# Rotary Positional Embeddings (RoPE)

This is an implementation of Rotary Positional Embeddings (RoPE) in PyTorch.

Rotary Positional Embeddings (RoPE) encode position information of tokens with a rotation matrix that naturally incorporates explicit relative position dependency.

Here's the training code for training a transformer model with RoPE on Tiny Shakespeare dataset.

25import torch
26from torch import nn
27
28from labml.logger import inspect
29from labml_nn.transformers.mha import MultiHeadAttention

## RoPE module

Rotary encoding transforms pairs of features by rotating in the 2D plane. That is, it organizes the features as pairs. Each pair can be considered a coordinate in a 2D plane, and the encoding will rotate it by an angle depending on the position of the token.

### For a pair of features

Let and be two features of the key or query of any head at position . Or for simplicity assume has only two features. Then the transformation is,

where is a constant angle. The other pairs of features are transformed similarly.

### Attention is relative

For a pair of features, dot-product attention score between two positions and would be

This shows that for dot-production attention the rotary encodings gives relative attention.

### For all features

The features are grouped into pairs and handled as above. They use a different for each pair.

The paper suggests using for the pairs of features.

We pair feature with feature . So for position we transform

to

32class RotaryPositionalEmbeddings(nn.Module):
• d is the number of features
• base is the constant used for calculating
118    def __init__(self, d: int, base: int = 10_000):
123        super().__init__()
125        self.theta = nn.Parameter(1. / (base ** (torch.arange(0, d, 2).float() / d)), requires_grad=False)
• x is the Tensor at the head of a key or a query with shape [seq_len, batch_size, n_heads, d]
127    def forward(self, x: torch.Tensor):

Extract the shape

132        seq_len, batch_size, n_heads, d = x.shape
135        d_2 = d // 2

Create position indexes [0, 1, ..., seq_len - 1]

138        seq_idx = torch.arange(seq_len, device=x.device).type_as(self.theta)

Calculate the product of position index and

141        idx_theta = torch.einsum('n,d->nd', seq_idx, self.theta)

Concatenate so that for row we have

145        idx_theta2 = torch.cat([idx_theta, idx_theta], dim=1)

Calculate

148        neg_half_x = torch.cat([-x[:, :, :, d_2:], x[:, :, :, :d_2]], dim=-1)

Calculate

for

160        rx = (x * idx_theta2.cos()[:, None, None, :]) + (neg_half_x * idx_theta2.sin()[:, None, None, :])
163        return rx

## Multi-head attention with rotary positional embeddings

We override multi-head attention from original transformer.

166class RotaryPEMultiHeadAttention(MultiHeadAttention):
173    def __init__(self, heads: int, d_model: int, dropout_prob: float = 0.1):

The linear transformations do not need a bias since we explicitly include it when calculating scores. However having a bias for value might make sense.

177        super().__init__(heads, d_model, dropout_prob, bias=False)

Rotary positional embedding layers

180        self.query_rotary_pe = RotaryPositionalEmbeddings(self.d_k)
181        self.key_rotary_pe = RotaryPositionalEmbeddings(self.d_k)

### Calculate scores between queries and keys

183    def get_scores(self, query: torch.Tensor, key: torch.Tensor):

Calculate dot-product with RoPE

189        return torch.einsum('ibhd,jbhd->ijbh', self.query_rotary_pe(query), self.key_rotary_pe(key))

Testing RoPE with a simple example

192def _test_rotary():
196    x = torch.tensor([[1, 2, 3, 4], [4, 5, 6, 7], [7, 8, 9, 10]], dtype=torch.float)
197    x = x[:, None, None, :]
198    inspect(x)
199
200    rotary_pe = RotaryPositionalEmbeddings(3)
201    inspect(rotary_pe(x))
202
203
204if __name__ == '__main__':
205    _test_rotary()