Rotary Place Embeddings (RoPE) is a method for encoding token positions in a sequence. It’s extensively utilized in many fashions and works properly for normal context lengths. Nonetheless, it requires adaptation for longer contexts. On this article, you’ll learn the way RoPE is tailored for lengthy context size.
Let’s get began.
Rotary Place Embeddings for Lengthy Context Size
Photograph by Nastya Dulhiier. Some rights reserved.
Overview
This text is split into two elements; they’re:
Easy RoPE
RoPE for Lengthy Context Size
Easy RoPE
In comparison with the sinusoidal place embeddings within the unique Transformer paper, RoPE mutates the enter tensor utilizing a rotation matrix:
$$
start{aligned}
X_{n,i} &= X_{n,i} cos(ntheta_i) – X_{n,frac{d}{2}+i} sin(ntheta_i)
X_{n,frac{d}{2}+i} &= X_{n,i} sin(ntheta_i) + X_{n,frac{d}{2}+i} cos(ntheta_i)
finish{aligned}
$$
the place $X_{n,i}$ is the $i$-th factor of the vector on the $n$-th place of the sequence of tensor $X$. The size of every vector (also called the hidden dimension or the mannequin dimension) is $d$. The amount $theta_i$ is the frequency of the $i$-th factor of the vector. It’s computed as:
$$
theta_i = frac{1}{N^{2i/d}}
$$
A easy implementation of RoPE seems like this:
import torch
import torch.nn as nn
def rotate_half(x: torch.Tensor) -> torch.Tensor:
“””Rotates half the hidden dims of the enter.
It is a helper operate for rotary place embeddings (RoPE).
For a tensor of form (…, d), it returns a tensor the place the final
d/2 dimensions are rotated by swapping and negating.
Args:
x: Enter tensor of form (…, d)
Returns:
Tensor of similar form with rotated final dimension
“””
x1, x2 = x.chunk(2, dim=-1)
return torch.cat((-x2, x1), dim=-1) # Concatenate with rotation
class RotaryPositionEncoding(nn.Module):
“””Rotary place encoding.”””
def __init__(self, dim: int, max_position_embeddings: int) -> None:
“””Initialize the RotaryPositionEncoding module
Args:
dim: The hidden dimension of the enter tensor to which RoPE is utilized
max_position_embeddings: The utmost sequence size of the enter tensor
“””
tremendous().__init__()
self.dim = dim
self.max_position_embeddings = max_position_embeddings
# compute a matrix of ntheta_i
N = 10_000.0
inv_freq = 1.0 / (N ** (torch.arange(0, dim, 2).float() / dim))
inv_freq = torch.cat((inv_freq, inv_freq), dim=-1)
place = torch.arange(max_position_embeddings).float()
sinusoid_inp = torch.outer(place, inv_freq)
# save cosine and sine matrices as buffers
self.register_buffer(“cos”, sinusoid_inp.cos())
self.register_buffer(“sin”, sinusoid_inp.sin())
def ahead(self, x: torch.Tensor) -> torch.Tensor:
“””Apply RoPE to tensor x
Args:
x: Enter tensor of form (batch_size, seq_length, num_heads, head_dim)
Returns:
Output tensor of form (batch_size, seq_length, num_heads, head_dim)
“””
batch_size, seq_len, num_heads, head_dim = x.form
dtype = x.dtype
# remodel the cosine and sine matrices to 4D tensor and the identical dtype as x
cos = self.cos.to(dtype)[:seq_len].view(1, seq_len, 1, -1)
sin = self.sin.to(dtype)[:seq_len].view(1, seq_len, 1, -1)
# apply RoPE to x
output = (x * cos) + (rotate_half(x) * sin)
return output
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import torch
import torch.nn as nn
def rotate_half(x: torch.Tensor) -> torch.Tensor:
“”“Rotates half the hidden dims of the enter.
It is a helper operate for rotary place embeddings (RoPE).
For a tensor of form (…, d), it returns a tensor the place the final
d/2 dimensions are rotated by swapping and negating.
Args:
x: Enter tensor of form (…, d)
Returns:
Tensor of similar form with rotated final dimension
““”
x1, x2 = x.chunk(2, dim=–1)
return torch.cat((–x2, x1), dim=–1) # Concatenate with rotation
class RotaryPositionEncoding(nn.Module):
“”“Rotary place encoding.”“”
def __init__(self, dim: int, max_position_embeddings: int) -> None:
“”“Initialize the RotaryPositionEncoding module
Args:
dim: The hidden dimension of the enter tensor to which RoPE is utilized
max_position_embeddings: The utmost sequence size of the enter tensor
““”
tremendous().__init__()
self.dim = dim
self.max_position_embeddings = max_position_embeddings
# compute a matrix of ntheta_i
N = 10_000.0
inv_freq = 1.0 / (N ** (torch.arange(0, dim, 2).float() / dim))
inv_freq = torch.cat((inv_freq, inv_freq), dim=–1)
place = torch.arange(max_position_embeddings).float()
sinusoid_inp = torch.outer(place, inv_freq)
# save cosine and sine matrices as buffers
self.register_buffer(“cos”, sinusoid_inp.cos())
self.register_buffer(“sin”, sinusoid_inp.sin())
def ahead(self, x: torch.Tensor) -> torch.Tensor:
“”“Apply RoPE to tensor x
Args:
x: Enter tensor of form (batch_size, seq_length, num_heads, head_dim)
Returns:
Output tensor of form (batch_size, seq_length, num_heads, head_dim)
““”
batch_size, seq_len, num_heads, head_dim = x.form
dtype = x.dtype
# remodel the cosine and sine matrices to 4D tensor and the identical dtype as x
cos = self.cos.to(dtype)[:seq_len].view(1, seq_len, 1, –1)
sin = self.sin.to(dtype)[:seq_len].view(1, seq_len, 1, –1)
# apply RoPE to x
output = (x * cos) + (rotate_half(x) * sin)
return output
The code above defines a tensor inv_freq because the inverse frequency of the RoPE, equivalent to the frequency time period $theta_i$ within the method. It’s referred to as inverse frequency within the RoPE literature as a result of it’s inversely proportional to the wavelength (i.e., the utmost distance) that RoPE can seize.
If you multiply two vectors from positions $p$ and $q$, as you’ll do within the scaled-dot product consideration, you discover that the end result relies on the relative place $p-q$ because of the trigonometric identities:
$$
start{aligned}
cos(a – b) = cos(a) cos(b) + sin(a) sin(b)
sin(a – b) = sin(a) cos(b) – cos(a) sin(b)
finish{aligned}
$$
In language fashions, relative place usually issues greater than absolute place. Due to this fact, RoPE is usually preferable to the unique sinusoidal positional embeddings.
RoPE for Lengthy Context Size
The features $sin kx$ and $cos kx$ are periodic with interval $2pi/ok$. In RoPE, the time period $theta_i$ known as the frequency time period as a result of it determines the periodicity. In a language mannequin, the high-frequency phrases are essential as a result of they assist perceive close by phrases in a sentence. The low-frequency phrases, nevertheless, are helpful for understanding context that spans throughout a number of sentences.
Due to this fact, while you design a mannequin with a protracted context size, you need it to carry out properly for brief sentences since they’re extra widespread, however you additionally need it to deal with lengthy contexts that your mannequin ought to help. You don’t want RoPE to deal with each sequence size equally.
The technique is to reallocate the RoPE scaling funds: apply a scaling issue to enhance long-range stability (at low frequencies of sine and cosine) whereas avoiding scaling when native place info is essential (at excessive frequencies of sine and cosine).
In Llama variations 1 and a pair of, RoPE is carried out with a most size of 4096, much like the earlier part. In Llama 3.1, the mannequin’s context size is expanded to 131K tokens, whereas RoPE is computed with a base size of 8192. The implementation is as follows:
import torch
import torch.nn as nn
import math
def rotate_half(x: Tensor) -> Tensor:
“””Rotates half the hidden dims of the enter.
It is a helper operate for rotary place embeddings (RoPE).
For a tensor of form (…, d), it returns a tensor the place the final
d/2 dimensions are rotated by swapping and negating.
Args:
x: Enter tensor of form (…, d)
Returns:
Tensor of similar form with rotated final dimension
“””
x1, x2 = x.chunk(2, dim=-1)
return torch.cat((-x2, x1), dim=-1) # Concatenate with rotation
class RotaryPositionEncoding(nn.Module):
“””Rotary place encoding.”””
def __init__(self, dim: int, max_position_embeddings: int, base_length: int = 8192) -> None:
“””Initialize the RotaryPositionEncoding module
Args:
dim: The hidden dimension of the enter tensor to which RoPE is utilized
max_position_embeddings: The utmost sequence size of the enter tensor
base_length: The bottom size of the RoPE
“””
tremendous().__init__()
self.dim = dim
self.max_position_embeddings = max_position_embeddings
# compute a matrix of ntheta_i
N = 10_000.0
scale_factor = 8.0
low_factor, high_factor = 1.0, 4.0
base_length = 8192
# Compute the inverse frequency based mostly on the usual RoPE method
inv_freq = 1.0 / (N ** (torch.arange(0, dim, 2).float().to(“cuda”) / dim))
# Compute the modified inverse frequency
# scaled if freq too low, orig if freq too excessive, smoothed if in between
wavelen = 2 * math.pi / inv_freq
max_wavelen = base_length / low_factor
min_wavelen = base_length / high_factor
smooth_factor = (base_length / wavelen – low_factor) / (high_factor – low_factor)
smoothed = (1 – smooth_factor) * inv_freq / scale_factor + smooth_factor * inv_freq
inv_freq = torch.the place(wavelen > max_wavelen, inv_freq / scale_factor,
torch.the place(wavelen < min_wavelen, inv_freq,
smoothed))
# multiply with sequence size
inv_freq = torch.cat((inv_freq, inv_freq), dim=-1)
place = torch.arange(max_position_embeddings).float()
sinusoid_inp = torch.outer(place, inv_freq)
# save cosine and sine matrices as buffers
self.register_buffer(“cos”, sinusoid_inp.cos())
self.register_buffer(“sin”, sinusoid_inp.sin())
def ahead(self, x: Tensor) -> Tensor:
“””Apply RoPE to tensor x
Args:
x: Enter tensor of form (batch_size, seq_length, num_heads, head_dim)
Returns:
Output tensor of form (batch_size, seq_length, num_heads, head_dim)
“””
batch_size, seq_len, num_heads, head_dim = x.form
dtype = x.dtype
# remodel the cosine and sine matrices to 4D tensor and the identical dtype as x
cos = self.cos.to(dtype)[:seq_len].view(1, seq_len, 1, -1)
sin = self.sin.to(dtype)[:seq_len].view(1, seq_len, 1, -1)
# apply RoPE to x
output = (x * cos) + (rotate_half(x) * sin)
return output
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import torch
import torch.nn as nn
import math
def rotate_half(x: Tensor) -> Tensor:
“”“Rotates half the hidden dims of the enter.
It is a helper operate for rotary place embeddings (RoPE).
For a tensor of form (…, d), it returns a tensor the place the final
d/2 dimensions are rotated by swapping and negating.
Args:
x: Enter tensor of form (…, d)
Returns:
Tensor of similar form with rotated final dimension
““”
x1, x2 = x.chunk(2, dim=–1)
return torch.cat((–x2, x1), dim=–1) # Concatenate with rotation
class RotaryPositionEncoding(nn.Module):
“”“Rotary place encoding.”“”
def __init__(self, dim: int, max_position_embeddings: int, base_length: int = 8192) -> None:
“”“Initialize the RotaryPositionEncoding module
Args:
dim: The hidden dimension of the enter tensor to which RoPE is utilized
max_position_embeddings: The utmost sequence size of the enter tensor
base_length: The bottom size of the RoPE
““”
tremendous().__init__()
self.dim = dim
self.max_position_embeddings = max_position_embeddings
# compute a matrix of ntheta_i
N = 10_000.0
scale_factor = 8.0
low_factor, high_factor = 1.0, 4.0
base_length = 8192
# Compute the inverse frequency based mostly on the usual RoPE method
inv_freq = 1.0 / (N ** (torch.arange(0, dim, 2).float().to(“cuda”) / dim))
# Compute the modified inverse frequency
# scaled if freq too low, orig if freq too excessive, smoothed if in between
wavelen = 2 * math.pi / inv_freq
max_wavelen = base_length / low_factor
min_wavelen = base_length / high_factor
smooth_factor = (base_length / wavelen – low_factor) / (high_factor – low_factor)
smoothed = (1 – smooth_factor) * inv_freq / scale_factor + smooth_factor * inv_freq
inv_freq = torch.the place(wavelen > max_wavelen, inv_freq / scale_factor,
torch.the place(wavelen < min_wavelen, inv_freq,
smoothed))
# multiply with sequence size
inv_freq = torch.cat((inv_freq, inv_freq), dim=–1)
place = torch.arange(max_position_embeddings).float()
sinusoid_inp = torch.outer(place, inv_freq)
# save cosine and sine matrices as buffers
self.register_buffer(“cos”, sinusoid_inp.cos())
self.register_buffer(“sin”, sinusoid_inp.sin())
def ahead(self, x: Tensor) -> Tensor:
“”“Apply RoPE to tensor x
Args:
x: Enter tensor of form (batch_size, seq_length, num_heads, head_dim)
Returns:
Output tensor of form (batch_size, seq_length, num_heads, head_dim)
““”
batch_size, seq_len, num_heads, head_dim = x.form
dtype = x.dtype
# remodel the cosine and sine matrices to 4D tensor and the identical dtype as x
cos = self.cos.to(dtype)[:seq_len].view(1, seq_len, 1, –1)
sin = self.sin.to(dtype)[:seq_len].view(1, seq_len, 1, –1)
# apply RoPE to x
output = (x * cos) + (rotate_half(x) * sin)
return output
The constructor of the RotaryPositionEncoding class makes use of a extra refined algorithm to compute the inv_freq tensor. The concept is to compute a wavelength for every frequency part, representing the utmost distance between two tokens that the corresponding RoPE part can seize. If the wavelength is simply too quick (or the frequency is simply too excessive), the frequency stays unchanged. Nonetheless, if the wavelength is simply too lengthy, the frequency is lowered by the scale_factor, successfully growing the utmost distance that the RoPE part can seize. To make sure stability, frequency elements between the low- and high-frequency thresholds are interpolated easily.
As an instance the impact of scaling, you may plot the ensuing inverse frequency with Matplotlib:
import matplotlib.pyplot as plt
import torch
import math
N = 10_000.0
dim = 256
scale_factor = 8.0
low_factor, high_factor = 1.0, 4.0
base_length = 8192
# Compute the inverse frequency based mostly on the usual RoPE method
inv_freq = 1.0 / (N ** (torch.arange(0, dim, 2).float() / dim))
# Compute the modified inverse frequency
# scaled if freq too low, orig if freq too excessive, smoothed if in between
wavelen = 2 * math.pi / inv_freq
max_wavelen = base_length / low_factor
min_wavelen = base_length / high_factor
smooth_factor = (base_length / wavelen – low_factor) / (high_factor – low_factor)
smoothed = (1 – smooth_factor) * inv_freq / scale_factor + smooth_factor * inv_freq
new_freq = torch.the place(wavelen > max_wavelen, inv_freq / scale_factor,
torch.the place(wavelen < min_wavelen, inv_freq,
smoothed))
# Plot the ensuing inverse frequency
plt.plot(inv_freq, label=”Unique”)
plt.plot(inv_freq / scale_factor, label=”Scaled”)
plt.plot(new_freq, label=”New Frequency”)
plt.grid(True)
plt.yscale(‘log’)
plt.xlabel(‘Dimension’)
plt.ylabel(‘Inverse Frequency’)
plt.legend()
plt.present()
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import matplotlib.pyplot as plt
import torch
import math
N = 10_000.0
dim = 256
scale_factor = 8.0
low_factor, high_factor = 1.0, 4.0
base_length = 8192
# Compute the inverse frequency based mostly on the usual RoPE method
inv_freq = 1.0 / (N ** (torch.arange(0, dim, 2).float() / dim))
# Compute the modified inverse frequency
# scaled if freq too low, orig if freq too excessive, smoothed if in between
wavelen = 2 * math.pi / inv_freq
max_wavelen = base_length / low_factor
min_wavelen = base_length / high_factor
smooth_factor = (base_length / wavelen – low_factor) / (high_factor – low_factor)
smoothed = (1 – smooth_factor) * inv_freq / scale_factor + smooth_factor * inv_freq
new_freq = torch.the place(wavelen > max_wavelen, inv_freq / scale_factor,
torch.the place(wavelen < min_wavelen, inv_freq,
smoothed))
# Plot the ensuing inverse frequency
plt.plot(inv_freq, label=‘Unique’)
plt.plot(inv_freq / scale_factor, label=‘Scaled’)
plt.plot(new_freq, label=‘New Frequency’)
plt.grid(True)
plt.yscale(‘log’)
plt.xlabel(‘Dimension’)
plt.ylabel(‘Inverse Frequency’)
plt.legend()
plt.present()
The plot is proven beneath:
Plot of inverse frequency earlier than and after RoPE scaling
You’ll be able to see that the unique RoPE frequency is preserved till the wavelength is roughly 2000 tokens (at an inverse frequency of round 0.003), after which it’s step by step scaled. The wavelength is scaled by an element of 8 when it exceeds 9000 tokens (i.e., the inverse frequency is beneath 6e-4).
From the x-axis of the plot, you may see that round 60% of the size seize dependencies inside 2000 tokens, whereas the remaining seize distances as much as 60000 tokens ($2pi N$ precisely; a bigger $N$ allows the mannequin to help longer context lengths).
This successfully gives larger decision for RoPE at quick distances and decrease decision at lengthy distances, in keeping with how language fashions ought to behave when understanding language.
Additional Studying
Under are some sources that you could be discover helpful:
Abstract
On this article, you realized how RoPE is tailored for lengthy context size. Particularly, you realized how Llama 3 helps longer context lengths by scaling the RoPE frequency on the low-frequency finish.
