.. _sec_attention-scoring-functions:
注意力评分函数
==============
:numref:`sec_nadaraya-watson`\ 使用了高斯核来对查询和键之间的关系建模。
:eq:`eq_nadaraya-watson-gaussian`\ 中的
高斯核指数部分可以视为\ *注意力评分函数*\ （attention scoring
function）， 简称\ *评分函数*\ （scoring function），
然后把这个函数的输出结果输入到softmax函数中进行运算。
通过上述步骤，将得到与键对应的值的概率分布（即注意力权重）。
最后，注意力汇聚的输出就是基于这些注意力权重的值的加权和。
从宏观来看，上述算法可以用来实现
:numref:`fig_qkv`\ 中的注意力机制框架。
:numref:`fig_attention_output`\ 说明了
如何将注意力汇聚的输出计算成为值的加权和，
其中\ :math:`a`\ 表示注意力评分函数。 由于注意力权重是概率分布，
因此加权和其本质上是加权平均值。
.. _fig_attention_output:
.. figure:: ../img/attention-output.svg
计算注意力汇聚的输出为值的加权和
用数学语言描述，假设有一个查询 :math:`\mathbf{q} \in \mathbb{R}^q`\ 和
:math:`m`\ 个“键－值”对
:math:`(\mathbf{k}_1, \mathbf{v}_1), \ldots, (\mathbf{k}_m, \mathbf{v}_m)`\ ，
其中\ :math:`\mathbf{k}_i \in \mathbb{R}^k`\ ，\ :math:`\mathbf{v}_i \in \mathbb{R}^v`\ 。
注意力汇聚函数\ :math:`f`\ 就被表示成值的加权和：
.. math:: f(\mathbf{q}, (\mathbf{k}_1, \mathbf{v}_1), \ldots, (\mathbf{k}_m, \mathbf{v}_m)) = \sum_{i=1}^m \alpha(\mathbf{q}, \mathbf{k}_i) \mathbf{v}_i \in \mathbb{R}^v,
:label: eq_attn-pooling
其中查询\ :math:`\mathbf{q}`\ 和键\ :math:`\mathbf{k}_i`\ 的注意力权重（标量）
是通过注意力评分函数\ :math:`a`\ 将两个向量映射成标量，
再经过softmax运算得到的：
.. math:: \alpha(\mathbf{q}, \mathbf{k}_i) = \mathrm{softmax}(a(\mathbf{q}, \mathbf{k}_i)) = \frac{\exp(a(\mathbf{q}, \mathbf{k}_i))}{\sum_{j=1}^m \exp(a(\mathbf{q}, \mathbf{k}_j))} \in \mathbb{R}.
:label: eq_attn-scoring-alpha
正如上图所示，选择不同的注意力评分函数\ :math:`a`\ 会导致不同的注意力汇聚操作。
本节将介绍两个流行的评分函数，稍后将用他们来实现更复杂的注意力机制。
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\diilbookstyleinputcell
.. code:: python
import math
import mindspore
from mindspore import nn, ops
from d2l import mindspore as d2l
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\diilbookstyleinputcell
.. code:: python
import math
import torch
from torch import nn
from d2l import torch as d2l
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\diilbookstyleinputcell
.. code:: python
#@save
def masked_softmax(X, valid_lens):
"""通过在最后一个轴上掩蔽元素来执行 softmax 操作"""
# X:3D张量，valid_lens:1D或2D张量
if valid_lens is None:
return nn.Softmax(-1)(X)
else:
shape = X.shape
if valid_lens.ndim == 1:
valid_lens = ops.repeat_interleave(valid_lens, shape[1])
else:
valid_lens = valid_lens.reshape(-1)
# 最后一轴上被掩蔽的元素使用一个非常大的负值替换，从而其softmax输出为0
X = d2l.sequence_mask(X.reshape(-1, shape[-1]), valid_lens,
value=-1e6)
return nn.Softmax(-1)(X.reshape(shape))
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\diilbookstyleinputcell
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#@save
def masked_softmax(X, valid_lens):
"""通过在最后一个轴上掩蔽元素来执行softmax操作"""
# X:3D张量，valid_lens:1D或2D张量
if valid_lens is None:
return nn.functional.softmax(X, dim=-1)
else:
shape = X.shape
if valid_lens.dim() == 1:
valid_lens = torch.repeat_interleave(valid_lens, shape[1])
else:
valid_lens = valid_lens.reshape(-1)
# 最后一轴上被掩蔽的元素使用一个非常大的负值替换，从而其softmax输出为0
X = d2l.sequence_mask(X.reshape(-1, shape[-1]), valid_lens,
value=-1e6)
return nn.functional.softmax(X.reshape(shape), dim=-1)
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\diilbookstyleinputcell
.. code:: python
masked_softmax(d2l.rand((2, 2, 4)), mindspore.tensor([2, 3], mindspore.int32))
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Tensor(shape=[2, 2, 4], dtype=Float32, value=
[[[ 3.83217573e-01, 6.16782486e-01, 0.00000000e+00, 0.00000000e+00],
[ 4.93376523e-01, 5.06623507e-01, 0.00000000e+00, 0.00000000e+00]],
[[ 2.93767989e-01, 2.91695654e-01, 4.14536387e-01, 0.00000000e+00],
[ 3.78261268e-01, 3.25547487e-01, 2.96191186e-01, 0.00000000e+00]]])
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\diilbookstyleinputcell
.. code:: python
masked_softmax(torch.rand(2, 2, 4), torch.tensor([2, 3]))
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tensor([[[0.5267, 0.4733, 0.0000, 0.0000],
[0.5901, 0.4099, 0.0000, 0.0000]],
[[0.2026, 0.4461, 0.3514, 0.0000],
[0.4005, 0.4087, 0.1908, 0.0000]]])
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\diilbookstyleinputcell
.. code:: python
masked_softmax(d2l.rand((2, 2, 4)), mindspore.tensor([[1, 3], [2, 4]], mindspore.int32))
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Tensor(shape=[2, 2, 4], dtype=Float32, value=
[[[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
[ 4.40855712e-01, 1.90578580e-01, 3.68565649e-01, 0.00000000e+00]],
[[ 5.13258040e-01, 4.86741990e-01, 0.00000000e+00, 0.00000000e+00],
[ 2.50914991e-01, 2.04868034e-01, 2.09382087e-01, 3.34834844e-01]]])
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
masked_softmax(torch.rand(2, 2, 4), torch.tensor([[1, 3], [2, 4]]))
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tensor([[[1.0000, 0.0000, 0.0000, 0.0000],
[0.4706, 0.2577, 0.2717, 0.0000]],
[[0.4517, 0.5483, 0.0000, 0.0000],
[0.3784, 0.1615, 0.2510, 0.2091]]])
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\diilbookstyleinputcell
.. code:: python
#@save
class AdditiveAttention(nn.Cell):
"""加性注意力"""
def __init__(self, key_size, query_size, num_hiddens, dropout, **kwargs):
super(AdditiveAttention, self).__init__(**kwargs)
self.W_k = nn.Dense(key_size, num_hiddens, has_bias=False)
self.W_q = nn.Dense(query_size, num_hiddens, has_bias=False)
self.w_v = nn.Dense(num_hiddens, 1, has_bias=False)
self.dropout = nn.Dropout(p=dropout)
def construct(self, queries, keys, values, valid_lens):
queries, keys = self.W_q(queries), self.W_k(keys)
# 在维度扩展后，
# queries的形状：(batch_size，查询的个数，1，num_hidden)
# key的形状：(batch_size，1，“键－值”对的个数，num_hiddens)
# 使用广播方式进行求和
features = d2l.expand_dims(queries, 2) + d2l.expand_dims(keys, 1)
features = d2l.tanh(features)
# self.w_v仅有一个输出，因此从形状中移除最后那个维度。
# scores的形状：(batch_size，查询的个数，“键-值”对的个数)
scores = self.w_v(features).squeeze(-1)
self.attention_weights = masked_softmax(scores, valid_lens)
# values的形状：(batch_size，“键－值”对的个数，值的维度)
return self.dropout(self.attention_weights).bmm(values)
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\diilbookstyleinputcell
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#@save
class AdditiveAttention(nn.Module):
"""加性注意力"""
def __init__(self, key_size, query_size, num_hiddens, dropout, **kwargs):
super(AdditiveAttention, self).__init__(**kwargs)
self.W_k = nn.Linear(key_size, num_hiddens, bias=False)
self.W_q = nn.Linear(query_size, num_hiddens, bias=False)
self.w_v = nn.Linear(num_hiddens, 1, bias=False)
self.dropout = nn.Dropout(dropout)
def forward(self, queries, keys, values, valid_lens):
queries, keys = self.W_q(queries), self.W_k(keys)
# 在维度扩展后，
# queries的形状：(batch_size，查询的个数，1，num_hidden)
# key的形状：(batch_size，1，“键－值”对的个数，num_hiddens)
# 使用广播方式进行求和
features = queries.unsqueeze(2) + keys.unsqueeze(1)
features = torch.tanh(features)
# self.w_v仅有一个输出，因此从形状中移除最后那个维度。
# scores的形状：(batch_size，查询的个数，“键-值”对的个数)
scores = self.w_v(features).squeeze(-1)
self.attention_weights = masked_softmax(scores, valid_lens)
# values的形状：(batch_size，“键－值”对的个数，值的维度)
return torch.bmm(self.dropout(self.attention_weights), values)
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\diilbookstyleinputcell
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queries, keys = d2l.normal((2, 1, 20), 0, 1), d2l.ones((2, 10, 2))
# values的小批量，两个值矩阵是相同的
values = d2l.tile(d2l.arange(40, dtype=mindspore.float32).reshape(1, 10, 4), (
2, 1, 1))
valid_lens = mindspore.tensor([2, 6])
attention = AdditiveAttention(key_size=2, query_size=20, num_hiddens=8,
dropout=0.1)
attention.set_train(False)
attention(queries, keys, values, valid_lens)
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Tensor(shape=[2, 1, 4], dtype=Float32, value=
[[[ 2.00000000e+00, 3.00000000e+00, 4.00000000e+00, 5.00000000e+00]],
[[ 1.00000000e+01, 1.10000000e+01, 1.20000000e+01, 1.30000000e+01]]])
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\diilbookstyleinputcell
.. code:: python
queries, keys = torch.normal(0, 1, (2, 1, 20)), torch.ones((2, 10, 2))
# values的小批量，两个值矩阵是相同的
values = torch.arange(40, dtype=torch.float32).reshape(1, 10, 4).repeat(
2, 1, 1)
valid_lens = torch.tensor([2, 6])
attention = AdditiveAttention(key_size=2, query_size=20, num_hiddens=8,
dropout=0.1)
attention.eval()
attention(queries, keys, values, valid_lens)
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tensor([[[ 2.0000, 3.0000, 4.0000, 5.0000]],
[[10.0000, 11.0000, 12.0000, 13.0000]]], grad_fn=)
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\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(attention.attention_weights.reshape((1, 1, 2, 10)),
xlabel='Keys', ylabel='Queries')
.. figure:: output_attention-scoring-functions_2a8fdc_57_0.png
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(attention.attention_weights.reshape((1, 1, 2, 10)),
xlabel='Keys', ylabel='Queries')
.. figure:: output_attention-scoring-functions_2a8fdc_60_0.svg
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\diilbookstyleinputcell
.. code:: python
#@save
class DotProductAttention(nn.Cell):
"""缩放点积注意力"""
def __init__(self, dropout, **kwargs):
super(DotProductAttention, self).__init__(**kwargs)
self.dropout = nn.Dropout(p=dropout)
# queries的形状：(batch_size，查询的个数，d)
# keys的形状：(batch_size，“键－值”对的个数，d)
# values的形状：(batch_size，“键－值”对的个数，值的维度)
# valid_lens的形状:(batch_size，)或者(batch_size，查询的个数)
def construct(self, queries, keys, values, valid_lens=None):
d = queries.shape[-1]
# 设置transpose_b=True为了交换keys的最后两个维度
scores = queries.bmm(keys.swapaxes(1, 2)) / math.sqrt(d)
self.attention_weights = masked_softmax(scores, valid_lens)
return self.dropout(self.attention_weights).bmm(values)
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\diilbookstyleinputcell
.. code:: python
#@save
class DotProductAttention(nn.Module):
"""缩放点积注意力"""
def __init__(self, dropout, **kwargs):
super(DotProductAttention, self).__init__(**kwargs)
self.dropout = nn.Dropout(dropout)
# queries的形状：(batch_size，查询的个数，d)
# keys的形状：(batch_size，“键－值”对的个数，d)
# values的形状：(batch_size，“键－值”对的个数，值的维度)
# valid_lens的形状:(batch_size，)或者(batch_size，查询的个数)
def forward(self, queries, keys, values, valid_lens=None):
d = queries.shape[-1]
# 设置transpose_b=True为了交换keys的最后两个维度
scores = torch.bmm(queries, keys.transpose(1,2)) / math.sqrt(d)
self.attention_weights = masked_softmax(scores, valid_lens)
return torch.bmm(self.dropout(self.attention_weights), values)
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\diilbookstyleinputcell
.. code:: python
queries = d2l.normal((2, 1, 2), 0, 1)
attention = DotProductAttention(dropout=0.5)
attention.set_train(False)
attention(queries, keys, values, valid_lens)
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Tensor(shape=[2, 1, 4], dtype=Float32, value=
[[[ 2.00000000e+00, 3.00000000e+00, 4.00000000e+00, 5.00000000e+00]],
[[ 1.00000000e+01, 1.10000000e+01, 1.20000000e+01, 1.30000000e+01]]])
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\diilbookstyleinputcell
.. code:: python
queries = torch.normal(0, 1, (2, 1, 2))
attention = DotProductAttention(dropout=0.5)
attention.eval()
attention(queries, keys, values, valid_lens)
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tensor([[[ 2.0000, 3.0000, 4.0000, 5.0000]],
[[10.0000, 11.0000, 12.0000, 13.0000]]])
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\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(attention.attention_weights.reshape((1, 1, 2, 10)),
xlabel='Keys', ylabel='Queries')
.. figure:: output_attention-scoring-functions_2a8fdc_84_0.png
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(attention.attention_weights.reshape((1, 1, 2, 10)),
xlabel='Keys', ylabel='Queries')
.. figure:: output_attention-scoring-functions_2a8fdc_87_0.svg
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`讨论 `__
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`讨论 `__
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