaluation results. Transformers have become a key component of NLP.

The digital world would never have existed without NLP. Natural Language Processing would have remained primitive and inefficient without artificial intelligence. However, the use of Recurrent Neural Networks (RNNs) and Convolutional Neural Networks (CNNs) comes at a tremendous cost in terms of calculations and machine power.

In this chapter, we will first start with the background of NLP that led to the rise of the Transformer. We will briefly go from early NLP to RNNs and CNNs. Then we will see how the Transformer overthrew the reign of RNNs and CNNs, which had prevailed for decades for sequence analysis.

Then we will open the hood of the Transformer model described by Vaswani et al. (2017) and examine the key components of its architecture. We will explore the fascinating world of attention and illustrate the key components of the Transformer.

This chapter covers the following topics:

  • The background of the Transformer
  • The architecture of the Transformer
  • The Transformer's self-attention model
  • The encoding and decoding stacks
  • Input and output embedding
  • Positional embedding
  • Self-attention
  • Multi-head attention
  • Masked multi-attention
  • Residual connections
  • Normalization
  • Feedforward network
  • Output probabilities

Our first step will be to explore the background of the Transformer.


The background of the Transformer

In this section, we will go through the background of NLP that led to the Transformer. The Transformer model invented by Google Research has toppled decades of Natural Language Processing research, development, and implementations.

Let us first see how that happened when NLP reached a critical limit that required a new approach.

Over the past 100+ years, many great minds have worked on sequence transduction and language modeling. Machines progressively learned how to predict probable sequences of words. It would take a whole book to cite all the giants that made this happen.

In this section, I will share my favorite researchers with you to lay the ground for the arrival of the Transformer.

In the early 20th century, Andrey Markov introduced the concept of random values and created a theory of stochastic processes. We know them in artificial intelligence (AI) as Markov Decision Processes (MDPs), Markov Chains, and Markov Processes. In 1902, Markov showed that we could predict the next element of a chain, a sequence, using only the last past element of that chain. In 1913, he applied this to a 20,000-letter dataset using past sequences to predict the future letters of a chain. Bear in mind that he had no computer but managed to prove his theory, which is still in use today in AI.

In 1948, Claude Shannon's The Mathematical Theory of Communication was published. He cites Andrey Markov's theory multiple times when building his probabilistic approach to sequence modeling. Claude Shannon laid the ground for a communication model based on a source encoder, a transmitter, and a received decoder or semantic decoder.

In 1950, Alan Turing published his seminal article: Computing Machinery and Intelligence. Alan Turing based this article on machine intelligence on the immensely successful Turing Machine that decrypted German messages. The expression artificial intelligence was first used by John McCarthy in 1956. However, Alan Turing was implementing artificial intelligence in the 1940s to decode encrypted encoded messages in German.

In 1954, the Georgetown-IBM experiment used computers to translate Russian sentences into English using a rule system. A rule system is a program that runs a list of rules that will analyze language structures. Rule systems still exist. However, creating rule lists for the billions of language combinations in our digital world is a challenge yet to be met. For the moment, it seems impossible. But who knows what will happen?

In 1982, John Hopfield introduced Recurrent Neural Networks (RNNs), known as Hopfield networks or "associative" neural networks. John Hopfield was inspired by W.A. Little, who wrote The Existence of Persistent States in the Brain in 1974. RNNs evolved, and LSTMs emerged as we know them. An RNN memorizes the persistent states of a sequence efficiently:

Figure 1.1: The RNN process

Each state Sn captures the information of Sn-1 When the end of the network is reached, a function F will perform an action: transduction, modeling, or any other type of sequence-based task.

In the 1980s, Yann Le Cun designed the multi-purpose Convolutional Neural Network (CNN). He applied CNNs to text sequences, and they have been widely used for sequence transduction and modeling as well. They are also based on persistent states that gather information layer by layer. In the 1990s, summing up several years of work, Yann Le Cun produced LeNet-5, which led to the many CNN models we know today. The CNN's otherwise efficient architecture faces problems when dealing with long-term dependencies in very long and complex sequences.

We could mention many other great names, papers, and models that would humble any AI specialist. It seemed that everybody in AI was on the right track for all these years. Markov Fields, RNNs, and CNNs evolved into multiple other models. The notion of attention appeared: peeking at other tokens in a sequence, not just the last one. It was added to the RNN and CNN models.

After that, if AI models needed to analyze longer sequences that required an increasing amount of computer power, AI developers used more powerful machines and found ways to optimize gradients.

It seemed that nothing else could be done to make more progress. Thirty years passed this way. And then, in December 2017, came the Transformer, the incredible innovation that seems to have come from a distant planet. The Transformer swept everything away, producing impressive scores on standard datasets.

Let's start our exploration of the architecture of the Transformer with the design of this alien NLP/NLU spaceship!


The rise of the Transformer: Attention Is All You Need

In December 2017, Vaswani et al. published their seminal paper, Attention Is All You Need. They performed their work at Google Research and Google Brain. I will refer to the model described in Attention Is All You Need as the "original Transformer model" throughout this chapter and book.

In this section, we will look at the Transformer model they built from the outside. In the following sections, we will explore what is inside each component of the model.

The original Transformer model is a stack of 6 layers. The output of layer l is the input of layer l+1 until the final prediction is reached. There is a 6-layer encoder stack on the left and a 6-layer decoder stack on the right:

Figure 1.2: The architecture of the Transformer

On the left, the inputs enter the encoder side of the Transformer through an attention sub-layer and FeedForward Network (FFN) sub-layer. On the right, the target outputs go into the decoder side of the Transformer through two attention sub-layers and an FFN sub-layer. We immediately notice that there is no RNN, LSTM, or CNN. Recurrence has been abandoned.

Attention has replaced recurrence, which requires an increasing number of operations as the distance between two words increases. The attention mechanism is a "word-to-word" operation. The attention mechanism will find how each word is related to all other words in a sequence, including the word being analyzed itself. Let's examine the following sequence:

The cat sat on the mat.

Attention will run dot products between word vectors and determine the strongest relationships of a word among all the other words, including itself ("cat" and "cat"):

Figure 1.3: Attending to all the words

The attention mechanism will provide a deeper relationship between words and produce better results.

For each attention sub-layer, the original Transformer model runs not one but eight attention mechanisms in parallel to speed up the calculations. We will explore this architecture in the following section, The encoder stack. This process is named "multi-head attention," providing:

  • A broader in-depth analysis of sequences
  • The preclusion of recurrence reducing calculation operations
  • The implementation of parallelization, which reduces training time
  • Each attention mechanism learns different perspectives of the same input sequence

    Attention replaced recurrence. However, there are several other creative aspects of the Transformer that are as critical as the attention mechanism, as you will see when we look inside the architecture.

We just looked at the Transformer from the outside. Let's now go into each component of the Transformer. We will start with the encoder.

The encoder stack

The layers of the encoder and decoder of the original Transformer model are stacks of layers. Each layer of the encoder stack has the following structure:

Figure 1.4: A layer of the encoder stack of the Transformer

The original encoder layer structure remains the same for all of the N=6 layers of the Transformer model. Each layer contains two main sub-layers: a multi-headed attention mechanism and a fully connected position-wise feedforward network.

Notice that a residual connection surrounds each main sub-layer, Sublayer(x), in the Transformer model. These connections transport the unprocessed input x of a sub-layer to a layer normalization function. This way, we are certain that key information such as positional encoding is not lost on the way. The normalized output of each layer is thus:

LayerNormalization (x + Sublayer(x))

Though the structure of each of the N=6 layers of the encoder is identical, the content of each layer is not strictly identical to the previous layer.

For example, the embedding sub-layer is only present at the bottom level of the stack. The other five layers do not contain an embedding layer, and this guarantees that the encoded input is stable through all the layers.

Also, the multi-head attention mechanisms perform the same functions from layer 1 to 6. However, they do not perform the same tasks. Each layer learns from the previous layer and explores different ways of associating the tokens in the sequence. It looks for various associations of words, just like how we look for different associations of letters and words when we solve a crossword puzzle.

The designers of the Transformer introduced a very efficient constraint. The output of every sub-layer of the model has a constant dimension, including the embedding layer and the residual connections. This dimension is dmodel and can be set to another value depending on your goals. In the original Transformer architecture, dmodel =512.

dmodel has a powerful consequence. Practically all the key operations are dot products. The dimensions remain stable, which reduces the number of operations to calculate, reduces machine consumption, and makes it easier to trace the information as it flows through the model.

This global view of the encoder shows the highly optimized architecture of the Transformer. In the following sections, we will zoom into each of the sub-layers and mechanisms.

We will begin with the embedding sub-layer.

Input embedding

The input embedding sub-layer converts the input tokens to vectors of dimension dmodel = 512 using learned embeddings in the original Transformer model. The structure of the input embedding is classical:

Figure 1.5: The input embedding sub-layer of the Transformer

The embedding sub-layer works like other standard transduction models. A tokenizer will transform a sentence into tokens. Each tokenizer has its methods, but the results are similar. For example, a tokenizer applied to the sequence "the Transformer is an innovative NLP model!" will produce the following tokens in one type of model:

['the', 'transform', 'er', 'is', 'a', 'revolutionary', 'n', 'l', 'p', 'model', '!']

You will notice that this tokenizer normalized the string to lower case and truncated it into subparts. A tokenizer will generally provide an integer representation that will be used for the embedding process. For example:

Text = "The cat slept on the couch.It was too tired to get up."
tokenized text= [1996, 4937, 7771, 2006, 1996, 6411, 1012, 2009, 2001, 2205, 5458, 2000, 2131, 2039, 1012]

There is not enough information in the tokenized text at this point to go further. The tokenized text must be embedded.

The Transformer contains a learned embedding sub-layer. Many embedding methods can be applied to the tokenized input.

I chose the skip-gram architecture of the word2vec embedding approach Google made available in 2013 to illustrate the embedding sublayer of the Transformer. A skip-gram will focus on a center word in a window of words and predicts context words. For example, if word(i) is the center word in a two-step window, a skip-gram model will analyze word(i-2), word(i-1), word(i+1), and word(i+2). Then the window will slide and repeat the process. A skip-gram model generally contains an input layer, weights, a hidden layer, and an output containing the word embeddings of the tokenized input words.

Suppose we need to perform embedding for the following sentence:

The black cat sat on the couch and the brown dog slept on the rug.

We will focus on two words, black and brown. The word embedding vectors of these two words should be similar.

Since we must produce a vector of size dmodel = 512 for each word, we will obtain a size 512 vector embedding for each word:

black=[[-0.01206071  0.11632373  0.06206119  0.01403395  0.09541149  0.10695464 0.02560172  0.00185677 -0.04284821  0.06146432  0.09466285  0.04642421 0.08680347  0.05684567 -0.00717266 -0.03163519  0.03292002 -0.11397766 0.01304929  0.01964396  0.01902409  0.02831945  0.05870414  0.03390711 -0.06204525  0.06173197 -0.08613958 -0.04654748  0.02728105 -0.07830904
0.04340003 -0.13192849 -0.00945092 -0.00835463 -0.06487109  0.05862355 -0.03407936 -0.00059001 -0.01640179  0.04123065 
-0.04756588  0.08812257 0.00200338 -0.0931043  -0.03507337  0.02153351 -0.02621627 -0.02492662 -0.05771535 -0.01164199 
-0.03879078 -0.05506947  0.01693138 -0.04124579 -0.03779858 
-0.01950983 -0.05398201  0.07582296  0.00038318 -0.04639162 
-0.06819214  0.01366171  0.01411388  0.00853774  0.02183574 
-0.03016279 -0.03184025 -0.04273562]]

The word black is now represented by 512 dimensions. Other embedding methods could be used and dmodel could have a higher number of dimensions.

The word embedding of brown is also represented by 512 dimensions:

brown=[[ 1.35794589e-02 -2.18823571e-02  1.34526128e-02  6.74355254e-02
   1.04376070e-01  1.09921647e-02 -5.46298288e-02 -1.18385479e-02
   4.41223830e-02 -1.84863899e-02 -6.84073642e-02  3.21860164e-02
   4.09143828e-02 -2.74433400e-02 -2.47369967e-02  7.74542615e-02
   9.80964210e-03  2.94299088e-02  2.93895267e-02 -3.29437815e-02
  7.20389187e-02  1.57317147e-02 -3.10291946e-02 -5.51304631e-02
  -7.03861639e-02  7.40829483e-02  1.04319192e-02 -2.01565702e-03
   2.43322570e-02  1.92969330e-02  2.57341694e-02 -1.13280728e-01
   8.45847875e-02  4.90090018e-03  5.33546880e-02 -2.31553353e-02
   3.87288055e-05  3.31782512e-02 -4.00604047e-02 -1.02028981e-01
   3.49597558e-02 -1.71501152e-02  3.55573371e-02 -1.77437533e-02
  -5.94457164e-02  2.21221056e-02  9.73121971e-02 -4.90022525e-02]]

To verify the word embedding produced for these two words, we can use cosine similarity to see if the word embeddings of the words black and brown are similar.

Cosine similarity uses Euclidean (L2) norm to create vectors in a unit sphere. The dot product of the vectors we are comparing is the cosine between the points of those two vectors. For more on the theory of cosine similarity, you can consult scikit-learn's documentation, among many other sources:

The cosine similarity between the black vector of size dmodel = 512 and brown vector of size dmodel = 512 in the embedding of the example is:

cosine_similarity(black, brown)= [[0.9998901]]

The skip-gram produced two vectors that are very close to each other. It detected that black and brown form a color subset of the dictionary of words.

The Transformer's subsequent layers do not start empty-handed. They have learned word embeddings that already provide information on how the words can be associated.

However, a big chunk of information is missing because no additional vector or information indicates a word's position in a sequence.

The designers of the Transformer came up with yet another innovative feature: positional encoding.

Let's see how positional encoding works.

Positional encoding

We enter this positional encoding function of the Transformer with no idea of the position of a word in a sequence:

Figure 1.6: Position encoding

We cannot create independent positional vectors that would have a high cost on the training speed of the Transformer and make attention sub-layers very complex to work with. The idea is to add a positional encoding value to the input embedding instead of having additional vectors to describe the position of a token in a sequence.

We also know that the Transformer expects a fixed size dmodel = 512 (or other constant value for the model) for each vector of the output of the positional encoding function.

If we go back to the sentence we used in the word embedding sub-layer, we can see that black and brown may be similar, but they are far apart:

The black cat sat on the couch and the brown dog slept on the rug.

The word black is in position 2, pos=2, and the word brown is in position 10, pos=10.

Our problem is to find a way to add a value to the word embedding of each word so that it has that information. However, we need to add a value to the dmodel = 512 dimensions! For each word embedding vector, we need to find a way to provide information to i in the range(0,512) dimensions of the word embedding vector of black and brown.

There are many ways to achieve this goal. The designers found a clever way to use a unit sphere to represent positional encoding with sine and cosine values that will thus remain small but very useful.

Vaswani et al. (2017) provide sine and cosine functions so that we can generate different frequencies for the positional encoding (PE) for each position and each dimension i of the dmodel = 512 of the word embedding vector:

If we start at the beginning of the word embedding vector, we will begin with a constant (512), i=0, and end with i=511. This means that the sine function will be applied to the even numbers and the cosine function to the odd numbers. Some implementations do it differently. In that case, the domain of the sine function can be and the domain of the cosine function can be . This will produce similar results.

In this section, we will use the functions the way they were described by Vaswani et al. (2017). A literal translation into Python produces the following code for a positional vector pe[0][i] for a position pos:

def positional_encoding(pos,pe):
for i in range(0, 512,2):
         pe[0][i] = math.sin(pos / (10000 ** ((2 * i)/d_model)))
         pe[0][i+1] = math.cos(pos / (10000 ** ((2 * i)/d_model)))
return pe

Before going further, you might want to see the plot of the sine function, for example, for pos=2.

You can Google the following plot, for example:

plot y=sin(2/10000^(2*x/512))

Just enter the plot request:

Figure 1.7: Plotting with Google

You will obtain the following graph:

Figure 1.8: The graph

If we go back to the sentence we are parsing in this section, we can see that black is in position pos=2 and brown is in position pos=10:

The black cat sat on the couch and the brown dog slept on the rug.

If we apply the sine and cosine functions literally for pos=2, we obtain a size=512 positional encoding vector:

[[ 9.09297407e-01 -4.16146845e-01  9.58144367e-01 -2.86285430e-01
   9.87046242e-01 -1.60435960e-01  9.99164224e-01 -4.08766568e-02
   9.97479975e-01  7.09482506e-02  9.84703004e-01  1.74241230e-01
   9.63226616e-01  2.68690288e-01  9.35118318e-01  3.54335666e-01
   9.02130723e-01  4.31462824e-01  8.65725577e-01  5.00518918e-01
   8.27103794e-01  5.62049210e-01  7.87237823e-01  6.16649508e-01
   7.46903539e-01  6.64932430e-01  7.06710517e-01  7.07502782e-01
   5.47683925e-08  1.00000000e+00  5.09659337e-08  1.00000000e+00
   4.74274735e-08  1.00000000e+00  4.41346799e-08  1.00000000e+00
   4.10704999e-08  1.00000000e+00  3.82190599e-08  1.00000000e+00
   3.55655878e-08  1.00000000e+00  3.30963417e-08  1.00000000e+00
   3.07985317e-08  1.00000000e+00  2.86602511e-08  1.00000000e+00
   2.66704294e-08  1.00000000e+00  2.48187551e-08  1.00000000e+00
   2.30956392e-08  1.00000000e+00  2.14921574e-08  1.00000000e+00]]

We also obtain a size=512, positional encoding vector for position 10, pos=10:

[[-5.44021130e-01 -8.39071512e-01  1.18776485e-01 -9.92920995e-01
   6.92634165e-01 -7.21289039e-01  9.79174793e-01 -2.03019097e-01
   9.37632740e-01  3.47627431e-01  6.40478015e-01  7.67976522e-01
   2.09077001e-01  9.77899194e-01 -2.37917677e-01  9.71285343e-01
  -6.12936735e-01  7.90131986e-01 -8.67519796e-01  4.97402608e-01
  -9.87655997e-01  1.56638563e-01 -9.83699203e-01 -1.79821849e-01
  2.73841977e-07  1.00000000e+00  2.54829672e-07  1.00000000e+00
   2.37137371e-07  1.00000000e+00  2.20673414e-07  1.00000000e+00
   2.05352507e-07  1.00000000e+00  1.91095296e-07  1.00000000e+00
   1.77827943e-07  1.00000000e+00  1.65481708e-07  1.00000000e+00
   1.53992659e-07  1.00000000e+00  1.43301250e-07  1.00000000e+00
   1.33352145e-07  1.00000000e+00  1.24093773e-07  1.00000000e+00
   1.15478201e-07  1.00000000e+00  1.07460785e-07  1.00000000e+00]]

When we look at the results we obtained with an intuitive literal translation of the Vaswani et al. (2017) functions into Python, we would now like to check whether the results are meaningful.

The cosine similarity function used for word embedding comes in handy for having a better visualization of the proximity of the positions:

cosine_similarity(pos(2), pos(10)= [[0.8600013]]

The similarity between the position of the words black and brown and the lexical field (groups of words that go together) similarity is different:

cosine_similarity(black, brown)= [[0.9998901]]

The encoding of the position shows a lower similarity value than the word embedding similarity.

The positional encoding has taken these words apart. Bear in mind that word embeddings will vary with the corpus used to train them.

The problem is now how to add the positional encoding to the word embedding vectors.

Adding positional encoding to the embedding vector

The authors of the Transformer found a simple way by merely adding the positional encoding vector to the word embedding vector:

Figure 1.9: Positional encoding

If we go back and take the word embedding of black, for example, and name it y1=black, we are ready to add it to the positional vector pe(2) we obtained with positional encoding functions. We will obtain the positional encoding pc(black) of the input word black:


The solution is straightforward. However, if we apply it as shown, we might lose the information of the word embedding, which will be minimized by the positional encoding vector.

There are many possibilities to increase the value of y1 to make sure that the information of the word embedding layer can be used efficiently in the subsequent layers.

One of the many possibilities is to add an arbitrary value to y1, the word embedding of black:


We can now add the positional vector to the embedding vector of the word black, both of which are the same size (512):

for i in range(0, 512,2):
          pe[0][i] = math.sin(pos / (10000 ** ((2 * i)/d_model)))
          pc[0][i] = (y[0][i]*math.sqrt(d_model))+ pe[0][i]
          pe[0][i+1] = math.cos(pos / (10000 ** ((2 * i)/d_model)))
          pc[0][i+1] = (y[0][i+1]*math.sqrt(d_model))+ pe[0][i+1]

The result obtained is the final positional encoding vector of dimension dmodel = 512:

[[ 9.09297407e-01 -4.16146845e-01  9.58144367e-01 -2.86285430e-01
   9.87046242e-01 -1.60435960e-01  9.99164224e-01 -4.08766568e-02
  4.74274735e-08  1.00000000e+00  4.41346799e-08  1.00000000e+00
   4.10704999e-08  1.00000000e+00  3.82190599e-08  1.00000000e+00
   2.66704294e-08  1.00000000e+00  2.48187551e-08  1.00000000e+00
   2.30956392e-08  1.00000000e+00  2.14921574e-08  1.00000000e+00]]

The same operation is applied to the word brown and all of the other words in a sequence. The output of this algorithm, which is not rule-based, might slightly vary during each run.

We can apply the cosine similarity function to the positional encoding vectors of black and brown:

cosine_similarity(pc(black), pc(brown)= [[0.9627094]]

We now have a clear view of the positional encoding process through the three cosine similarity functions we applied to the three states representing the words black and brown:

[[0.99987495]] word similarity
[[0.8600013]] positional encoding vector similarity
[[0.9627094]] final positional encoding similarity

We saw that the initial word similarity of their embeddings was very high, with a value of 0.99. Then we saw the positional encoding vector of positions 2 and 10 drew these two words apart with a lower similarity value of 0.86.

Finally, we added the word embedding vector of each word to its respective positional encoding vector. We saw that this brought the cosine similarity of the two words to 0.96.

The positional encoding of each word now contains the initial word embedding information and the positional encoding values.

Hugging Face and Google Brain Trax both, among others, provide ready-to-use libraries for functionality we explored in the word embedding section and the present positional encoding section. Thus, you do not need to run the program I used in this chapter to check the Transformer equations, and this section is self-contained. However, if you wish to explore the code, you will find it in the Google Colaboratory positional_encoding.ipynb notebook and the text.txt file in this chapter's GitHub repository.

The output of positional encoding is the multi-head attention sub-layer.

Sub-layer 1: Multi-head attention

The multi-head attention sub-layer contains eight heads and is followed by post-layer normalization, which will add residual connections to the output of the sub-layer and normalize it:

Figure 1.10: Multi-head attention sub-layer

This section begins with the architecture of an attention layer. Then, an example of multi-attention is implemented in a small module in Python. Finally, post-layer normalization is described.

Let's start with the architecture of multi-head attention.

The architecture of multi-head attention

The input of the multi-attention sub-layer of the first layer of the encoder stack is a vector that contains the embedding and the positional encoding of each word. The next layers of the stack do not start these operations over.

The dimension of the vector of each word xn of an input sequence is dmodel = 512:

pe(xn)=[d1=9.09297407e-01, d2=9.09297407e-01,.., d512 = 1.00000000e+00]

The representation of each word xn has become a vector of dmodel = 512 dimensions.

Each word is mapped to all the other words to determine how it fits in a sequence.

In the following sentence, we can see that "it" could be related to "cat" and "rug" in the sequence:

Sequence =The cat sat on the rug and it was dry-cleaned.

The model will train to find out if "it" is related to "cat" or "rug." We could run a huge calculation by training the model using the dmodel = 512 dimensions as they are now.

However, we would only get one point of view at a time by analyzing the sequence with one dmodel block. Furthermore, it would take quite some calculation time to find other perspectives.

A better way is to divide the dmodel = 512 dimensions of each word xn of x (all of the words of a sequence) into 8 dk = 64 dimensions.

We then can run the 8 "heads" in parallel to speed up the training and obtain 8 different representation subspaces of how each word relates to another:

Figure 1.11: Multi-head representations

You can see that there are now 8 heads running in parallel. One head might decide that "it" fits well with "cat" and another that "it" fits well with "rug" and another that "rug" fits well with "dry-cleaned."

The output of each head is a matrix zi with a shape of x* dk The output of a multi-attention head is Z defined as:

Z = (z0, z1, z2, z3, z4, z5, z6, z7,)

However, Z must be concatenated so that the output of the multi-head sub-layer is not a sequence of dimensions but one lines of xm*dmodel matrix.

Before exiting the multi-head attention sub-layer, the elements of Z are concatenated:

MultiHead(output) = Concat(z0, z1, z2, z3, z4, z5, z6, z7,) = x, dmodel

Notice that each head is concatenated into z that has a dimension of dmodel = 512. The output of the multi-headed layer respects the constraint of the original Transformer model.

Inside each head hn of the attention mechanism, each word vector has three representations:

  • A query vector (Q) that has a dimension of dq = 64, which is activated and trained when a word vector xn seeks all of the key-value pairs of the other word vectors, including itself in self-attention
  • A key vector (K) that has a dimension of dk = 64, which will be trained to provide an attention value
  • A value vector (V) that has a dimension of dv = 64, which will be trained to provide another attention value

Attention is defined as "Scaled Dot-Product Attention," which is represented in the following equation in which we plug Q, K, and V:

The vectors all have the same dimension making it relatively simple to use a scaled dot product to obtain the attention values for each head and then concatenate the output Z of the 8 heads.

To obtain Q, K, and V, we must train the model with their respective weight matrices Qw, Kw and Vw, which have dk = 64 columns and dmodel = 512 rows. For example, Q is obtained by a dot-product between x and Qw. Q will have a dimension of dk = 64.

You can modify all of the parameters such as the number of layers, heads, dmodel, dk, and other variables of the Transformer to fit your model. This chapter describes the original Transformer parameters by Vaswani et al. (2017). It is essential to understand the original architecture before modifying it or exploring variants of the original model designed by others.

Hugging Face and Google Brain Trax, among others, provide ready-to-use frameworks, libraries, and modules that we will be using throughout this book.

However, let's open the hood of the Transformer model and get our hands dirty in Python to illustrate the architecture we just explored in order to visualize the model in code and show it with intermediate images.

We will use basic Python code with only numpy and a softmax function in 10 steps to run the key aspects of the attention mechanism.

Let's now start building Step 1 of our model to represent the input.

Step 1: Represent the input

Save Multi_Head_Attention_Sub_Layer.ipynb to your Google Drive (make sure you have a Gmail account) and then open it in Google Colaboratory. The notebook is in the GitHub repository for this chapter.

We will start by only using minimal Python functions to understand the Transformer at a low level with the inner workings of an attention head. We will explore the inner workings of the multi-head attention sub-layer using basic code:

import numpy as np
from scipy.special import softmax

The input of the attention mechanism we are building is scaled down to dmodel = 4 instead of dmodel = 512. This brings the dimensions of the vector of an input x down to dmodel = 4, which is easier to visualize.

x contains 3 inputs with 4 dimensions each instead of 512:

print("Step 1: Input : 3 inputs, d_model=4")
x =np.array([[1.0, 0.0, 1.0, 0.0],   # Input 1
             [0.0, 2.0, 0.0, 2.0],   # Input 2
             [1.0, 1.0, 1.0, 1.0]])  # Input 3

The output shows that we have 3 vectors of dmodel = 4.

Step 1: Input : 3 inputs, d_model=4
[[1. 0. 1. 0.]
 [0. 2. 0. 2.]
 [1. 1. 1. 1.]]

The first step of our model is ready:

Figure 1.12: Input of a multi-head attention sub-layer

We will now add the weight matrices to our model.

Step 2: Initializing the weight matrices

Each input has 3 weight matrices:

  • Qw to train the queries
  • Kw to train the keys
  • Vw to train the values

These 3 weight matrices will be applied to all the inputs in this model.

The weight matrices described by Vaswani et al. (2017) are dk = 64 dimensions. However, let's scale the matrices down to dk = 3. The dimensions are scaled down to 3*4 weight matrices to be able to visualize the intermediate results more easily and perform dot products with the input x.

The three weight matrices are initialized starting with the query weight matrix:

print("Step 2: weights 3 dimensions x d_model=4")
w_query =np.array([[1, 0, 1],
                   [1, 0, 0],
                   [0, 0, 1],
                   [0, 1, 1]])

The output is the w_query weight matrix:

Step 2: weights 3 dimensions x d_model=4
[[1 0 1]
 [1 0 0]
 [0 0 1]
 [0 1 1]]

We will now initialize the key weight matrix:

w_key =np.array([[0, 0, 1],
                 [1, 1, 0],
                 [0, 1, 0],
                 [1, 1, 0]])

The output is the key weight matrix:

[[0 0 1]
 [1 1 0]
 [0 1 0]
 [1 1 0]]

Finally, we initialize the value weight matrix:

w_value = np.array([[0, 2, 0],
                    [0, 3, 0],
                    [1, 0, 3],
                    [1, 1, 0]])

The output is the value weight matrix:

[[0 2 0]
 [0 3 0]
 [1 0 3]
 [1 1 0]]

The second step of our model is ready:

Figure 1.13: Weight matrices added to the model

We will now multiply the weights by the input vectors to obtain Q, K, and V.

Step 3: Matrix multiplication to obtain Q, K, V

We will now multiply the input vectors by the weight matrices to obtain a query, key, and value vector for each input.

In this model, we will assume that there is one w_query, w_key, and w_value weight matrix for all inputs. Other approaches are possible.

Let's first multiply the input vectors by the w_query weight matrix:

print("Step 3: Matrix multiplication to obtain Q,K,V")
print("Query: x * w_query")

The output is a vector for Q1= [1, 0, 2],Q2= [2,2, 2], and Q3= [2,1, 3]:

Step 3: Matrix multiplication to obtain Q,K,V
Query: x * w_query
[[1. 0. 2.]
 [2. 2. 2.]
 [2. 1. 3.]]

We now multiply the input vectors by the w_key weight matrix:

print("Key: x * w_key")

We obtain a vector for K1= [0, 1, 1],K2= [4, 4, 0], and K3= [2 ,3, 1]:

Key: x * w_key
[[0. 1. 1.]
 [4. 4. 0.]
 [2. 3. 1.]]

Finally, we multiply the input vectors by the w_value weight matrix:

print("Value: x * w_value")

We obtain a vector for V1= [1, 2, 3],V2= [2, 8, 0], and V3= [2 ,6, 3]:

Value: x * w_value
[[1. 2. 3.]
 [2. 8. 0.]
 [2. 6. 3.]]

The third step of our model is ready:

Figure 1.14: Q, K, and V are generated

We have the Q, K, and V values we need to calculate the attention scores.

Step 4: Scaled attention scores

The attention head now implements the original Transformer equation:

Step 4 focuses on Q and K:

For this model, we will round and plug the values into the Q and K part of the equation:

print("Step 4: Scaled Attention Scores")
k_d = 1   #square root of k_d=3 rounded down to 1 for this example
attention_scores = (Q @ K.transpose())/k_d

The intermediate result is displayed:

Step 4: Scaled Attention Scores
[[ 2.  4.  4.]
 [ 4. 16. 12.]
 [ 4. 12. 10.]]

Step 4 is now complete. For example, the score for x1 is [2,4,4] across the K vectors across the head as displayed:

Figure 1.15: Scaled attention scores for input #1

The attention equation will now apply softmax to the intermediate scores for each vector.

Step 5: Scaled softmax attention scores for each vector

We now apply a softmax function to each intermediate attention score. Instead of doing a matrix multiplication, let's zoom down to each individual vector:

print("Step 5: Scaled softmax attention_scores for each vector")

We obtain scaled softmax attention scores for each vector:

Step 5: Scaled softmax attention_scores for each vector
[0.06337894 0.46831053 0.46831053]
[6.03366485e-06 9.82007865e-01 1.79861014e-02]
[2.95387223e-04 8.80536902e-01 1.19167711e-01] 

Step 5 is now complete. For example, the softmax of the score of x1 for all the keys is:

Figure 1.16: Softmax score of input #1 for all of the keys

We can now calculate the final attention values with the complete equation.

Step 6: The final attention representations

We now can finalize the attention equation by plugging V in:

We will first calculate the attention score of input x1 for Steps 6 and 7. We calculate one attention value for one word vector. When we reach Step 8, we will generalize the attention calculation to the other two input vectors.

To obtain Attention (Q,K,V) for x1, we multiply the intermediate attention score by the 3 value vectors one by one to zoom down into the inner workings of the equation:

print("Step 6: attention value obtained by score1/k_d * V")
print("Attention 1")
print("Attention 2")
print("Attention 3")
Step 6: attention value obtained by score1/k_d * V
[1. 2. 3.]
[2. 8. 0.]
[2. 6. 3.]
Attention 1
[0.06337894 0.12675788 0.19013681]
Attention 2
[0.93662106 3.74648425 0.        ]
Attention 3
[0.93662106 2.80986319 1.40493159]

Step 6 is complete. For example, the 3 attention values for x1 for each input have been calculated:

Figure 1.17: Attention representations

The attention values now need to be summed up.

Step 7: Summing up the results

The 3 attention values of input #1 obtained will now be summed to obtain the first line of the output matrix:

print("Step7: summed the results to create the first line of the output matrix")

The output is the first line of the output matrix for input #1:

Step 7: summed the results to create the first line of the output matrix
[1.93662106 6.68310531 1.59506841]]

The second line will be for the output of the next input, input #2, for example.

We can see the summed attention value for x1 in Figure 1.18:

Figure 1.18: Summed results for one input

We have completed the steps for input #1. We now need to add the results of all the inputs to the model.

Step 8: Steps 1 to 7 for all the inputs

The Transformer can now produce the attention values of input #2 and input #3 using the same method described from Step 1 to Step 7 for one attention head.

From this step onward, we will assume we have 3 attention values with learned weights with dmodel = 64. We now want to see what the original dimensions look like when they reach the sub-layer's output.

We have seen the attention representation process in detail with a small model. Let's go directly to the result and assume we have generated the 3 attention representations with a dimension of dmodel = 64:

print("Step 8: Step 1 to 7 for inputs 1 to 3")
#We assume we have 3 results with learned weights (they were not trained in this example)
#We assume we are implementing the original Transformer paper.We will have 3 results of 64 dimensions each
attention_head1=np.random.random((3, 64))

The following output displays the simulation of z0, which represents the 3 output vectors of dmodel = 64 dimensions for head 1:

Step 8: Step 1 to 7 for inputs 1 to 3
[[0.31982626 0.99175996…(61 squeezed values)  ... 0.16233212]
 [0.99584327 0.55528662…(61 squeezed values)  ... 0.70160307]
 [0.14811583 0.50875291…(61 squeezed values)  ... 0.83141355]]

The results will vary when you run the notebook because of the random generation of the vectors.

The Transformer now has the output vectors for the inputs of one head. The next step is to generate the outputs of the 8 heads to create the final output of the attention sub-layer.

Step 9: The output of the heads of the attention sub-layer

We assume that we have trained the 8 heads of the attention sub-layer. The transformer now has 3 output vectors (of the 3 input vectors that are words or word pieces) of dmodel = 64 dimensions each:

print("Step 9: We assume we have trained the 8 heads of the attention sub-layer")
z0h1=np.random.random((3, 64))
z1h2=np.random.random((3, 64))
z2h3=np.random.random((3, 64))
z3h4=np.random.random((3, 64))
z4h5=np.random.random((3, 64))
z5h6=np.random.random((3, 64))
z6h7=np.random.random((3, 64))
z7h8=np.random.random((3, 64))
print("shape of one head",z0h1.shape,"dimension of 8 heads",64*8)

The output shows the shape of one of the heads:

Step 9: We assume we have trained the 8 heads of the attention sub-layer
shape of one head (3, 64) dimension of 8 heads 512

The 8 heads have now produced Z:

Z = (z0, z1, z2, z3, z4, z5, z6, z7,)

The Transformer will now concatenate the 8 elements of Z for the final output of the multi-head attention sub-layer.

Step 10: Concatenation of the output of the heads

The Transformer concatenates the 8 elements of Z:

MultiHead(output) = Concat(z0, z1, z2, z3, z4, z5, z6, z7,) W0 = x, dmodel

Note that Z is multiplied by W0, which is a weight matrix that is trained as well. In this model, we will assume W0 is trained and integrated into the concatenation function.

z0 to z7 is concantenated:

print("Step 10: Concantenation of heads 1 to 8 to obtain the original 8x64=512 ouput dimension of the model")

The output is the concatenation of Z:

Step 10: Concatenation of heads 1 to 8 to obtain the original 8x64=512 output dimension of the model
[[0.65218495 0.11961095 0.9555153  ... 0.48399266 0.80186221 0.16486792]
 [0.95510952 0.29918492 0.7010377  ... 0.20682832 0.4123836  0.90879359]
 [0.20211378 0.86541746 0.01557758 ... 0.69449636 0.02458972 0.889699  ]]

The concatenation can be visualized as stacking the elements of Z side by side:

Figure 1.19: Attention sub-layer output

The concatenation produced a standard dmodel = 512 dimensional output:

Figure 1.20: Concatenation of the output of the 8 heads

Layer normalization will now process the attention sub-layer.

Post-layer normalization

Each attention sub-layer and each feedforward sub-layer of the Transformer is followed by post-layer normalization (Post-LN):

Figure 1.21: Post-layer normalization

The Post-LN contains an add function and a layer normalization process. The add function processes the residual connections that come from the input of the sub-layer. The goal of the residual connections is to make sure critical information is not lost. The Post-LN or layer normalization can thus be described as follows:


Sublayer(x) is the sub-layer itself. x is the information available at the input step of Sublayer(x).

The input of LayerNorm is a vector v resulting from x + Sublayer(x). dmodel = 512 for every input and output of the Transformer, which standardizes all the processes.

Many layer normalization methods exist, and variations exist from one model to another. The basic concept for v= x + Sublayer(x) can be defined by LayerNorm(v):


The variables are:

  • is the mean of v of dimension d. As such:
  • is the standard deviation v of dimension d. As such:
  • is a scaling parameter.
  • is a bias vector.

This version of LayerNorm(v) shows the general idea of the many possible Post-LN methods.

The next sub-layer can now process the output of the Post-LN or LayerNorm(v). In this case, the sub-layer is a feedforward network.

Sub-layer 2: Feedforward network

The input of the FFN is the dmodel = 512 output of the Post-LN of the previous sub-layer:

Figure 1.22: Feedforward sub-layer

The FFN sub-layer can be described as follows:

  • The FFNs in the encoder and decoder are fully connected.
  • The FFN is a position-wise network. Each position is processed separately and in an identical way.
  • The FFN contains two layers and applies a ReLU activation function.
  • The input and output of the FFN layers is dmodel = 512, but the inner layer is larger with dff = 2048
  • The FFN can be viewed as performing two kernel size 1 convolutions.

Taking this description into account, we can describe the optimized and standardized FFN as follows:

FFN(x) = max(0, xW1 + b1)W2 =b2

The output of the FFN goes to the Post-LN, as described in the previous section. Then the output is sent to the next layer of the encoder stack and the multi-head attention layer of the decoder stack.

Let's now explore the decoder stack.

The decoder stack

The layers of the decoder of the Transformer model are stacks of layers like the encoder layers. Each layer of the decoder stack has the following structure:

Figure 1.23: A layer of the decoder stack of the Transformer

The structure of the decoder layer remains the same as the encoder for all the N=6 layers of the Transformer model. Each layer contains three sub-layers: a multi-headed masked attention mechanism, a multi-headed attention mechanism, and a fully connected position-wise feedforward network.

The decoder has a third main sub-layer, which is the masked multi-head attention mechanism. In this sub-layer output, at a given position, the following words are masked so that the Transformer bases its assumptions on its inferences without seeing the rest of the sequence. That way, in this model, it cannot see future parts of the sequence.

A residual connection, Sublayer(x), surrounds each of the three main sub-layers in the Transformer model like in the encoder stack:

LayerNormalization(x + Sublayer(x))

The embedding layer sub-layer is only present at the bottom level of the stack, like for the encoder stack. The output of every sub-layer of the decoder stack has a constant dimension, dmodel like in the encoder stack, including the embedding layer and the output of the residual connections.

We can see that the designers worked hard to create symmetrical encoder and decoder stacks.

The structure of each sub-layer and function of the decoder is similar to the encoder. In this section, we can refer to the encoder for the same functionality when we need to. We will only focus on the differences between the decoder and the encoder.

Output embedding and position encoding

The structure of the sub-layers of the decoder is mostly the same as the sub-layers of the encoder. The output embedding layer and position encoding function are the same as in the encoder stack.

In the Transformer usage we are exploring through the model presented by Vaswani et al. (2017), the output is a translation we need to learn. I chose to use a French translation:

Output=Le chat noir était assis sur le canapé et le chien marron dormait sur le tapis 

This output is the French translation of the English input sentence:

Input=The black cat sat on the couch and the brown dog slept on the rug.

The output words go through the word embedding layer, and then the positional encoding function, like in the first layer of the encoder stack.

Let's see the specific properties of the multi-head attention layers of the decoder stack.

The attention layers

The Transformer is an auto-regressive model. It uses the previous output sequences as an additional input. The multi-head attention layers of the decoder use the same process as the encoder.

However, the masked multi-head attention sub-layer 1 only lets attention apply to the positions up to and including the current position. The future words are hidden from the Transformer, and this forces it to learn how to predict.

A post-layer normalization process follows the masked multi-head attention sub-layer 1 as in the encoder.

The multi-head attention sub-layer 2 also only attends to the positions up to the current position the Transformer is predicting to avoid seeing the sequence it must predict.

The multi-head attention sub-layer 2 draws information from the encoder by taking encoder (K, V) into account during the dot-product attention operations. This sub-layer also draws information from the masked multi-head attention sub-layer 1 (masked attention) by also taking sub-layer 1(Q) into account during the dot-product attention operations. The decoder thus uses the trained information of the encoder. We can define the input of the self-attention multi-head sub-layer of a decoder as:

Input_Attention=(Output_decoder_sub_layer-1(Q), Output_encoder_layer(K,V))

A post-layer normalization process follows the masked multi-head attention sub-layer 1 as in the encoder.

The Transformer then goes to the FFN sub-layer, followed by a Post-LN and the linear layer.

The FFN sub-layer, the Post-LN, and the linear layer

The FFN sub-layer has the same structure as the FFN of the encoder stack. The Post-LN of the FFN works as the layer normalization of the encoder stack.

The Transformer produces an output sequence of only one element at a time:

Output sequence= (y1, y2, … yn)

The linear layer produces an output sequence with a linear function that varies per model but relies on the standard method:

y = w*x + b

x and b are learned parameters.

The linear layer will thus produce the next probable elements of a sequence that a softmax function will convert into a probable element.

The decoder layer as the encoder layer will then go from layer l to layer l+1 up to the top layer of the N=6-layer transformer stack.

Let's now see how the Transformer was trained and the performance it obtained.


Training and performance

The original Transformer was trained on a 4.5-million-sentence-pair English-German dataset and a 36-million-sentence English-French dataset.

The datasets come from Workshops on Machine Translation (WMT), which can be found at the following link if you wish to explore the WMT datasets:

The training of the original Transformer base models took 12 hours to train for 100,000 steps on a machine with 8 NVIDIA P100 GPUs. The big models took 3.5 days for 300,000 steps.

The original Transformer outperformed all the previous machine translation models with a BLEU score of 41.8. The result was obtained on the WMT English-to-French dataset.

BLEU stands for Bilingual Evaluation Understudy. It is an algorithm that evaluates the quality of the results of machine translations.

The Google Research and Google Brain team applied optimization strategies to improve the performance of the Transformer. For example, the Adam optimizer was used, but the learning rate varied by first going through warmup states with a linear rate and decreasing the rate afterward.

Different types of regularization techniques were applied, such as residual dropout and dropouts, to the sums of embeddings. Also, the Transformer applies label smoothing, which avoids overfitting with overconfident one-hot outputs. It introduces less accurate evaluations and forces the model to train more and better.

Several other Transformer model variations have led to other models and usages that we will explore in subsequent chapters.

Before we leave, let's get a feel for the simplicity of ready-to-use transformer models in Hugging Face.

Before we end the chapter

Everything you saw in this chapter can be condensed into a ready-to-use Hugging Face transformer model. Bear in mind that Hugging Face, like all other solutions, is evolving at full speed to keep up with the research labs so you might encounter deprecation messages in the future.

With Hugging Face, you can implement machine translation in three lines of code!

Open Multi_Head_Attention_Sub_Layer.ipynb in Google Colaboratory. Save the notebook in your Google Drive (make sure you have a Gmail account). Go to the last two cells.

We first ensure that Hugging Face's transformers are installed:

!pip -qq install transformers

The first cell imports the Hugging Face pipeline, which contains several transformer usages:

#@title Retrieve pipeline of modules and choose English to French translation
from transformers import pipeline

We then implement the Hugging Face pipeline that contains several transformer usages. The pipeline contains ready-to-use functions. In our case, to illustrate the Transformer model of this chapter, we activate the translator model and enter a sentence to translate from English to French:

translator = pipeline("translation_en_to_fr")
#One line of code!
print(translator("It is easy to translate languages with transformers", max_length=40))

And voilà! The translation is displayed:

[{'translation_text': 'Il est facile de traduire des langues avec des transformateurs.'}]

Hugging Face shows how transformer architectures can be used in ready-to-use models.



In this chapter, we first got started by examining the mind-blowing long-distance dependencies transformer architectures can uncover. Transformers can perform transduction from written and oral sequences to meaningful representations as never before in the history of Natural Language Understanding (NLU).

These two dimensions, the expansion of transduction and the simplification of implementation, are taking artificial intelligence to a level never seen before.

We explored the bold approach of removing RNNs, LSTMs, and CNNs from transduction problems and sequence modeling to build the Transformer architecture. The symmetrical design of the standardized dimensions of the encoder and decoder makes the flow from one sub-layer to another nearly seamless.

We saw that beyond removing recurrent network models, transformers introduce parallelized layers that reduce training time. We discovered other innovations, such as positional encoding and masked multi-headed attention.

The flexible, original Transformer architecture provides the basis for many other innovative variations that open the way for yet more powerful transduction problems and language modeling.

We will zoom in more depth into some aspects of the Transformer's architecture in the following chapters when describing the many variants of the original model.

The arrival of the Transformer marks the beginning of a new generation of ready-to-use artificial intelligence models. For example, Hugging Face and Google Brain make artificial intelligence easy to implement with a few lines of code.

In the next chapter, Fine-Tuning BERT Models, we will explore the powerful evolutions of the original Transformer model.



  1. NLP transduction can encode and decode text representations. (True/False)
  2. Natural Language Understanding (NLU) is a subset of Natural Language Processing (NLP). (True/False)
  3. Language modeling algorithms generate probable sequences of words based on input sequences. (True/False)
  4. A transformer is a customized LSTM with a CNN layer. (True/False)
  5. A transformer does not contain an LSTM or CNN layers. (True/False)
  6. Attention examines all of the tokens in a sequence, not just the last one. (True/False)
  7. A transformer uses a positional vector, not positional encoding. (True/False)
  8. A transformer contains a feedforward network. (True/False)
  9. The masked multi-headed attention component of the decoder of a transformer prevents the algorithm parsing a given position from seeing the rest of a sequence that is being processed. (True/False)
  10. Transformers can analyze long-distance dependencies better than LSTMs. (True/False)


About the Author

  • Denis Rothman

    Denis Rothman graduated from Sorbonne University and Paris-Diderot University, patenting one of the very first word2matrix embedding solutions. Denis Rothman is the author of three cutting-edge AI solutions: one of the first AI cognitive chatbots more than 30 years ago; a profit-orientated AI resource optimizing system; and an AI APS (Advanced Planning and Scheduling) solution based on cognitive patterns used worldwide in aerospace, rail, energy, apparel, and many other fields. Designed initially as a cognitive AI bot for IBM, it then went on to become a robust APS solution used to this day.

    Browse publications by this author

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