relative attention notes

labml_nn/transformers/feedback/__init__.py

@@ -18,38 +18,17 @@ from labml_nn.transformers.models import FeedForward
 from labml_nn.utils import clone_module_list
 
 
-class PrepareQueryForMultiHeadAttention(Module):
-    """
-    ## Prepare query for multi-head attention
-
-    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.
-    """
-
-    def __init__(self, d_model: int, heads: int, d_k: int, bias: bool):
-        super().__init__()
-        # Linear layer for linear transform
-        self.linear = nn.Linear(d_model, heads * d_k, bias=bias)
-        # Number of heads
-        self.heads = heads
-        # Number of dimensions in vectors in each head
-        self.d_k = d_k
-
-    def __call__(self, x: torch.Tensor):
-        # Input has shape `[seq_len, batch_size, d_model]`
-        batch_size, _ = x.shape
-
-        # Linear transform
-        x = self.linear(x)
-        # Split into heads
-        x = x.view(batch_size, self.heads, self.d_k)
-
-        # Output has shape `[seq_len, batch_size, heads, d_k]`
-        return x
-
-
 class FeedbackAttention(Module):
+    """
+    ## Feedback Attention
+
+    This is very similar to [Relative Multi-Head Attention](../relative_mha.html)
+    but with some modifications.
+
+    📝 Decided not to extend from [Relative Multi-Head Attention](../relative_mha.html)
+     or [Multi-Head Attention](../mha.html) to improve readability.
+    """
+
     def __init__(self, heads: int, d_model: int, dropout_prob: float = 0.1):
         super().__init__()
 
@@ -57,7 +36,7 @@ class FeedbackAttention(Module):
         self.heads = heads
 
         # These transform the `query`, `key` and `value` vectors for multi-headed attention.
-        self.query = PrepareQueryForMultiHeadAttention(d_model, heads, self.d_k, False)
+        self.query = PrepareForMultiHeadAttention(d_model, heads, self.d_k, False)
         self.key = PrepareForMultiHeadAttention(d_model, heads, self.d_k, False)
         self.value = PrepareForMultiHeadAttention(d_model, heads, self.d_k, False)
 
@@ -68,17 +47,60 @@ class FeedbackAttention(Module):
         # Scaling factor before the softmax
         self.scale = 1 / math.sqrt(self.d_k)
 
+        # Softmax for attention along the time dimension of `key`
+        self.softmax = nn.Softmax(dim=0)
+
+        # Number of relative positions
+        self.P = 2 ** 12
+
+        # Relative positional embeddings for key relative to the query.
+        self.key_pos_embeddings = nn.Parameter(torch.zeros((self.P, heads, self.d_k)), requires_grad=True)
+        # Relative positional embedding bias for key relative to the query.
+        self.key_pos_bias = nn.Parameter(torch.zeros((self.P, heads)), requires_grad=True)
+        # Positional embeddings for the query is independent of the position of the query
+        self.query_pos_bias = nn.Parameter(torch.zeros((heads, self.d_k)), requires_grad=True)
+
         # We store attentions so that it can used for logging, or other computations if needed
         self.attn = None
 
-        self.P = 2 ** 12
-
-        self.key_pos_embeddings = nn.Parameter(torch.zeros((self.P, heads, self.d_k)), requires_grad=True)
-        self.key_pos_bias = nn.Parameter(torch.zeros((self.P, heads)), requires_grad=True)
-        self.query_pos_bias = nn.Parameter(torch.zeros((heads, self.d_k)), requires_grad=True)
-        self.softmax = nn.Softmax(dim=0)
-
     def get_scores(self, query: torch.Tensor, key: torch.Tensor):
+        """
+        ### Get relative attention scores
+
+        With absolute attention
+
+        \begin{align}
+        A^{abs}_{j} &= lin_q(\color{cyan}{X^q_i + P_i})^T lin_k(\color{lightgreen}{X^k_j + P_j}) \\
+                      &= \color{cyan}{Q_i^T} \color{lightgreen}{K_j} +
+                         \color{cyan}{Q_i^T} \color{lightgreen}{U_j} +
+                         \color{cyan}{V_i^T} \color{lightgreen}{K_j} +
+                         \color{cyan}{V_i^T} \color{lightgreen}{U_j}
+        \end{align}
+
+        where $\color{cyan}{Q_i}, \color{lightgreen}{K_j}$, are linear transformations of
+         original embeddings $\color{cyan}{X^q_i}, \color{lightgreen}{X^k_j}$
+         and $\color{cyan}{V_i}, \color{lightgreen}{U_j}$ are linear transformations of
+         absolute positional encodings $\color{cyan}{P_i}, \color{lightgreen}{P_j}$.
+
+        They reason out that the attention to a given key should be the same regardless of
+        the position of query. Hence replace $\color{cyan}{V_i^T} \color{lightgreen}{K_j}$
+        with a constant $\color{orange}{v^T} \color{lightgreen}{K_j}$.
+        🤔 May be worthwhile testing without this assumption.
+
+        For the second and third terms relative positional encodings are introduced.
+        So $\color{cyan}{Q_i^T} \color{lightgreen}{U_j}$ is
+        replaced with $\color{cyan}{Q_i^T} \color{orange}{R_{i - j}}$
+        and $\color{cyan}{V_i^T} \color{lightgreen}{U_j}$ with $\color{orange}{S_{i-j}}$.
+
+        \begin{align}
+        A^{rel}_{i,j} &= \color{cyan}{Q_i^T} \color{lightgreen}{K_j} +
+                         \color{cyan}{Q_i^T} \color{orange}{R_{i - j}} +
+                         \color{orange}{v^T} \color{lightgreen}{K_j} +
+                         \color{orange}{S_{i-j}}
+        \end{align}
+        """
+
+        # $\color{orange}{R_{i - j}}$
         key_pos_emb = self.key_pos_embeddings[-key.shape[0]:]
         key_pos_bias = self.key_pos_bias[-key.shape[0]:]
         query_pos_bias = self.query_pos_bias[None, :, :]

labml_nn/transformers/mha.py

@@ -18,10 +18,10 @@ import math
 from typing import Optional
 
 import torch
+from torch import nn as nn
+
 from labml import tracker
 from labml_helpers.module import Module
-from torch import nn as nn
-from torch.nn import functional as F
 
 
 class PrepareForMultiHeadAttention(Module):
@@ -43,29 +43,28 @@ class PrepareForMultiHeadAttention(Module):
         self.d_k = d_k
 
     def __call__(self, x: torch.Tensor):
-        # Input has shape `[seq_len, batch_size, d_model]`
+        # Input has shape `[seq_len, batch_size, d_model]` or `[batch_size, d_model]`.
-        seq_len, batch_size, _ = x.shape
+        # We apply the linear transformation of the last dimension and splits that into
+        # the heads
+        head_shape = x.shape[:-1]
 
         # Linear transform
         x = self.linear(x)
-        # Split into heads
-        x = x.view(seq_len, batch_size, self.heads, self.d_k)
 
-        # Output has shape `[seq_len, batch_size, heads, d_k]`
+        # Split last dimension into heads
+        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]`
         return x
 
 
 class MultiHeadAttention(Module):
-    def __init__(self, heads: int, d_model: int, dropout_prob: float = 0.1, bias: bool = True):
+    r"""
-        """
     ## Multi-Head Attention Module
 
-        * `heads` is the number of heads.
-        * `d_model` is the number of features in the `query`, `key` and `value` vectors.
-
     This computes scaled multi-headed attention for given `query`, `key` and `value` vectors.
 
-        $$\mathop{Attention}(Q, K, V) = \mathop{softmax}\Bigg(\frac{Q K^T}{\sqrt{d_k}}\Bigg)V$$
+    $$\mathop{Attention}(Q, K, V) = \underset{seq}{\mathop{softmax}}\Bigg(\frac{Q K^T}{\sqrt{d_k}}\Bigg)V$$
 
     In simple terms, it finds keys that matches the query, and get the values of
      those keys.
@@ -78,8 +77,17 @@ class MultiHeadAttention(Module):
     Softmax is calculate along the axis of of the sequence (or time).
     """
 
+    def __init__(self, heads: int, d_model: int, dropout_prob: float = 0.1, bias: bool = True):
+        """
+        * `heads` is the number of heads.
+        * `d_model` is the number of features in the `query`, `key` and `value` vectors.
+        """
+
         super().__init__()
+
+        # Number of features per head
         self.d_k = d_model // heads
+        # Number of heads
         self.heads = heads
 
         # These transform the `query`, `key` and `value` vectors for multi-headed attention.
@@ -87,6 +95,9 @@ class MultiHeadAttention(Module):
         self.key = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias)
         self.value = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias)
 
+        # Softmax for attention along the time dimension of `key`
+        self.softmax = nn.Softmax(dim=1)
+
         # Output layer
         self.output = nn.Linear(d_model, d_model)
         # Dropout
@@ -153,7 +164,7 @@ class MultiHeadAttention(Module):
 
         # $softmax$ attention along the key sequence dimension
         # $\underset{seq}{softmax}\Bigg(\frac{Q K^T}{\sqrt{d_k}}\Bigg)$
-        attn = F.softmax(scores, dim=1)
+        attn = self.softmax(scores)
 
         # Save attentions if debugging
         tracker.debug('attn', attn)

labml_nn/transformers/relative_mha.py

@@ -51,56 +51,77 @@ class RelativeMultiHeadAttention(MultiHeadAttention):
         # The linear transformations doesn't need a bias since we take care of it when
         # calculating scores.
         # However having a bias for `value` might make sense.
-        super().__init__(heads, d_model, dropout_prob, False)
+        super().__init__(heads, d_model, dropout_prob, bias=False)
 
+        # Number of relative positions
         self.P = 2 ** 12
 
+        # Relative positional embeddings for key relative to the query.
+        # We need $2P$ embeddings because the keys can be before or after the query.
         self.key_pos_embeddings = nn.Parameter(torch.zeros((self.P * 2, heads, self.d_k)), requires_grad=True)
-        self.query_pos_bias = nn.Parameter(torch.zeros((heads, self.d_k)), requires_grad=True)
+        # Relative positional embedding bias for key relative to the query.
         self.key_pos_bias = nn.Parameter(torch.zeros((self.P * 2, heads)), requires_grad=True)
+        # Positional embeddings for the query is independent of the position of the query
+        self.query_pos_bias = nn.Parameter(torch.zeros((heads, self.d_k)), requires_grad=True)
 
     def get_scores(self, query: torch.Tensor, key: torch.Tensor):
-        """
+        r"""
+        ### Get relative attention scores
+
         With absolute attention
 
         \begin{align}
-        A^{abs}_{i,j} &= lin_q(X^q_i + P_i)^T lin_k(X^k_j + P_j) \\
+        A^{abs}_{j} &= lin_q(\color{cyan}{X^q_i + P_i})^T lin_k(\color{lightgreen}{X^k_j + P_j}) \\
-                      &= Q_i^T K_j + Q_i^T U_j + V_i^T K_j + V_i^T U_j
+                      &= \color{cyan}{Q_i^T} \color{lightgreen}{K_j} +
+                         \color{cyan}{Q_i^T} \color{lightgreen}{U_j} +
+                         \color{cyan}{V_i^T} \color{lightgreen}{K_j} +
+                         \color{cyan}{V_i^T} \color{lightgreen}{U_j}
         \end{align}
 
-        where $Q_i$, $K_j$, $V_i$, and $U_j$ are linear transformations of
+        where $\color{cyan}{Q_i}, \color{lightgreen}{K_j}$, are linear transformations of
-         orginal embeddings and positional encodings.
+         original embeddings $\color{cyan}{X^q_i}, \color{lightgreen}{X^k_j}$
+         and $\color{cyan}{V_i}, \color{lightgreen}{U_j}$ are linear transformations of
+         absolute positional encodings $\color{cyan}{P_i}, \color{lightgreen}{P_j}$.
 
         They reason out that the attention to a given key should be the same regardless of
-        the position of query. Hence replace $V_i^T K_j$ with a constant $v^T K_j$.
+        the position of query. Hence replace $\color{cyan}{V_i^T} \color{lightgreen}{K_j}$
+        with a constant $\color{orange}{v^T} \color{lightgreen}{K_j}$.
         🤔 May be worthwhile testing without this assumption.
 
         For the second and third terms relative positional encodings are introduced.
-        So $Q_i^T U_j$ is replaced with $Q_i^T R_{i - j}$ and $V_i^T U_j$ with $S_{i-j}$.
+        So $\color{cyan}{Q_i^T} \color{lightgreen}{U_j}$ is
+        replaced with $\color{cyan}{Q_i^T} \color{orange}{R_{i - j}}$
+        and $\color{cyan}{V_i^T} \color{lightgreen}{U_j}$ with $\color{orange}{S_{i-j}}$.
 
         \begin{align}
-        A^{rel}_{i,j} &= Q_i^T K_j + Q_i^T R_{i - j} + v^T K_j + S_{i-j}
+        A^{rel}_{i,j} &= \underset{\mathbf{A}}{\color{cyan}{Q_i^T} \color{lightgreen}{K_j}} +
+                         \underset{\mathbf{B}}{\color{cyan}{Q_i^T} \color{orange}{R_{i - j}}} +
+                         \underset{\mathbf{C}}{\color{orange}{v^T} \color{lightgreen}{K_j}} +
+                         \underset{\mathbf{D}}{\color{orange}{S_{i-j}}}
         \end{align}
-
         """
 
-        # $R_{i-j}$ pre-shift
+        # $\color{orange}{R_k}$
         key_pos_emb = self.key_pos_embeddings[self.P - query.shape[0]:self.P + key.shape[0]]
-        # $S_{i-j}$ pre-shift
+        # $\color{orange}{S_k}$
         key_pos_bias = self.key_pos_bias[self.P - query.shape[0]:self.P + key.shape[0]]
-        # $v^T$
+        # $\color{orange}{v^T}$
         query_pos_bias = self.query_pos_bias[None, None, :, :]
 
-        # $Q_i^T K_j + v^T K_j$
+        # ${(\mathbf{A} + \mathbf{C})}_{i,j} = \color{cyan}{Q_i^T} \color{lightgreen}{K_j} +
+        # \color{orange}{v^T} \color{lightgreen}{K_j}$
         ac = torch.einsum('ibhd,jbhd->ijbh', query + query_pos_bias, key)
-        # $Q_i^T R_{i - j}$ pre-shift
+        # $\mathbf{B'}_{i,k} = \color{cyan}{Q_i^T} \color{orange}{R_k}$
         b = torch.einsum('ibhd,jhd->ijbh', query, key_pos_emb)
-        # $S_{i-j}$ pre-shift
+        # $\mathbf{D'}_{i,k} = \color{orange}{S_k}$
         d = key_pos_bias[None, :, None, :]
-        # $Q_i^T R_{i - j} + S_{i-j}$
+        # Shift the rows of $\mathbf{(B' + D')}_{i,k}$
+        # to get $$\mathbf{(B + D)}_{i,j} = \mathbf{(B' + D')}_{i,i - j}$$
         bd = shift_right(b + d)
+        # Remove extra positions
         bd = bd[:, -key.shape[0]:]
 
+        # Return the sum $\mathbf{(A + B + C + D)}_{i,j}$
         return ac + bd