import torch
from torch import nn
from d2l import torch as d2l
生成数据集
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
50
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)
非参数注意力汇聚
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)
注意力权重
d2l.show_heatmaps(
attention_weights.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs', ylabel='Sorted testing inputs')
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 $(n,a,c)$
X = torch.ones((2, 1, 4))
Y = torch.ones((2, 4, 6))
torch.bmm(X, Y).shape
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
weights = torch.ones((2, 10)) * 0.1
values = torch.arange(20.0).reshape((2, 10))
torch.bmm(weights.unsqueeze(1), values.unsqueeze(-1))
tensor([[[ 4.5000]], [[14.5000]]])
带参数的注意力汇聚
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
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
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()))
预测结果绘制
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)
曲线在注意力权重较大的区域变得更不平滑
d2l.show_heatmaps(
net.attention_weights.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs', ylabel='Sorted testing inputs')