RoPE (Rotary positional embeddings) explained: The positional workhorse of modern LLMs

DeepLearning Hero2023-08-02

deep learning#llms#llm#chatgpt#transformer models#machine learning

10K views|1 years ago

💫 Short Summary

Rope Embeddings: A Detailed Overview" discusses the limitations of sinusoidal positional embeddings in Transformer models and introduces rope embeddings as a solution. Rope embeddings involve rotating query vectors based on their position in the sequence, addressing the chaotic behavior of sinusoidal embeddings and providing more stability and interpretability in the model. The video explains the mathematical principles and benefits of rope embeddings compared to sinusoidal embeddings. Rope embeddings are a technique used in large language models to embed the position of tokens in a sequence. By breaking the rotation matrix into blocks and rotating each block independently, rope embeddings provide a more predictable and resilient alternative to sinusoidal embeddings, especially when dealing with sequence lengths beyond the training length. This makes rope embeddings better at generalizing and adapting to different sequence lengths. While there are other techniques like Alibi, rope embeddings are currently the most popular positional embedding for large language models.

✨ Highlights

📊 Transcript

✦

Transformer models rely on positional embeddings to maintain the sequential nature of the data.

00:00

The original Transformer model introduced sine positional embeddings.

Most modern Transformer models use rotary positional embeddings (rope embeddings) for their performance and generalizability.

✦

The self-attention layer in Transformer models revolves around queries, keys, and values, with a focus on encoding token and positional similarity.

01:06

Queries and keys are used to compute the attention matrix.

Token embeddings' similarity is captured by the radial component, while the positional similarity is influenced by the angles of the query and key vectors.

✦

Rope embeddings are a technique used to rotate vectors in a d-dimensional space, providing a more resilient and predictable way of embedding positional information.

10:00

Rope embeddings break the input matrix into blocks of two, with each block associated with its own rotation matrix.

Each rotation block has its own Theta constant, which depends on the index along the hidden dimension.

Multiplying a d-dimensional vector by a block diagonal rotation matrix results in each block in the input matrix being rotated by a unique angle.

Rope embeddings are demonstrated to be more resilient to out-of-time predictions and generally perform better on log likelihood loss compared to sinusoidal embeddings.

✦

Rope embeddings involve rotating q and k by angles dependent on their position in the sequence, allowing them to adapt better to sequence lengths beyond the training length.

12:00

The equations for rope embeddings involve introducing Theta parameters for q and k, arranging the thetas to be close together, and rotating q and k by the corresponding angle.

Rope embeddings are designed to adapt better to sequence lengths beyond the training length by behaving in a predictable manner as the position changes.

✦

Other techniques, such as Alibi, claim to be superior to rope embeddings, but rope is currently the most popular positional embedding for large language models (LLMs).

13:00

Alibi is mentioned as another technique that claims to be even more superior than rope embeddings.

Rope embeddings are described as the most commoditized positional embedding for LLMs at the moment.

00:00 Transformer models rely on positional

00:03 embeddings to help them uphold the

00:06 sequential nature of the data when they

00:09 process it

00:11 the original Transformer model

00:13 introduced sine soil positional

00:15 embeddings where each hidden Dimension

00:18 is modeled via a sine soil curve

00:22 since then different solutions have come

00:25 and gone

00:26 in this modern era of llms most models

00:31 have found one type of embeddings to be

00:33 performant and generalizable that is

00:37 rope embeddings rope stands for rotary

00:41 positional embeddings and as the name

00:44 suggests its primary functionality is

00:47 rotating query and key vectors based on

00:51 their position in the sequence

00:54 as we go through the video we will see

00:56 why this is important and things will

00:59 fall into place

01:01 so with no pun intended let's get the

01:05 wheel spinning

01:06 the flagship computation of a transform

01:09 model is the self-attention layer

01:12 computation which revolves around three

01:15 ingredients queries keys and values out

01:20 of these queries and keys are used to

01:23 compute the attention Matrix if you'd

01:26 like to learn about the end-to-end

01:28 computation in Transformer models check

01:31 out my video on the Transformer model

01:35 each position has a query and a key in

01:39 our case let's assume a dimensionality

01:42 of two

01:43 so each query or key is a 2d Vector when

01:48 Computing the attention Matrix we wanted

01:51 to encode two characteristics

01:54 tokens that have similar token

01:56 embeddings should have a higher score

01:59 which is a no-brainer for example if the

02:02 two tokens are cat and dog it makes

02:06 sense to give them a higher score as

02:08 they appear in very similar contexts

02:12 next the score should be higher for

02:15 words that are close together

02:17 again quite sensible

02:20 the further part two words are the less

02:23 likely for them to be related you can

02:25 see the effect of dispositional behavior

02:28 from the bottom attention Matrix where

02:31 the intensity is highest along the

02:33 diagonal the diagonal is where the query

02:36 and the key share the same position and

02:40 the score declines as you go in the

02:42 orthogonal direction to the diagonal

02:44 each entry in this attention Matrix is

02:48 the dot product of sub Q M and K N

02:52 since we are working with 2D vectors we

02:55 can visualize the query and the key on a

02:58 2d plane

03:00 let's then switch to the polar

03:02 coordinates now here's one way to think

03:05 about the relationships between the Q

03:08 and K vectors and the two versions of

03:11 the attention Matrix that needs to be

03:13 captured the magnitude or the radial

03:16 component of the vector contributes to

03:19 the token's similarity Matrix that is

03:22 the similarity between radial components

03:25 denoted by Norm of Q dot Norm of K

03:29 corresponds to the Token embedding

03:31 similarity at M comma and position the

03:35 angles of Q and K Theta M and Theta n

03:39 contributes to the positional similarity

03:42 at M comma and position here's something

03:45 important about this design

03:48 the angle component only depends on the

03:51 position of the vector in this equals

03:54 and is independent of the actual token

03:57 embeddings

03:58 on the other hand the token similarity

04:01 is captured by the radial component of

04:03 the vector and is independent of the

04:06 angle it creates

04:08 remember this as this school of thought

04:11 plays the intuition for rope embeddings

04:16 next we will discuss how we can design

04:19 functions to capture these properties

04:21 during the attention computation

04:24 we are not so much worried about the

04:26 token similarity as we are about the

04:30 positional similarity

04:31 so for the rest of this talk we will

04:34 assume same token everywhere

04:37 so the token similarity attention Matrix

04:40 has the same value everywhere now the

04:43 only differentiator is how we design the

04:46 positional embeddings

04:48 before we type headfirst to the Rope

04:51 computation let's motivate ourselves by

04:54 learning what's wrong with the sinusoid

04:57 embeddings here's what the sinusoidal

04:59 membrings look like for 2D query key

05:02 vectors all we do is push these dots

05:06 along the curves and take measurements

05:08 at each position

05:10 so at position 0 it's 0 comma one and

05:14 position one it's 0.88 comma 0.61 and so

05:21 on

05:22 if you compute the actual values at m

05:25 equals 1 2 3 and so on you will get just

05:30 slightly different results due to

05:33 Precision issues in our computation this

05:36 looks like a sensible thing for a

05:37 positional embedding the values seem to

05:40 gracefully go up and down based on their

05:43 position

05:44 so what's the issue once the position

05:47 embeddings are computed they are added

05:50 to X so let's see how this affects an

05:53 example Vector for Simplicity let's pick

05:57 one comma one ah here's what's wrong the

06:01 conceptually it looks good not so much

06:04 in practice the vector seems to move a

06:07 bit chaotically as the position changes

06:11 it's difficult to see a pattern in the

06:14 way it moves both the magnitude and the

06:17 angle changes significantly from each

06:19 positional shift

06:21 also remember this is only two

06:24 dimensions

06:25 standard Transformer models have

06:27 hundreds of them and things are probably

06:29 getting unimaginably complex when you

06:32 scale up to those numbers here's some

06:35 organizational view of what might be

06:37 happening in this model

06:39 first you give the model some data

06:42 the talk appears at position 4 bark at

06:47 six and per at 20. then the model asks

06:51 what are these positions

06:54 then you tell the model that design

06:56 using specialized sinusoids to provide a

06:59 vector at each position

07:01 but all that the model sees is a table

07:04 of numbers

07:07 as we saw it's hard to see the pattern

07:10 through those numbers the model does the

07:12 best it can given the situation it's

07:15 going to memorize these positions and

07:18 learning the pattern here's an analogy

07:20 imagine you have an exam coming up and

07:23 you haven't studied for

07:24 so the night before the exam you hit the

07:27 boots and try to study the material but

07:30 concepts are tough to learn and you're

07:33 running out of time so what do you do

07:35 you try to memorize the answers instead

07:39 the Transformer model opts into

07:42 something similar but unlike you the

07:45 Transformer model has its reasons to

07:48 memorize answers first reason these

07:51 models have billions of parameters and

07:53 as you know the more parameters the more

07:57 overfitting or memorization takes place

08:00 second reason when being trained on

08:03 trillions of tokens it probably sees

08:06 most of the words at all possible

08:09 positions in the sequence so there's

08:11 hardly any reason for the model to go

08:13 above and beyond to learn patterns

08:16 continuing on our conversation you train

08:19 the model this way using 20 positions

08:22 and then at the inference time you ask

08:25 about position 21 this will confuse the

08:29 model so much because it has no notion

08:31 of a position 21. the model asks what is

08:35 position 21

08:37 and output something erratic

08:39 research has already confirmed this

08:41 Behavior the perplexity of the models

08:44 trained using sinusoidals explored past

08:48 the training length now let's learn how

08:51 Rob fixes this rope is built on a simple

08:55 concept instead of trying to match the

08:58 token and position embeddings into a

09:00 paste by adding the two together

09:03 and Computing Q let's find a

09:06 transformation for Q and its position M

09:09 that gives a new vector

09:11 the idea they came up with is rotating Q

09:14 by a factor of M Theta where Theta is a

09:18 constant

09:20 it depends on the index of the Hidden

09:22 Dimension d

09:24 since we only have D equals 2 Theta will

09:28 be 1. if you had a hidden dimensionality

09:31 greater than 2 you'd have several thetas

09:34 causing each pair of Dimensions to

09:37 rotate at different speeds we'll chat

09:40 about D greater than 2 K's later that's

09:43 what all these equations on the left do

09:46 rotating Q by m Theta

09:49 the uppercase or denotes the standard

09:52 rotation Matrix

09:53 when you multiply a matrix or a vector

09:56 by this you're simply rotating that

09:58 Matrix or vector by some angle here's

10:01 what you get for different positions of

10:03 Q when you use this equation

10:06 you can already see how well-behaved Q

10:09 is with this new embedding Q rotates

10:12 around counterclockwise by some Theta

10:14 every time you shift the position to the

10:17 right but one question Still Remains

10:20 how does it scale to D greater than two

10:24 since we have an r that can only rotate

10:27 a vector on a 2d plane

10:30 for this they use a trick

10:32 they break Q or k into blocks of two

10:36 so for D equals four we have two blocks

10:39 and more generally for a d dimensional

10:43 Matrix you'd have D divided by two

10:46 blocks then we can repeat the same

10:49 operations we did for D equals two for

10:52 each block independently

10:54 specifically we convert R to a block

10:57 diagonal rotation Matrix

11:00 each block is associated with one block

11:03 of Q or k each rotation block will have

11:06 its own Theta constant and Theta depends

11:10 on the index along the hidden Dimension

11:12 multiplying a d dimensional vector by a

11:16 block diagonal R means each block in Q

11:19 or k will be rotated by the angle unique

11:22 to that block in r

11:25 to see how this works we will break our

11:27 four-dimensional Vector 2 to the vectors

11:30 and see how the rotation changes

11:33 you can see that the new Vector in green

11:37 moves ever so slightly every time the

11:40 yellow Vector makes a big jump

11:43 now you have the full idea

11:45 each block of qrk will rotate at

11:49 different speeds

11:50 but they rotate in a consistent

11:52 predictable way always in combination it

11:57 gives the positionally embedded version

11:58 of Q or k as you can see in this graph

12:02 rope embeddings are much more resilient

12:05 to Out of Time predictions than

12:07 Transformers with the sinusoidal

12:10 embeddings

12:11 they also generally perform better on

12:13 log likelihood loss

12:15 there's also a paper that explains how

12:18 to increase the context window length of

12:21 an already trained model using

12:23 interpolation techniques but that's a

12:26 story for another day let's also look at

12:29 how the equations are changing from

12:32 sinusoidal positional embeddings to rope

12:35 embeddings first we're gonna ditch the

12:38 positional embeddings which leaves us

12:41 with wqxm transpose Dot wkxn

12:46 then we're going to introduce our Theta

12:49 1 for Q and 1 for k then we can arrange

12:53 this in a way that the r thetas are

12:56 close together and finally we arrive at

12:59 the equation that is in the paper this

13:02 brings us to the end of this video

13:04 though the raw paper is full of

13:06 mathematics conceptually this is all

13:09 that happens rotating q and K by some

13:12 angle which is dependent on their

13:14 position in the sequence as he saw the

13:18 erratic nature in which sinusoidal

13:20 positional embeddings move reduces

13:23 models that aren't able to generalize

13:25 Beyond training sequence length this

13:27 lies the motivation for rope due to rope

13:31 behaving in a predictable manner as the

13:33 position changes it's able to adapt much

13:36 better to sequence lengths Beyond

13:39 training length there's another

13:41 technique called Alibi which claims to

13:43 be even more Superior than rope but

13:46 right now rope is the most commoditized

13:50 positional embedding for llms if you

13:53 enjoyed this video and like to see more

13:55 videos about machine learning and large

13:58 language models subscribe to my channel

14:01 deep learning hero I will see you soon

14:03 with another video

💫 FAQs about This YouTube Video

1. What are rope embeddings and how do they differ from sinusoidal positional embeddings in Transformer models?

Rope embeddings, or rotary positional embeddings, are a type of positional embedding used in Transformer models to help uphold the sequential nature of data. They differ from sinusoidal positional embeddings by focusing on rotating query and key vectors based on their position in the sequence, addressing the issues of chaotic movement and memorization that can occur with sinusoidal embeddings.

2. How do rope embeddings contribute to capturing positional similarity in Transformer models?

Rope embeddings contribute to capturing positional similarity in Transformer models by rotating query and key vectors based on their position in the sequence. This allows for a more stable and well-behaved representation of positional information, addressing the issues of chaotic movement and memorization that can occur with other types of positional embeddings.

3. What is the primary functionality of rope embeddings in Transformer models?

The primary functionality of rope embeddings in Transformer models is to rotate query and key vectors based on their position in the sequence. This helps to capture positional similarity in a more stable and well-behaved manner, addressing the issues of chaotic movement and memorization that can occur with other types of positional embeddings.

4. Why are rope embeddings considered performant and generalizable in Transformer models?

Rope embeddings are considered performant and generalizable in Transformer models due to their ability to effectively capture positional similarity by rotating query and key vectors based on their position in the sequence. This results in a more stable and well-behaved representation of positional information, addressing issues that can occur with other types of positional embeddings such as chaotic movement and memorization.

5. How do rope embeddings address the issues of chaotic movement and memorization in Transformer models?

Rope embeddings address the issues of chaotic movement and memorization in Transformer models by providing a more stable and well-behaved representation of positional information. By rotating query and key vectors based on their position in the sequence, rope embeddings effectively capture positional similarity, offering a solution to the problems associated with other types of positional embeddings.

6. What are rope embeddings in the context of machine learning and large language models?

Rope embeddings are a positional embedding technique that allows for better adaptation to sequence lengths beyond the training length in machine learning and large language models. They behave in a predictable manner as the position changes, making them more resilient to out-of-time predictions and generally perform better on log likelihood loss compared to traditional sinusoidal positional embeddings.

7. How do rope embeddings improve the performance of transformers in machine learning?

Rope embeddings improve the performance of transformers in machine learning by providing better adaptation to sequence lengths beyond the training length. They allow for more resilient predictions and generally perform better on log likelihood loss, making them a preferred choice for large language models.

8. What is the key feature of rope embeddings for machine learning and language models?

The key feature of rope embeddings is their ability to adapt to sequence lengths beyond the training length in machine learning and language models. They behave in a predictable manner as the position changes, making them more resilient to out-of-time predictions and generally perform better on log likelihood loss, thus improving the overall performance of transformers.

9. Why are rope embeddings considered important for machine learning and large language models?

Rope embeddings are considered important for machine learning and large language models because they offer better adaptation to sequence lengths beyond the training length. Their ability to behave in a predictable manner as the position changes makes them more resilient to out-of-time predictions and generally results in better performance on log likelihood loss, which is crucial for the effectiveness of transformers.

10. How do rope embeddings contribute to the resilience of large language models in machine learning?

Rope embeddings contribute to the resilience of large language models in machine learning by providing better adaptation to sequence lengths beyond the training length. They behave in a predictable manner as the position changes, making them more resilient to out-of-time predictions and generally perform better on log likelihood loss, thereby improving the overall performance of transformers.

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