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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 $d$ features as $\frac{d}{2}$ 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 $x_{m}^{(1)}$ and $x_{m}^{(2)}$ be two features of the key or query of any head at position $m$ . Or for simplicity assume $x$ has only two features. Then the transformation is,

R o PE (x_{m}^{(1)}, x_{m}^{(2)}, m) = (cos m θ sin m θ - sin m θ cos m θ) (x_{m}^{(1)} x_{m}^{(2)}) = (x_{m}^{(1)} cos m θ - x_{m}^{(2)} sin m θ x_{m}^{(2)} cos m θ + x_{m}^{(1)} sin m θ)

where $θ$ 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 $m$ and $n$ would be

⟨ R o PE (x_{m}^{(1)}, x_{m}^{(2)}, m), R o PE (x_{n}^{(1)}, x_{n}^{(2)}, n) ⟩ (x_{m}^{(1)} cos m θ - x_{m}^{(2)} sin m θ) (x_{n}^{(1)} cos n θ - x_{n}^{(2)} sin n θ) (x_{m}^{(2)} cos m θ + x_{m}^{(1)} sin m θ) (x_{n}^{(2)} cos n θ + x_{n}^{(1)} sin n θ) x_{m}^{(1)} x_{n}^{(1)} (cos m θ cos n θ + sin m θ sin n θ) x_{m}^{(1)} x_{n}^{(2)} (- cos m θ sin n θ + sin m θ cos n θ) x_{m}^{(2)} x_{n}^{(1)} (- sin m θ cos n θ + cos m θ sin n θ) x_{m}^{(2)} x_{n}^{(2)} (sin m θ sin n θ + cos m θ cos n θ) x_{m}^{(1)} x_{n}^{(1)} cos (m - n) θ + x_{m}^{(1)} x_{n}^{(2)} sin (m - n) θ - x_{m}^{(2)} x_{n}^{(1)} sin (m - n) θ + x_{m}^{(2)} x_{n}^{(1)} cos (m - n) θ (x_{m}^{(1)} cos (m - n) θ - x_{m}^{(2)} sin (m - n) θ) x_{n}^{(1)} (x_{m}^{(2)} cos (m - n) m θ + x_{m}^{(1)} sin (m - n) θ) x_{n}^{(2)} ⟨ R o PE (x_{m}^{(1)}, x_{m}^{(2)}, m - n), R o PE (x_{n}^{(1)}, x_{n}^{(2)}, 0) ⟩ = + = + + + = + = + =

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 $θ$ for each pair.

The paper suggests using $Θ = θ_{i} = 1000 0^{\frac{2 ( i - 1 )}{d}}, i \in [1, 2, ..., \frac{d}{2}]$ for the $\frac{d}{2}$ pairs of features.

We pair feature $i$ with feature $i + \frac{d}{2}$ . So for position $m$ we transform

(x_{m}^{(i)} x_{m}^{(i + \frac{d}{2})})

to

(x_{m}^{(i)} cos m θ_{i} - x_{m}^{(i + \frac{d}{2})} sin m θ_{i} x_{m}^{(i + \frac{d}{2})} cos m θ_{i} + x_{m}^{(i)} sin m θ_{i})

32class RotaryPositionalEmbeddings(nn.Module):

#

d is the number of features $d$
base is the constant used for calculating $Θ$

118    def __init__(self, d: int, base: int = 10_000):

#

123        super().__init__()

#

$Θ = θ_{i} = 1000 0^{\frac{2 ( i - 1 )}{d}}, i \in [1, 2, ..., \frac{d}{2}]$

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

#

$\frac{d}{2}$

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 $θ_{i}$

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

#

Concatenate so that for row $m$ we have $[m θ_{0}, m θ_{1}, ..., m θ_{\frac{d}{2}}, m θ 0, m θ 1, ..., m θ_{\frac{d}{2}}]$

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

#

Calculate $[- x^{(\frac{d}{2} + 1)}, - x^{(\frac{d}{2} + 2)}, ..., - x^{(d)}, x^{(1)}, x^{(2)}, ..., - x^{(\frac{d}{2})}]$

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

#

Calculate

(x_{m}^{(i)} cos m θ_{i} - x_{m}^{(i + \frac{d}{2})} sin m θ_{i} x_{m}^{(i + \frac{d}{2})} cos m θ_{i} + x_{m}^{(i)} sin m θ_{i})

for $i \in 1, 2, ..., \frac{d}{2}$

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()