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上一篇文章我们介绍了causal attention和attention机制中的dropout，本文将继续介绍multi-head attention。
multi-head是指把attention机制分成多个head，每个head单独计算，每个head会关注当前单词与输入序列中其他单词在不同方面的上下文关系。上一篇文章中介绍的causal attention就可以看作是一个single-head attention。
multi-head，顾名思义，就是把多个single-head attention叠加起来。下面实现一个叠加多个CausalAttention类的multi-head attention:
class MultiHeadAttentionWrapper(nn.Module):def __init__(self, d_in, d_out, context_length, dropout, num_heads, qkv_bias=False):super().__init__()self.heads = nn.ModuleList([CausalAttention(d_in, d_out, context_length, dropout, qkv_bias) for _ in range(num_heads)])def forward(self, x):return torch.cat([head(x) for head in self.heads], dim=-1)torch.manual_seed(123)context_length = batch.shape[1] # This is the number of tokens
d_in, d_out = 3, 2
mha = MultiHeadAttentionWrapper(d_in, d_out, context_length, 0.0, num_heads=2
)context_vecs = mha(batch)print(context_vecs)
print("context_vecs.shape:", context_vecs.shape)
# 输出：
#tensor([[[-0.4519, 0.2216, 0.4772, 0.1063],
# [-0.5874, 0.0058, 0.5891, 0.3257],
# [-0.6300, -0.0632, 0.6202, 0.3860],
# [-0.5675, -0.0843, 0.5478, 0.3589],
# [-0.5526, -0.0981, 0.5321, 0.3428],
# [-0.5299, -0.1081, 0.5077, 0.3493]],# [[-0.4519, 0.2216, 0.4772, 0.1063],
# [-0.5874, 0.0058, 0.5891, 0.3257],
# [-0.6300, -0.0632, 0.6202, 0.3860],
# [-0.5675, -0.0843, 0.5478, 0.3589],
# [-0.5526, -0.0981, 0.5321, 0.3428],
# [-0.5299, -0.1081, 0.5077, 0.3493]]], grad_fn=<CatBackward0>)
# context_vecs.shape: torch.Size([2, 6, 4])
可以看到，输出结果的最后一个维度是4，而上一篇文章最后一个维度的输出是2，这是因为2个single-head attention的结果叠加(concat)起来了，示意图如下：
上面这种方法从功能上讲，完全可以实现multi-head attention的功能，但是每一个single-head都有自己的Wq, Wk, Wv权重矩阵，这不仅增加了参数量而且增加了计算量。下面介绍一种只有一套Wq, Wk, Wv权重矩阵的multi-head attention实现方法：
class MultiHeadAttention(nn.Module):def __init__(self, d_in, d_out, context_length, dropout, num_heads, qkv_bias=False):super().__init__()assert (d_out % num_heads == 0), \"d_out must be divisible by num_heads"self.d_out = d_outself.num_heads = num_headsself.head_dim = d_out // num_heads # Reduce the projection dim to match desired output dimself.W_query = nn.Linear(d_in, d_out, bias=qkv_bias)self.W_key = nn.Linear(d_in, d_out, bias=qkv_bias)self.W_value = nn.Linear(d_in, d_out, bias=qkv_bias)self.out_proj = nn.Linear(d_out, d_out) # Linear layer to combine head outputsself.dropout = nn.Dropout(dropout)self.register_buffer("mask",torch.triu(torch.ones(context_length, context_length),diagonal=1))def forward(self, x):b, num_tokens, d_in = x.shape# As in `CausalAttention`, for inputs where `num_tokens` exceeds `context_length`, # this will result in errors in the mask creation further below. # In practice, this is not a problem since the LLM (chapters 4-7) ensures that inputs # do not exceed `context_length` before reaching this forwarkeys = self.W_key(x) # Shape: (b, num_tokens, d_out)queries = self.W_query(x)values = self.W_value(x)# We implicitly split the matrix by adding a `num_heads` dimension# Unroll last dim: (b, num_tokens, d_out) -> (b, num_tokens, num_heads, head_dim)keys = keys.view(b, num_tokens, self.num_heads, self.head_dim) values = values.view(b, num_tokens, self.num_heads, self.head_dim)queries = queries.view(b, num_tokens, self.num_heads, self.head_dim)# Transpose: (b, num_tokens, num_heads, head_dim) -> (b, num_heads, num_tokens, head_dim)keys = keys.transpose(1, 2)queries = queries.transpose(1, 2)values = values.transpose(1, 2)# Compute scaled dot-product attention (aka self-attention) with a causal maskattn_scores = queries @ keys.transpose(2, 3) # Dot product for each head# Original mask truncated to the number of tokens and converted to booleanmask_bool = self.mask.bool()[:num_tokens, :num_tokens]# Use the mask to fill attention scoresattn_scores.masked_fill_(mask_bool, -torch.inf)attn_weights = torch.softmax(attn_scores / keys.shape[-1]**0.5, dim=-1)attn_weights = self.dropout(attn_weights)# Shape: (b, num_tokens, num_heads, head_dim)context_vec = (attn_weights @ values).transpose(1, 2) # Combine heads, where self.d_out = self.num_heads * self.head_dimcontext_vec = context_vec.contiguous().view(b, num_tokens, self.d_out)context_vec = self.out_proj(context_vec) # optional projectionreturn context_vectorch.manual_seed(123)batch_size, context_length, d_in = batch.shape
d_out = 2
mha = MultiHeadAttention(d_in, d_out, context_length, 0.0, num_heads=2)context_vecs = mha(batch)print(context_vecs)
print("context_vecs.shape:", context_vecs.shape)
# 输出：
#tensor([[[0.3190, 0.4858],
# [0.2943, 0.3897],
# [0.2856, 0.3593],
# [0.2693, 0.3873],
# [0.2639, 0.3928],
# [0.2575, 0.4028]],# [[0.3190, 0.4858],
# [0.2943, 0.3897],
# [0.2856, 0.3593],
# [0.2693, 0.3873],
# [0.2639, 0.3928],
# [0.2575, 0.4028]]], grad_fn=<ViewBackward0>)
# context_vecs.shape: torch.Size([2, 6, 2])
上面这中实现方法，只用了一套Wq, Wk, Wv，就实现了multi-head attention。其核心是将W权重矩阵分割成num_heads份，每一份单独计算，最后再合并。也就是说，上一种方法计算attention score时，每个单词向量的全部都参与计算，比如attention_scores(6×6) = queries(6×2) @ keys.T(2×6)；第二种方法计算attention score时，每个单词向量被分割成了num_heads份，每个子份计算自己的attention score，即attention_scores(6×6) = queries(6×1) @ keys.T(1×6)。每个子份算出自己的上下文向量后再合并成总的上下文向量。
下面再以一个简单的例子，进一步理解上面例子中的多维矩阵的乘法：
# (b, num_heads, num_tokens, head_dim) = (1, 2, 3, 4)
a = torch.tensor([[[[0.2745, 0.6584, 0.2775, 0.8573],[0.8993, 0.0390, 0.9268, 0.7388],[0.7179, 0.7058, 0.9156, 0.4340]],[[0.0772, 0.3565, 0.1479, 0.5331],[0.4066, 0.2318, 0.4545, 0.9737],[0.4606, 0.5159, 0.4220, 0.5786]]]])print(a @ a.transpose(2, 3))
# 输出：
#tensor([[[[1.3208, 1.1631, 1.2879],
# [1.1631, 2.2150, 1.8424],
# [1.2879, 1.8424, 2.0402]],# [[0.4391, 0.7003, 0.5903],
# [0.7003, 1.3737, 1.0620],
# [0.5903, 1.0620, 0.9912]]]])
例子中a是一个四维矩阵，因此矩阵相乘会在最后2维(num_tokens, head_dim)上进行，然后每个head都会做一次矩阵相乘。下面对每个head进行分拆，分别单独计算：
first_head = a[0, 0, :, :]
first_res = first_head @ first_head.T
print("First head:\n", first_res)second_head = a[0, 1, :, :]
second_res = second_head @ second_head.T
print("\nSecond head:\n", second_res)
# 输出：
# First head:
# tensor([[1.3208, 1.1631, 1.2879],
# [1.1631, 2.2150, 1.8424],
# [1.2879, 1.8424, 2.0402]])# Second head:
# tensor([[0.4391, 0.7003, 0.5903],
# [0.7003, 1.3737, 1.0620],
# [0.5903, 1.0620, 0.9912]])
第二种方法MultiHeadAttention相比第一种方法MultiHeadAttentionWrapper，多了一些矩阵reshape和transpose的操作，更复杂但是更高效。因为我们只需要一套Wq, Wk, Wv计算queries, keys, values，分别只需要进行一次矩阵乘法(e.g. queries = self.W_query(x))。而在MultiHeadAttentionWrapper中，每个attention head都有一套Wq, Wk, Wv，都要重复进行矩阵乘法，而矩阵乘法是最耗资源的操作之一。因此第二种方法会更高效。
参考资料：
《Build a Large Language Model from Scratch》