Overview of Relative Positional Encoding and examples of algorithms and implementations

Overview of Relative Positional Encoding

Relative Positional Encoding (RPE) is a method for neural network models that use the transformer architecture to incorporate relative positional information of words and tokens into the model. Although transformers have been very successful in many tasks such as natural language processing and image recognition, they are not good at directly modeling the relative positional relationships between tokens. Therefore, RPE is used to provide relative location information to the model.

Typically, transformer models capture position information with absolute position encoding (usually using sinus/cosine functions). This approach has the advantage that it is scalable with respect to sequence length and allows the transformer to adapt to different tasks. However, to accurately model relative positional information, Relative Positional Encoding is required.

Relative Positional Encoding is used to encode the relative positional relationship between words (tokens), and this encoding is usually calculated as follows

1. calculate the absolute positional encoding of each token. This will be the normal positional encoding.

2. calculate the relative position difference from the absolute position difference of the tokens. This takes into account the relative positional relationship between tokens.

3. Various mathematical techniques are used to encode relative position differences. In general, methods similar to absolute positional encoding using sinus/cosine functions are used, but based on relative positional information.

By using Relative Positional Encoding, the model can take into account the relative positional relationships between tokens, allowing for better performance on tasks that are sensitive to the order of elements in a sequence. This has been applied to a variety of natural language processing tasks, including machine translation, question answering, and text generation.

Algorithm used for Relative Positional Encoding

To implement Relative Positional Encoding (RPE), the following algorithms are usually used.

1. computation of absolute positional encoding:

First, the absolute positional encoding of each token is computed as in the normal transformer positional encoding. This is usually expressed by the following equation.

\[PE(pos, 2i) = sin(pos / 10000^(2i / d_model))\\
PE(pos, 2i+1) = cos(pos / 10000^(2i / d_model))\]

where \(pos\) is the absolute position of the token, \(i\) is the dimension index of the encoding, and \(d_model\) is the embedding dimension of the model.

2. Calculation of relative position differences:

Next, the relative position difference of each token pair is computed. This is done by deriving the relative position difference from the absolute position of each token, typically calculated as follows

\[rel_pos = pos1 – pos2\]

where \(pos1\) is the absolute position of one token and (pos2) is the absolute position of the other token.

3. computation of relative position encoding:

Based on the relative position difference, the relative position encoding is calculated. This is usually done using the same sinus/cosine function-based method as for absolute position encoding. Relative position encoding varies with position difference.

\[RPE(rel_pos, 2i) = sin(rel_pos / 10000^(2i / d_model))\\
RPE(rel_pos, 2i+1) = cos(rel_pos / 10000^(2i / d_model))\]

where \(rel_pos\) is the relative position difference, \(i\) is the dimension index of the encoding, and \(d_model\) is the embedding dimension of the model.

4. Combining absolute and relative position encodings:

Finally, the absolute and relative position encodings are combined to obtain the position encoding of each token. Usually, this concatenation is an element-by-element addition.

\[Token_Encoding = PE(pos) + RPE(rel_pos)\]

This incorporates both absolute and relative positional information in the encoding of each token, allowing the model to account for relative positional relationships among tokens. This method helps improve the model’s sensitivity to token ordering in natural language processing tasks.

Application Examples of Relative Positional Encoding

Relative Positional Encoding (RPE) is used in Natural Language Processing (NLP) and related tasks to consider relative positional information between tokens in the transformer model. The following are examples of RPE applications

1. machine translation:

In machine translation, the relative positional relationship between words is important between input and output sentences; RPE is used to capture the relative positional information between source and target language tokens to help produce more accurate translation results.

2. question answering:

In question-answering systems, the answer to a question often resides at a specific position in the sentence; RPE provides the model with relative positional information between the question and the sentence to help it find the correct answer.

3. text generation:

In the text generation task, the order of sentences is important. Incorporating relative positional information between tokens in a sentence into the model helps to maintain flow and consistency of the sentence. This is used in sentence summarization, sentence generation, and dialogue generation.

4. document classification:

RPE is used to incorporate relationships between sections or paragraphs within a document into the model. This allows the document classification model to more accurately understand the context and helps to take into account the structure of the document.

5. relationship analysis between tokens:

RPE can also be used as a tool to analyze the relative positional relationships among tokens. This is useful for tasks such as relationships between entities in a document, token dependencies, and processing graph data.

6. linguistic modeling:

The introduction of RPEs in language models can help improve the quality of text generation because models can take into account relative positional information in the probability of occurrence of tokens.

RPE is an approach that improves the performance of natural language processing tasks by incorporating relative positional information between tokens into the model, allowing for context-aware processing.

Example implementation of Relative Positional Encoding

The implementation of Relative Positional Encoding (RPE) depends on the programming language and deep learning framework, but the general approach is as follows. The following example is a simple implementation of RPE using Python and PyTorch.

import torch
import torch.nn.functional as F

# RPE Implementation
def relative_positional_encoding(queries, keys, max_relative_position=10, num_heads=8):
    batch_size, num_tokens, hidden_dim = queries.size()
    position_ids = torch.arange(0, num_tokens)
    position_ids = position_ids.unsqueeze(0).expand_as(queries)  # Location ID for each token

    # Calculate relative position differences
    relative_positions = position_ids[:, :, None] - position_ids[:, None, :]
    
    # Create weight matrix for RPE
    num_positions = 2 * max_relative_position + 1
    relative_position_weights = torch.arange(-max_relative_position, max_relative_position + 1).unsqueeze(0)

    # Calculate RPE using sinus and cosine
    relative_position_encodings = torch.sin(relative_positions / 10000 ** (2 * (relative_position_weights // 2) / hidden_dim))

    return relative_position_encodings

# Token embedding
embedding_dim = 256
num_tokens = 50
queries = torch.rand(batch_size, num_tokens, embedding_dim)
keys = torch.rand(batch_size, num_tokens, embedding_dim)

# RPE Application
rpe = relative_positional_encoding(queries, keys)
queries_with_rpe = queries + rpe  # Add RPE to query

# Here you can use queries_with_rpe to apply the transformer layer

This code shows the basic steps for calculating the RPE. Here, the relative positional differences between tokens are calculated and the corresponding relative positional encodings are generated. These RPEs can then be added to the token embeddings for use within the transformer model.

Relative Positional Encoding Challenges

Relative Positional Encoding (RPE) is a useful method for incorporating relative positional information in transformer models, but there are several challenges and limitations. The following is a discussion of some of the challenges of RPE.

1. memory usage:

Because RPE encodes location information on a token-by-token basis, memory usage increases as sequence length increases. Especially for long sequences, the cost is high for training and inference of large models.

2. positional distance constraints:

RPEs are usually constrained to a fixed relative position distance. That is, RPEs can only model positional relationships up to a certain maximum distance. This constraint may not be applicable to very long sentences or documents.

3. learning difficulty:

Tuning and designing the exact hyperparameters of the RPE can be difficult, and in particular, setting the optimal relative position distance and finding the dimensionality of the relative position encoding can be challenging because they are task-dependent.

4. computational cost:

Implementing RPE increases computational cost. Especially for large models and long sequences, RPE computation can become a bottleneck.

5. lack of generality:

RPEs may lack generality because they need to be tailored to specific tasks or data sets. Different tasks require different RPE designs.

These issues are perspectives that should be considered when using RPE. It is important to understand these challenges and pay attention to model design and hyperparameter tuning before using RPE for a specific task or dataset, and RPE may be considered for use in combination with other location encoding methods or model architectures.

Addressing Relative Positional Encoding Challenge

There are several strategies to address the challenges of Relative Positional Encoding (RPE). They are described below.

1. controlling memory usage:

If memory usage is an issue, the dimension of the RPE may be adjusted according to the length of the sequence. For longer sequences, using higher dimensional RPEs may provide appropriate information to the model, and changing the dense representation of RPEs to a sparse representation is also being considered to reduce memory usage.

2. relaxing positional distance constraints:

The maximum distance can be increased to make relative position encoding more flexible. Soft constraints could also be introduced when the model learns relative positions at different distances. Optimal relative position constraints should be designed for different tasks.

3. addressing learning difficulties:

Hyperparameter tuning and search algorithms can be used to tune the hyperparameters of the RPE. Alternatively, pre-trained RPEs from pre-trained models could be used, which would allow the selection of the best RPE for the task.

4. reduction of computational costs:

To reduce computational cost, the RPE could be processed using faster hardware or acceleration. In addition, algorithms that optimize the computation of the RPE or parallel processing can be used to make the computation more efficient.

5. improving generality:

To improve the generality of the RPE, it is important to try several different RPE designs to find the best RPE that fits the task and data set. It is also helpful to consider how to combine the transformer model with the RPE.

Reference Information and Reference Books

For more information on natural language processing in general, see “Natural Language Processing Technology” and “Overview of Natural Language Processing and Examples of Various Implementations.

Reference books include “Natural language processing (NLP): Unleashing the Power of Human Communication through Machine Intelligence“.

“Practical Natural Language Processing: A Comprehensive Guide to Building Real-World NLP Systems“

“Natural Language Processing With Transformers: Building Language Applications With Hugging Face“