This is a tutorial/implementation of multi-headed attention from paper Attention Is All You Need in PyTorch. The implementation is inspired from Annotated Transformer.

Here is the training code that uses a basic transformer with MHA for NLP auto-regression.

```
20import math
21from typing import Optional
22
23import torch
24from torch import nn as nn
25
26from labml import tracker
27from labml_helpers.module import Module
```

This module does a linear transformation and splits the vector into given
number of heads for multi-head attention.
This is used to transform **key**, **query**, and **value** vectors.

`30class PrepareForMultiHeadAttention(Module):`

```
41 def __init__(self, d_model: int, heads: int, d_k: int, bias: bool):
42 super().__init__()
```

Linear layer for linear transform

`44 self.linear = nn.Linear(d_model, heads * d_k, bias=bias)`

Number of heads

`46 self.heads = heads`

Number of dimensions in vectors in each head

`48 self.d_k = d_k`

`50 def forward(self, x: torch.Tensor):`

Input has shape `[seq_len, batch_size, d_model]`

or `[batch_size, d_model]`

.
We apply the linear transformation to the last dimension and split that into
the heads.

`54 head_shape = x.shape[:-1]`

Linear transform

`57 x = self.linear(x)`

Split last dimension into heads

`60 x = x.view(*head_shape, self.heads, self.d_k)`

Output has shape `[seq_len, batch_size, heads, d_k]`

or `[batch_size, d_model]`

`63 return x`

This computes scaled multi-headed attention for given `query`

, `key`

and `value`

vectors.

In simple terms, it finds keys that matches the query, and gets the values of those keys.

It uses dot-product of query and key as the indicator of how matching they are. Before taking the $softmax$ the dot-products are scaled by $\frac{1}{\sqrt{d_k}}$. This is done to avoid large dot-product values causing softmax to give very small gradients when $d_k$ is large.

Softmax is calculated along the axis of of the sequence (or time).

`66class MultiHeadAttention(Module):`

`heads`

is the number of heads.`d_model`

is the number of features in the`query`

,`key`

and`value`

vectors.

`87 def __init__(self, heads: int, d_model: int, dropout_prob: float = 0.1, bias: bool = True):`

`93 super().__init__()`

Number of features per head

`96 self.d_k = d_model // heads`

Number of heads

`98 self.heads = heads`

These transform the `query`

, `key`

and `value`

vectors for multi-headed attention.

```
101 self.query = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias=bias)
102 self.key = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias=bias)
103 self.value = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias=True)
```

Softmax for attention along the time dimension of `key`

`106 self.softmax = nn.Softmax(dim=1)`

Output layer

`109 self.output = nn.Linear(d_model, d_model)`

Dropout

`111 self.dropout = nn.Dropout(dropout_prob)`

Scaling factor before the softmax

`113 self.scale = 1 / math.sqrt(self.d_k)`

We store attentions so that it can be used for logging, or other computations if needed

`116 self.attn = None`

This method can be overridden for other variations like relative attention.

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

Calculate $Q K^\top$ or $S_{ijbh} = \sum_d Q_{ibhd} K_{jbhd}$

`126 return torch.einsum('ibhd,jbhd->ijbh', query, key)`

`query`

, `key`

and `value`

are the tensors that store
collection of *query*, *key* and *value* vectors.
They have shape `[seq_len, batch_size, d_model]`

.

`mask`

has shape `[seq_len, seq_len, batch_size]`

and
`mask[i, j, b]`

indicates whether for batch `b`

,
query at position `i`

has access to key-value at position `j`

.

```
128 def forward(self, *,
129 query: torch.Tensor,
130 key: torch.Tensor,
131 value: torch.Tensor,
132 mask: Optional[torch.Tensor] = None):
```

`query`

, `key`

and `value`

have shape `[seq_len, batch_size, d_model]`

```
144 seq_len, batch_size, _ = query.shape
145
146 if mask is not None:
```

`mask`

has shape `[seq_len_q, seq_len_k, batch_size]`

,
where first dimension is the query dimension.
If the query dimension is equal to $1$ it will be broadcasted.

```
150 assert mask.shape[0] == 1 or mask.shape[0] == query.shape[0]
151 assert mask.shape[1] == key.shape[0]
152 assert mask.shape[2] == 1 or mask.shape[2] == query.shape[1]
```

Same mask applied to all heads.

`155 mask = mask.unsqueeze(-1)`

Prepare `query`

, `key`

and `value`

for attention computation.
These will then have shape `[seq_len, batch_size, heads, d_k]`

.

```
159 query = self.query(query)
160 key = self.key(key)
161 value = self.value(value)
```

Compute attention scores $Q K^\top$.
This gives a tensor of shape `[seq_len, seq_len, batch_size, heads]`

.

`165 scores = self.get_scores(query, key)`

Scale scores $\frac{Q K^\top}{\sqrt{d_k}}$

`168 scores *= self.scale`

Apply mask

```
171 if mask is not None:
172 scores = scores.masked_fill(mask == 0, float('-inf'))
```

$softmax$ attention along the key sequence dimension $\underset{seq}{softmax}\Bigg(\frac{Q K^\top}{\sqrt{d_k}}\Bigg)$

`176 attn = self.softmax(scores)`

Save attentions if debugging

`179 tracker.debug('attn', attn)`

Apply dropout

`182 attn = self.dropout(attn)`

Multiply by values

`186 x = torch.einsum("ijbh,jbhd->ibhd", attn, value)`

Save attentions for any other calculations

`189 self.attn = attn.detach()`

Concatenate multiple heads

`192 x = x.reshape(seq_len, batch_size, -1)`

Output layer

`195 return self.output(x)`