#

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.

23import torch
24from torch import nn
25
26from labml.logger import inspect
27from 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) = (c o s m θ s i n m θ - s i n m θ c o s m θ) (x_{m}^{(1)} x_{m}^{(2)}) = (x_{m}^{(1)} c o s m θ - x_{m}^{(2)} s i n m θ x_{m}^{(2)} c o s m θ + x_{m}^{(1)} s i n 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)} c o s m θ - x_{m}^{(2)} s i n m θ) (x_{n}^{(1)} c o s n θ - x_{n}^{(2)} s i n n θ) (x_{m}^{(2)} c o s m θ + x_{m}^{(1)} s i n m θ) (x_{n}^{(2)} c o s n θ + x_{n}^{(1)} s i n n θ) x_{m}^{(1)} x_{n}^{(1)} (c o s m θ c o s n θ + s i n m θ s i n n θ) x_{m}^{(1)} x_{n}^{(2)} (- c o s m θ s i n n θ + s i n m θ c o s n θ) x_{m}^{(2)} x_{n}^{(1)} (- s i n m θ c o s n θ + c o s m θ s i n n θ) x_{m}^{(2)} x_{n}^{(2)} (s i n m θ s i n n θ + c o s m θ c o s n θ) x_{m}^{(1)} x_{n}^{(1)} c o s (m - n) θ + x_{m}^{(1)} x_{n}^{(2)} s i n (m - n) θ - x_{m}^{(2)} x_{n}^{(1)} s i n (m - n) θ + x_{m}^{(2)} x_{n}^{(2)} c o s (m - n) θ (x_{m}^{(1)} c o s (m - n) θ - x_{m}^{(2)} s i n (m - n) θ) x_{n}^{(1)} (x_{m}^{(2)} c o s (m - n) m θ + x_{m}^{(1)} s i n (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)} c o s m θ_{i} - x_{m}^{(i + \frac{d}{2})} s i n m θ_{i} x_{m}^{(i + \frac{d}{2})} c o s m θ_{i} + x_{m}^{(i)} s i n m θ_{i})

30class RotaryPositionalEmbeddings(nn.Module):

#

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

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

#

122        super().__init__()
123
124        self.base = base
125        self.d = d
126        self.cos_cached = None
127        self.sin_cached = None

#

Cache $c o s$ and $s i n$ values

129    def _build_cache(self, x: torch.Tensor):

#

Return if cache is already built

134        if self.cos_cached is not None and x.shape[0] <= self.cos_cached.shape[0]:
135            return

#

Get sequence length

138        seq_len = x.shape[0]

#

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

141        theta = 1. / (self.base ** (torch.arange(0, self.d, 2).float() / self.d)).to(x.device)

#

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

144        seq_idx = torch.arange(seq_len, device=x.device).float().to(x.device)

#

Calculate the product of position index and $θ_{i}$

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

#

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

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

#

Cache them

154        self.cos_cached = idx_theta2.cos()[:, None, None, :]
155        self.sin_cached = idx_theta2.sin()[:, None, None, :]

#

157    def _neg_half(self, x: torch.Tensor):

#

$\frac{d}{2}$

159        d_2 = self.d // 2

#

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

162        return torch.cat([-x[:, :, :, d_2:], x[:, :, :, :d_2]], dim=-1)

#

x is the Tensor at the head of a key or a query with shape [seq_len, batch_size, n_heads, d]

164    def forward(self, x: torch.Tensor):

#

Cache $c o s$ and $s i n$ values

169        self._build_cache(x)

#

Split the features, we can choose to apply rotary embeddings only to a partial set of features.

172        x_rope, x_pass = x[..., :self.d], x[..., self.d:]

#

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

176        neg_half_x = self._neg_half(x_rope)

#

Calculate

(x_{m}^{(i)} c o s m θ_{i} - x_{m}^{(i + \frac{d}{2})} s i n m θ_{i} x_{m}^{(i + \frac{d}{2})} c o s m θ_{i} + x_{m}^{(i)} s i n m θ_{i})

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

188        x_rope = (x_rope * self.cos_cached[:x.shape[0]]) + (neg_half_x * self.sin_cached[:x.shape[0]])

#

191        return torch.cat((x_rope, x_pass), dim=-1)

#

Multi-head attention with rotary positional embeddings

We override multi-head attention from original transformer.

194class RotaryPEMultiHeadAttention(MultiHeadAttention):

#

201    def __init__(self, heads: int, d_model: int, rope_percentage: float = 0.5, dropout_prob: float = 0.0):
202        super().__init__(heads, d_model, dropout_prob)

#

Rotary positional embedding layers

205        d_rope = int(self.d_k * rope_percentage)
206        self.query_rotary_pe = RotaryPositionalEmbeddings(d_rope)
207        self.key_rotary_pe = RotaryPositionalEmbeddings(d_rope)

#

Calculate scores between queries and keys

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

#

Calculate dot-product with RoPE

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

#

Testing RoPE with a simple example

218def _test_rotary():

#

222    x = torch.tensor([[1, 2, 3, 4], [4, 5, 6, 7], [7, 8, 9, 10]], dtype=torch.float)
223    x = x[:, None, None, :]
224    inspect(x)
225
226    rotary_pe = RotaryPositionalEmbeddings(4)
227    inspect(rotary_pe(x))
228
229
230if __name__ == '__main__':
231    _test_rotary()