D2L-53 注意力分数
Published:
本文介绍注意力分数的计算与掩码 Softmax，对比加性注意力和缩放点积注意力，并给出相应的 PyTorch 实现与权重可视化示例。
\[f(x)=\sum_{i=1}^nsoftmax\left(-{1\over2}((x-x_i))^2\right)y_i\]- 其中，被normalize之前的 \((-{1\over2}((x-x_i))^2\)，称之为注意力分数 (attention scoring function)
拓展到高维度
- 假设 query \({\bf q}\in\mathbb R^n\)，\(m\) 对 key-value \((\bf k_1,v_1),...\)，这里 \({\bf k_i}\in\mathbb R^k\)，\({\bf v_i}\in\mathbb R^v\)
注意力池化层：
\[f({\bf q,(k_1,v_1),...,(k_m,v_m)})=\sum_{i=1}^m\alpha({\bf q,k_i}){\bf v_i}\in\mathbb R^v\] \[\alpha({\bf q,k_i})=softmax(a({\bf q,k_i}))={\exp(a({\bf q,k_i}))\over\sum_{j=1}^m\exp(a({\bf q,k_j})}\in\mathbb R\]
Additive Attention
- 可学参数：\({\bf W_k}\in\mathbb R^{h\times k},{\bf W_q}\in\mathbb R^{h\times q},{\bf w_v}\in\mathbb R^h\) \(a({\bf k,q})={\bf w_v}^Ttanh({\bf W_kk+W_qq })\)
- 等价于将 key 和 value 合并起来后放入到一个隐藏层大小为 h 输出大小为1的单隐藏层MLP
- 输出一个标量
- \(\bf q,k,v\) 可以是不同长度
- 拓展维度：\({\bf Q}\in\mathbb R^{n\times q},\ {\bf K}\in\mathbb R^{m\times k},\ V^{m\times v}\)
- \[a({\bf K,Q})={\bf w_v}^Ttanh({\bf KW_k^T+QW_q^T })\]
Scaled Dot_product Attention
- 如果 query 和 key 都是同样的长度，\({\bf q,k_i}\in\mathbb R^d\)，那么可以： \(a({\bf q_i,k})=<{\bf q,k_i}>/\sqrt d\)
- 除以根号 d 使对长度不敏感
- 向量化版本
- \[{\bf Q}\in\mathbb R^{n\times d},\ {\bf K}\in\mathbb R^{m\times d},\ V^{m\times v}\]
- 注意力分数：\(a({\bf Q,K})={\bf QK}^T/\sqrt d\in\mathbb R^{n \times m}\)
- 注意力池化：\(f=softmax(a({\bf Q,K})){\bf V}\in\mathbb R^{n \times v}\)
- 有 \(n\) 个 query， \(m\) 个 key，每个 key 有 \(v\) 个 value
总结
- 注意力分数是 query 和 key 的相似度，注意力权重是分数的softmax结果
- 两种常见的分数计算：
- 将 query 和 key 合并起来进入一个单输出单隐藏层的MLP
- 直接将 query 和 key 做内积
代码实现
定义padding屏蔽的辅助函数
d2l.show_heatmaps(attention.attention_weights.reshape((1, 1, 2, 10)),
xlabel='Keys', ylabel='Queries')
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:
# dim(): Returns the number of dimensions of self tensor.
# dim=1,a vector
valid_lens = torch.repeat_interleave(valid_lens, shape[1])
# 复制X行数的validlens shape=(shape[1]*len(valid_lens,)
else:
valid_lens = valid_lens.reshape(-1)
# 维度大于1，就拉成一行
X = d2l.sequence_mask(X.reshape(-1, shape[-1]), valid_lens,
value=-1e6)
# 把X变为二维按行堆积
# 把超出len的替换成非常大的负数
return nn.functional.softmax(X.reshape(shape), dim=-1)
# 维度可能会大于二，所以对指定对最后的列做softmax
masked_softmax(torch.rand(2, 2, 4), torch.tensor([2, 3]))
masked_softmax(torch.rand(2, 2, 4), torch.tensor([[1, 3], [2, 4]]))
加性注意力
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):
# valid_lens代表对于每一个query有效的key-value pair
queries, keys = self.W_q(queries), self.W_k(keys)
#(batch_size, num_quaries, num_hiddens)
#(bacth_size, num_keys, num_hiddens)
#中间维度不相同
features = queries.unsqueeze(2) + keys.unsqueeze(1)
#(batch_size, num_quaries, 1， num_hiddens)
#(bacth_size, 1， num_keys, num_hiddens)
#(bacth_size, num_quaries， num_keys, num_hiddens)
features = torch.tanh(features)
scores = self.w_v(features).squeeze(-1)
#和self.w做矩阵乘法长度h被消除，拿掉空维度
#(batch_size, num_quaries, num_keys)
self.attention_weights = masked_softmax(scores, valid_lens)
#过滤掉不需要的padding
return torch.bmm(self.dropout(self.attention_weights), values)
#用dropot随机把更多的权重变为0
queries, keys = torch.normal(0, 1, (2, 1, 20)), torch.ones((2, 10, 2))
# (2, 1, 20),(2, 10, 2)
values = torch.arange(40, dtype=torch.float32).reshape(1, 10, 4).repeat(
2, 1, 1)
# (2, 10, 4)
valid_lens = torch.tensor([2, 6])
attention = AdditiveAttention(key_size=2, query_size=20, num_hiddens=8,
dropout=0.1)
# (2, 1, 8),(2, 10, 8)
# (2, 1, 10 ,8)
# (2, 1, 10)
# (2, 1, 4)
attention.eval()
attention(queries, keys, values, valid_lens)
d2l.show_heatmaps(attention.attention_weights.reshape((1, 1, 2, 10)),
xlabel='Keys', ylabel='Queries')
#valid_lens.shape=(2, 6)
缩放点积注意力
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)
def forward(self, queries, keys, values, valid_lens=None):
# queries的形状：(batch_size，num_queries，d)
# keys的形状：(batch_size，num_kvpairs，d)
# values的形状：(batch_size，num_kvpairs，d_values)
# valid_lens的形状:(batch_size，)或者(batch_size，num_queries)
d = queries.shape[-1]
scores = torch.bmm(queries, keys.transpose(1,2)) / math.sqrt(d)
#(batch_size，num_queries，num_kvpairs)
self.attention_weights = masked_softmax(scores, valid_lens)
return torch.bmm(self.dropout(self.attention_weights), values)
#(batch_size，num_queries，d_values)
queries = torch.normal(0, 1, (2, 1, 2))
#keys=(2, 10, 2),values=(2, 10, 4)
attention = DotProductAttention(dropout=0.5)
#(2, 1, 4)
attention.eval()
attention(queries, keys, values, valid_lens)
d2l.show_heatmaps(attention.attention_weights.reshape((1, 1, 2, 10)),
xlabel='Keys', ylabel='Queries')
#超出valid_len的部分没有权重
