动手学pytorch-注意力机制和Seq2Seq模型

1. 基本概念

Attention 是一种通用的带权池化方法，输入由两部分构成：询问（query）和键值对（key-value pairs）。\(??_??∈?^{??_??}, ??_??∈?^{??_??}\). Query \(??∈?^{??_??}\) , attention layer得到输出与value的维度一致 \(??∈?^{??_??}\). 对于一个query来说，attention layer 会与每一个key计算注意力分数并进行权重的归一化，输出的向量\(o\)则是value的加权求和，而每个key计算的权重与value一一对应。

为了计算输出，我们首先假设有一个函数\(\alpha\) 用于计算query和key的相似性，然后可以计算所有的 attention scores \(a_1, \ldots, a_n\) by

\[ a_i = \alpha(\mathbf q, \mathbf k_i). \]

我们使用 softmax函数 获得注意力权重：

\[ b_1, \ldots, b_n = \textrm{softmax}(a_1, \ldots, a_n). \]

最终的输出就是value的加权求和：

\[ \mathbf o = \sum_{i=1}^n b_i \mathbf v_i. \]

不同的attetion layer的区别在于score函数的选择，下面主要讨论两个常用的注意层 Dot-product Attention 和 Multilayer Perceptron Attention

Softmax屏蔽

在深入研究实现之前，首先介绍softmax操作符的一个屏蔽操作，主要目的是屏蔽无关信息。

def SequenceMask(X, X_len,value=-1e6):
    maxlen = X.size(1)
    #print(X.size(),torch.arange((maxlen),dtype=torch.float)[None, :],'\n',X_len[:, None] )
    mask = torch.arange((maxlen),dtype=torch.float)[None, :] >= X_len[:, None]   
    #print(mask)
    X[mask]=value
    return X

def masked_softmax(X, valid_length):
    # X: 3-D tensor, valid_length: 1-D or 2-D tensor
    softmax = nn.Softmax(dim=-1)
    if valid_length is None:
        return softmax(X)
    else:
        shape = X.shape
        if valid_length.dim() == 1:
            try:
                valid_length = torch.FloatTensor(valid_length.numpy().repeat(shape[1], axis=0))#[2,2,3,3]
            except:
                valid_length = torch.FloatTensor(valid_length.cpu().numpy().repeat(shape[1], axis=0))#[2,2,3,3]
        else:
            valid_length = valid_length.reshape((-1,))
        # fill masked elements with a large negative, whose exp is 0
        X = SequenceMask(X.reshape((-1, shape[-1])), valid_length)
 
        return softmax(X).reshape(shape)

masked_softmax(torch.rand((2,2,4),dtype=torch.float), torch.FloatTensor([2,3]))

tensor([[[0.4047, 0.5953, 0.0000, 0.0000],
         [0.4454, 0.5546, 0.0000, 0.0000]],

        [[0.3397, 0.3389, 0.3213, 0.0000],
         [0.3526, 0.3318, 0.3156, 0.0000]]])

超出2维矩阵的乘法

\(X\) 和 \(Y\) 是维度分别为\((b,n,m)\) 和\((b, m, k)\)的张量，进行 \(b\) 次二维矩阵乘法后得到 \(Z\), 维度为 \((b, n, k)\)。

\[ Z[i,:,:] = dot(X[i,:,:], Y[i,:,:])\qquad for\ i= 1,…,n\ . \]

torch.bmm(torch.ones((2,1,3), dtype = torch.float), torch.ones((2,3,2), dtype = torch.float))

tensor([[[3., 3.]],

        [[3., 3.]]])

2. 两种常用的attention层

2.1点积注意力

The dot product 假设query和keys有相同的维度, 即 $\forall i, ??,??_?? ∈ ?_?? $. 通过计算query和key转置的乘积来计算attention score,通常还会除去 \(\sqrt{d}\) 减少计算出来的score对维度??的依赖性，如下

\[ ??(??,??)=???,???/ \sqrt{d} \]

假设 $ ??∈?^{??×??}$ 有 \(m\) 个query，\(??∈?^{??×??}\) 有 \(n\) 个keys. 我们可以通过矩阵运算的方式计算所有 \(mn\) 个score：

\[ ??(??,??)=????^??/\sqrt{d} \]

下面来实现这个层，它支持一批查询和键值对。此外，它支持作为正则化随机删除一些注意力权重.

# Save to the d2l package.
class DotProductAttention(nn.Module): 
    def __init__(self, dropout, **kwargs):
        super(DotProductAttention, self).__init__(**kwargs)
        self.dropout = nn.Dropout(dropout)

    # query: (batch_size, #queries, d)
    # key: (batch_size, #kv_pairs, d)
    # value: (batch_size, #kv_pairs, dim_v)
    # valid_length: either (batch_size, ) or (batch_size, xx)
    def forward(self, query, key, value, valid_length=None):
        d = query.shape[-1]
        # set transpose_b=True to swap the last two dimensions of key
        
        scores = torch.bmm(query, key.transpose(1,2)) / math.sqrt(d)
        attention_weights = self.dropout(masked_softmax(scores, valid_length))
        print("attention_weight\n",attention_weights)
        return torch.bmm(attention_weights, value)

测试
创建两个batch，每个batch有一个query和10个key-values对。通过valid_length指定，对于第一批，只关注前2个键-值对，而对于第二批，我们将检查前6个键-值对。尽管这两个批处理具有相同的查询和键值对，但获得的输出是不同的。

atten = DotProductAttention(dropout=0)

keys = torch.ones((2,10,2),dtype=torch.float)
values = torch.arange((40), dtype=torch.float).view(1,10,4).repeat(2,1,1)
print(values.shape)
atten(torch.ones((2,1,2),dtype=torch.float), keys, values, torch.FloatTensor([2, 6]))

torch.Size([2, 10, 4])
attention_weight
 tensor([[[0.5000, 0.5000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000,
          0.0000, 0.0000]],

        [[0.1667, 0.1667, 0.1667, 0.1667, 0.1667, 0.1667, 0.0000, 0.0000,
          0.0000, 0.0000]]])

tensor([[[ 2.0000,  3.0000,  4.0000,  5.0000]],

        [[10.0000, 11.0000, 12.0000, 13.0000]]])

2.2 多层感知机注意力

在多层感知器中，我们首先将 query and keys 投影到 \(?^?\) .为了更具体，我们将可以学习的参数做如下映射
\(??_??∈?^{?×??_??}\) , \(??_??∈?^{?×??_??}\) , and \(??∈?^h\) . 将score函数定义
\[ ??(??,??)=??^??tanh(??_????+??_????) \]
.
然后将key 和 value 在特征的维度上合并（concatenate），然后送至 a single hidden layer perceptron 这层中 hidden layer 为 ? and 输出的size为 1 .隐层激活函数为tanh，无偏置.

# Save to the d2l package.
class MLPAttention(nn.Module):  
    def __init__(self, units,ipt_dim,dropout, **kwargs):
        super(MLPAttention, self).__init__(**kwargs)
        # Use flatten=True to keep query's and key's 3-D shapes.
        self.W_k = nn.Linear(ipt_dim, units, bias=False)
        self.W_q = nn.Linear(ipt_dim, units, bias=False)
        self.v = nn.Linear(units, 1, bias=False)
        self.dropout = nn.Dropout(dropout)

    def forward(self, query, key, value, valid_length):
        query, key = self.W_k(query), self.W_q(key)
        #print("size",query.size(),key.size())
        # expand query to (batch_size, #querys, 1, units), and key to
        # (batch_size, 1, #kv_pairs, units). Then plus them with broadcast.
        features = query.unsqueeze(2) + key.unsqueeze(1)
        #print("features:",features.size())  #--------------开启
        scores = self.v(features).squeeze(-1) 
        attention_weights = self.dropout(masked_softmax(scores, valid_length))
        return torch.bmm(attention_weights, value)

测试
尽管MLPAttention包含一个额外的MLP模型，但如果给定相同的输入和相同的键，我们将获得与DotProductAttention相同的输出

atten = MLPAttention(ipt_dim=2,units = 8, dropout=0)
atten(torch.ones((2,1,2), dtype = torch.float), keys, values, torch.FloatTensor([2, 6]))

torch.Size([2, 1, 10])

tensor([[[ 2.0000,  3.0000,  4.0000,  5.0000]],

        [[10.0000, 11.0000, 12.0000, 13.0000]]], grad_fn=<BmmBackward>)

3. 带注意力机制的Seq2seq模型

解码器

在解码的每个时间步，使用解码器的最后一个RNN层的输出作为注意层的query。然后，将注意力模型的输出与输入嵌入向量连接起来，输入到RNN层。虽然RNN层隐藏状态也包含来自解码器的历史信息，但是attention model的输出显式地选择了enc_valid_len以内的编码器输出，这样attention机制就会尽可能排除其他不相关的信息。

class Seq2SeqAttentionDecoder(d2l.Decoder):
    def __init__(self, vocab_size, embed_size, num_hiddens, num_layers,
                 dropout=0, **kwargs):
        super(Seq2SeqAttentionDecoder, self).__init__(**kwargs)
        self.attention_cell = MLPAttention(num_hiddens,num_hiddens, dropout)
        self.embedding = nn.Embedding(vocab_size, embed_size)
        self.rnn = nn.LSTM(embed_size+ num_hiddens,num_hiddens, num_layers, dropout=dropout)
        self.dense = nn.Linear(num_hiddens,vocab_size)

    def init_state(self, enc_outputs, enc_valid_len, *args):
        outputs, hidden_state = enc_outputs
#         print("first:",outputs.size(),hidden_state[0].size(),hidden_state[1].size())
        # Transpose outputs to (batch_size, seq_len, hidden_size)
        return (outputs.permute(1,0,-1), hidden_state, enc_valid_len)
        #outputs.swapaxes(0, 1)
        
    def forward(self, X, state):
        enc_outputs, hidden_state, enc_valid_len = state
        #("X.size",X.size())
        X = self.embedding(X).transpose(0,1)
#         print("Xembeding.size2",X.size())
        outputs = []
        for l, x in enumerate(X):
#             print(f"\n{l}-th token")
#             print("x.first.size()",x.size())
            # query shape: (batch_size, 1, hidden_size)
            # select hidden state of the last rnn layer as query
            query = hidden_state[0][-1].unsqueeze(1) # np.expand_dims(hidden_state[0][-1], axis=1)
            # context has same shape as query
#             print("query enc_outputs, enc_outputs:\n",query.size(), enc_outputs.size(), enc_outputs.size())
            context = self.attention_cell(query, enc_outputs, enc_outputs, enc_valid_len)
            # Concatenate on the feature dimension
#             print("context.size:",context.size())
            x = torch.cat((context, x.unsqueeze(1)), dim=-1)
            # Reshape x to (1, batch_size, embed_size+hidden_size)
#             print("rnn",x.size(), len(hidden_state))
            out, hidden_state = self.rnn(x.transpose(0,1), hidden_state)
            outputs.append(out)
        outputs = self.dense(torch.cat(outputs, dim=0))
        return outputs.transpose(0, 1), [enc_outputs, hidden_state,
                                        enc_valid_len]

encoder = Seq2SeqEncoder(vocab_size=10, embed_size=8,
                            num_hiddens=16, num_layers=2)
# encoder.initialize()
decoder = Seq2SeqAttentionDecoder(vocab_size=10, embed_size=8,
                                  num_hiddens=16, num_layers=2)
X = torch.zeros((4, 7),dtype=torch.long)
print("batch size=4\nseq_length=7\nhidden dim=16\nnum_layers=2\n")
print('encoder output size:', encoder(X)[0].size())
print('encoder hidden size:', encoder(X)[1][0].size())
print('encoder memory size:', encoder(X)[1][1].size())
state = decoder.init_state(encoder(X), None)
out, state = decoder(X, state)
out.shape, len(state), state[0].shape, len(state[1]), state[1][0].shape

batch size=4
seq_length=7
hidden dim=16
num_layers=2

encoder output size: torch.Size([7, 4, 16])
encoder hidden size: torch.Size([2, 4, 16])
encoder memory size: torch.Size([2, 4, 16])

(torch.Size([4, 7, 10]), 3, torch.Size([4, 7, 16]), 2, torch.Size([2, 4, 16]))

4. 实验