注意力汇聚:Nadaraya-Watson 核回归

In [1]:
import torch
from torch import nn
from d2l import torch as d2l

生成数据集

In [3]:
n_train = 50
x_train, _ = torch.sort(torch.rand(n_train) * 5)

def f(x):
    return 2 * torch.sin(x) + x**0.8

y_train = f(x_train) + torch.normal(0.0, 0.5, (n_train,))
x_test = torch.arange(0, 5, 0.1)
y_truth = f(x_test)
n_test = len(x_test)
n_test
Out[3]:
50
In [5]:
def plot_kernel_reg(y_hat):
    d2l.plot(x_test, [y_truth, y_hat], 'x', 'y', legend=['Truth', 'Pred'],
             xlim=[0, 5], ylim=[-1, 5])
    d2l.plt.plot(x_train, y_train, 'o', alpha=0.5);

y_hat = torch.repeat_interleave(y_train.mean(), n_test)
plot_kernel_reg(y_hat)
2021-07-24T07:35:54.034688 image/svg+xml Matplotlib v3.4.0rc1, https://matplotlib.org/

非参数注意力汇聚

In [6]:
X_repeat = x_test.repeat_interleave(n_train).reshape((-1, n_train))
attention_weights = nn.functional.softmax(-(X_repeat - x_train)**2 / 2, dim=1)
y_hat = torch.matmul(attention_weights, y_train)
plot_kernel_reg(y_hat)
2021-07-24T07:35:54.300280 image/svg+xml Matplotlib v3.4.0rc1, https://matplotlib.org/

注意力权重

In [7]:
d2l.show_heatmaps(
    attention_weights.unsqueeze(0).unsqueeze(0),
    xlabel='Sorted training inputs', ylabel='Sorted testing inputs')
2021-07-24T07:35:54.546768 image/svg+xml Matplotlib v3.4.0rc1, https://matplotlib.org/

带参数注意力汇聚 假定两个张量的形状分别是 $(n,a,b)$ 和 $(n,b,c)$ ,它们的批量矩阵乘法输出的形状为 $(n,a,c)$

In [8]:
X = torch.ones((2, 1, 4))
Y = torch.ones((2, 4, 6))
torch.bmm(X, Y).shape
Out[8]:
torch.Size([2, 1, 6])

使用小批量矩阵乘法来计算小批量数据中的加权平均值

In [9]:
weights = torch.ones((2, 10)) * 0.1
values = torch.arange(20.0).reshape((2, 10))
torch.bmm(weights.unsqueeze(1), values.unsqueeze(-1))
Out[9]:
tensor([[[ 4.5000]],

        [[14.5000]]])

带参数的注意力汇聚

In [10]:
class NWKernelRegression(nn.Module):
    def __init__(self, **kwargs):
        super().__init__(**kwargs)
        self.w = nn.Parameter(torch.rand((1,), requires_grad=True))

    def forward(self, queries, keys, values):
        queries = queries.repeat_interleave(keys.shape[1]).reshape(
            (-1, keys.shape[1]))
        self.attention_weights = nn.functional.softmax(
            -((queries - keys) * self.w)**2 / 2, dim=1)
        return torch.bmm(self.attention_weights.unsqueeze(1),
                         values.unsqueeze(-1)).reshape(-1)

将训练数据集转换为键和值

In [11]:
X_tile = x_train.repeat((n_train, 1))
Y_tile = y_train.repeat((n_train, 1))
keys = X_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape(
    (n_train, -1))
values = Y_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape(
    (n_train, -1))

训练带参数的注意力汇聚模型

In [12]:
net = NWKernelRegression()
loss = nn.MSELoss(reduction='none')
trainer = torch.optim.SGD(net.parameters(), lr=0.5)
animator = d2l.Animator(xlabel='epoch', ylabel='loss', xlim=[1, 5])

for epoch in range(5):
    trainer.zero_grad()
    l = loss(net(x_train, keys, values), y_train) / 2
    l.sum().backward()
    trainer.step()
    print(f'epoch {epoch + 1}, loss {float(l.sum()):.6f}')
    animator.add(epoch + 1, float(l.sum()))
2021-07-24T07:35:55.822655 image/svg+xml Matplotlib v3.4.0rc1, https://matplotlib.org/

预测结果绘制

In [13]:
keys = x_train.repeat((n_test, 1))
values = y_train.repeat((n_test, 1))
y_hat = net(x_test, keys, values).unsqueeze(1).detach()
plot_kernel_reg(y_hat)
2021-07-24T07:35:56.044553 image/svg+xml Matplotlib v3.4.0rc1, https://matplotlib.org/

曲线在注意力权重较大的区域变得更不平滑

In [14]:
d2l.show_heatmaps(
    net.attention_weights.unsqueeze(0).unsqueeze(0),
    xlabel='Sorted training inputs', ylabel='Sorted testing inputs')
2021-07-24T07:35:56.238256 image/svg+xml Matplotlib v3.4.0rc1, https://matplotlib.org/