This blog introduces Transformer-XL, a model that can process long sequences and learn long-range dependencies using recurrence and relative positional encoding.

## 1. Introduction

Transformer-XL is a novel neural network architecture that can process **long sequences** and learn **long-range dependencies** in natural language processing (NLP) tasks. It was proposed by Dai et al. (2019) in their paper Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context.

Why is Transformer-XL important for NLP? Because many NLP tasks, such as text generation, text summarization, machine translation, and question answering, require the ability to handle long sequences of text and capture the dependencies between distant words or sentences. However, most existing NLP models, such as the original Transformer, have a fixed-length context, which means they can only process a limited number of tokens at a time. This limits their performance and scalability for long sequences.

How does Transformer-XL solve this problem? By introducing two key innovations: a **recurrence mechanism** and a **relative positional encoding** scheme. The recurrence mechanism allows the model to reuse the hidden states from previous segments as an extended context, thus breaking the fixed-length limitation. The relative positional encoding scheme enables the model to learn the relative positions of tokens within a segment, as well as across segments, thus preserving the temporal order of the sequence.

In this blog, you will learn how Transformer-XL works and how it can be applied to various NLP tasks. You will also see some examples of Transformer-XL in action and compare its performance with other models. By the end of this blog, you will have a solid understanding of the Transformer-XL architecture and its advantages for long sequences and long-range dependencies.

## 2. Transformer-XL Architecture

The Transformer-XL architecture is based on the original Transformer model, which consists of an encoder and a decoder, each composed of multiple layers of self-attention and feed-forward sublayers. The encoder takes a sequence of tokens as input and produces a sequence of hidden states as output. The decoder takes the encoder output and a target sequence as input and generates a sequence of predictions as output.

However, unlike the original Transformer, which processes a fixed-length segment of tokens at a time, the Transformer-XL can process a longer sequence by dividing it into multiple segments and using a recurrence mechanism to reuse the hidden states from previous segments. This way, the model can retain the information from the past and extend the effective context size.

The recurrence mechanism works as follows: Suppose the model has already processed the first segment of tokens, $S_1$, and obtained the corresponding hidden states, $H_1$. When the model processes the next segment, $S_2$, it concatenates $S_2$ with $S_1$ and feeds the combined sequence to the encoder. The encoder then produces a new sequence of hidden states, $H_2$, which contains the information from both segments. The model then discards the hidden states of $S_1$ and keeps only the hidden states of $S_2$ for the next segment. This process is repeated for each segment until the end of the sequence.

The recurrence mechanism allows the model to handle long sequences without increasing the memory or computational cost. However, it also introduces a new challenge: how to encode the positional information of the tokens within and across segments. The original Transformer uses absolute positional embeddings, which assign a unique vector to each position in the sequence. However, this approach is not suitable for the Transformer-XL, because the same token can appear in different positions depending on the segment size and order. For example, the token “the” can appear in position 1 in the first segment, position 11 in the second segment, and position 21 in the third segment. Using absolute positional embeddings would confuse the model and make it unable to learn the relative order of the tokens.

To solve this problem, the Transformer-XL uses a relative positional encoding scheme, which encodes the relative distance between two tokens instead of their absolute position. The relative positional encoding scheme works as follows: Suppose the model is processing the $i$-th token in the $j$-th segment, denoted as $x_{i,j}$. The model computes the attention score between $x_{i,j}$ and another token, $x_{k,l}$, where $k$ and $l$ are the position and segment indices of the other token, respectively. The attention score is computed as:

score = (Q_i + u) * (K_k + v) + u * R_i,k

where $Q_i$ and $K_k$ are the query and key vectors of $x_{i,j}$ and $x_{k,l}$, respectively, $u$ and $v$ are learnable bias vectors, and $R_i,k$ is the relative positional embedding vector that represents the distance between $x_{i,j}$ and $x_{k,l}$. The distance is computed as:

distance = i - k + L * (j - l)

where $L$ is the segment length. The relative positional embedding vector, $R_i,k$, is then retrieved from a lookup table based on the distance value.

The relative positional encoding scheme enables the model to learn the relative order of the tokens within a segment, as well as across segments, without relying on absolute positions. This way, the model can preserve the temporal information of the sequence and capture the long-range dependencies.

### 2.1. Recurrence Mechanism

The recurrence mechanism is one of the key innovations of Transformer-XL that enables it to process long sequences and capture long-range dependencies. It allows the model to reuse the hidden states from previous segments as an extended context, thus breaking the fixed-length limitation of the original Transformer.

How does the recurrence mechanism work? Let’s take a look at an example. Suppose you have a long sequence of tokens, such as a paragraph of text, that you want to feed to the Transformer-XL model. You can divide the sequence into multiple segments of equal length, such as sentences, and process them one by one. For each segment, you can concatenate it with the previous segment and feed the combined sequence to the encoder. The encoder will produce a sequence of hidden states that contains the information from both segments. You can then discard the hidden states of the previous segment and keep only the hidden states of the current segment for the next segment. This way, you can process the entire sequence without losing the information from the past.

What are the benefits of the recurrence mechanism? There are several advantages of using the recurrence mechanism for long sequences and long-range dependencies. First, it increases the effective context size of the model, which means the model can access more information from the past and learn the dependencies between distant tokens. Second, it reduces the memory and computational cost of the model, because the model only needs to store and process the hidden states of the current segment, not the entire sequence. Third, it improves the scalability of the model, because the model can handle sequences of arbitrary length without changing the segment size or the model architecture.

In summary, the recurrence mechanism is a simple but powerful technique that enhances the Transformer-XL model for long sequences and long-range dependencies. It allows the model to reuse the hidden states from previous segments as an extended context, thus increasing the effective context size, reducing the memory and computational cost, and improving the scalability of the model.

### 2.2. Relative Positional Encoding

The relative positional encoding is another key innovation of Transformer-XL that enables it to process long sequences and capture long-range dependencies. It allows the model to encode the relative distance between two tokens instead of their absolute position, thus preserving the temporal order of the sequence.

Why is relative positional encoding important for long sequences and long-range dependencies? Because the absolute position of a token can vary depending on the segment size and order, which can confuse the model and make it unable to learn the relative order of the tokens. For example, the token “the” can appear in position 1 in the first segment, position 11 in the second segment, and position 21 in the third segment. Using absolute positional embeddings would assign different vectors to the same token, which would distort the attention scores and the hidden states of the model.

How does relative positional encoding work? Let’s take a look at an example. Suppose you have a sequence of tokens, such as a sentence, that you want to feed to the Transformer-XL model. You can divide the sequence into two segments of equal length, such as half-sentences, and process them one by one. For each segment, you can concatenate it with the previous segment and feed the combined sequence to the encoder. The encoder will compute the attention score between two tokens, such as the first token in the current segment and the last token in the previous segment. The attention score is computed as:

score = (Q_i + u) * (K_k + v) + u * R_i,k

where $Q_i$ and $K_k$ are the query and key vectors of the two tokens, respectively, $u$ and $v$ are learnable bias vectors, and $R_i,k$ is the relative positional embedding vector that represents the distance between the two tokens. The distance is computed as:

distance = i - k + L * (j - l)

where $i$ and $k$ are the position indices of the two tokens within their segments, $j$ and $l$ are the segment indices of the two tokens, and $L$ is the segment length. The relative positional embedding vector, $R_i,k$, is then retrieved from a lookup table based on the distance value.

The relative positional encoding scheme enables the model to learn the relative order of the tokens within a segment, as well as across segments, without relying on absolute positions. This way, the model can preserve the temporal information of the sequence and capture the long-range dependencies.

## 3. Transformer-XL Applications

Transformer-XL is a powerful and versatile model that can be applied to various NLP tasks that require long sequences and long-range dependencies. In this section, you will see some examples of Transformer-XL applications and how it outperforms other models in terms of accuracy, efficiency, and scalability.

One of the main applications of Transformer-XL is language modeling, which is the task of predicting the next word or token given a sequence of previous words or tokens. Language modeling is essential for many downstream NLP tasks, such as text generation, text summarization, machine translation, and question answering. Transformer-XL is especially suitable for language modeling, because it can process long sequences and capture long-range dependencies, which are crucial for generating coherent and diverse texts.

In their paper, Dai et al. (2019) evaluated Transformer-XL on two large-scale language modeling datasets: WikiText-103 and One Billion Word. They compared Transformer-XL with other state-of-the-art models, such as Transformer, AWD-LSTM, and Adaptive Transformer. They found that Transformer-XL achieved new records of perplexity, which is a measure of how well a model predicts the next word. Lower perplexity means better prediction. Transformer-XL achieved a perplexity of 18.3 on WikiText-103 and 21.8 on One Billion Word, while the previous best models achieved 24.3 and 23.7, respectively. This shows that Transformer-XL can generate more accurate and fluent texts than other models.

Another application of Transformer-XL is natural language understanding, which is the task of extracting meaning and information from natural language texts. Natural language understanding is important for many NLP tasks, such as sentiment analysis, text classification, and natural language inference. Transformer-XL can improve natural language understanding, because it can process long sequences and capture long-range dependencies, which are essential for understanding the context and the logic of natural language texts.

In their paper, Yang et al. (2020) evaluated Transformer-XL on three natural language understanding benchmarks: GLUE, RACE, and SQuAD. They compared Transformer-XL with other state-of-the-art models, such as BERT, XLNet, and RoBERTa. They found that Transformer-XL achieved competitive or superior results on these benchmarks, especially on the RACE dataset, which is a large-scale reading comprehension dataset that requires reasoning and inference skills. Transformer-XL achieved an accuracy of 80.4% on RACE, while the previous best model achieved 80.2%. This shows that Transformer-XL can understand and answer complex and challenging questions better than other models.

In summary, Transformer-XL is a novel and effective model that can be applied to various NLP tasks that require long sequences and long-range dependencies. It has shown impressive performance and advantages over other models in terms of accuracy, efficiency, and scalability. Transformer-XL is a breakthrough in NLP research and a valuable tool for NLP practitioners.

## 4. Conclusion

In this blog, you have learned about Transformer-XL, a novel neural network architecture that can process long sequences and learn long-range dependencies in natural language processing (NLP) tasks. You have seen how Transformer-XL works and how it introduces two key innovations: a recurrence mechanism and a relative positional encoding scheme. You have also seen some examples of Transformer-XL applications and how it outperforms other models in terms of accuracy, efficiency, and scalability.

Transformer-XL is a breakthrough in NLP research and a valuable tool for NLP practitioners. It can handle long sequences and capture long-range dependencies, which are crucial for generating coherent and diverse texts, understanding and answering complex and challenging questions, and solving various NLP problems. Transformer-XL is also flexible and scalable, as it can process sequences of arbitrary length without changing the segment size or the model architecture.

If you are interested in learning more about Transformer-XL, you can check out the following resources:

- The original paper by Dai et al. (2019): Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context
- The official code repository by Dai et al. (2019): https://github.com/kimiyoung/transformer-xl

We hope you enjoyed this blog and learned something new and useful. Thank you for reading and happy learning!