.. _sec_nadaraya-waston:
*Pooling* de Atenção: Regressão de Kernel de Nadaraya-Watson
============================================================
Agora você conhece os principais componentes dos mecanismos de atenção
sob a estrutura em :numref:`fig_qkv`. Para recapitular, as interações
entre consultas (dicas volitivas) e chaves (dicas não volitivas)
resultam em *concentração de atenção*. O *pooling* de atenção agrega
valores seletivamente (entradas sensoriais) para produzir a saída. Nesta
secção, vamos descrever o agrupamento de atenção em mais detalhes para
lhe dar uma visão de alto nível como os mecanismos de atenção funcionam
na prática. Especificamente, o modelo de regressão do kernel de
Nadaraya-Watson proposto em 1964 é um exemplo simples, mas completo para
demonstrar o aprendizado de máquina com mecanismos de atenção.
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from mxnet import autograd, gluon, np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l
npx.set_np()
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import torch
from torch import nn
from d2l import torch as d2l
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n_train = 50 # No. of training examples
x_train = np.sort(np.random.rand(n_train) * 5) # Training inputs
def f(x):
return 2 * np.sin(x) + x**0.8
y_train = f(x_train) + np.random.normal(0.0, 0.5, (n_train,)) # Training outputs
x_test = np.arange(0, 5, 0.1) # Testing examples
y_truth = f(x_test) # Ground-truth outputs for the testing examples
n_test = len(x_test) # No. of testing examples
n_test
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50
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n_train = 50 # No. of training examples
x_train, _ = torch.sort(torch.rand(n_train) * 5) # Training inputs
def f(x):
return 2 * torch.sin(x) + x**0.8
y_train = f(x_train) + torch.normal(0.0, 0.5, (n_train,)) # Training outputs
x_test = torch.arange(0, 5, 0.1) # Testing examples
y_truth = f(x_test) # Ground-truth outputs for the testing examples
n_test = len(x_test) # No. of testing examples
n_test
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50
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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);
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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);
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y_hat = y_train.mean().repeat(n_test)
plot_kernel_reg(y_hat)
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y_hat = torch.repeat_interleave(y_train.mean(), n_test)
plot_kernel_reg(y_hat)
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# Shape of `X_repeat`: (`n_test`, `n_train`), where each row contains the
# same testing inputs (i.e., same queries)
X_repeat = x_test.repeat(n_train).reshape((-1, n_train))
# Note that `x_train` contains the keys. Shape of `attention_weights`:
# (`n_test`, `n_train`), where each row contains attention weights to be
# assigned among the values (`y_train`) given each query
attention_weights = npx.softmax(-(X_repeat - x_train)**2 / 2)
# Each element of `y_hat` is weighted average of values, where weights are
# attention weights
y_hat = np.dot(attention_weights, y_train)
plot_kernel_reg(y_hat)
.. figure:: output_nadaraya-waston_736177_39_0.svg
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# Shape of `X_repeat`: (`n_test`, `n_train`), where each row contains the
# same testing inputs (i.e., same queries)
X_repeat = x_test.repeat_interleave(n_train).reshape((-1, n_train))
# Note that `x_train` contains the keys. Shape of `attention_weights`:
# (`n_test`, `n_train`), where each row contains attention weights to be
# assigned among the values (`y_train`) given each query
attention_weights = nn.functional.softmax(-(X_repeat - x_train)**2 / 2, dim=1)
# Each element of `y_hat` is weighted average of values, where weights are
# attention weights
y_hat = torch.matmul(attention_weights, y_train)
plot_kernel_reg(y_hat)
.. figure:: output_nadaraya-waston_736177_42_0.svg
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d2l.show_heatmaps(np.expand_dims(np.expand_dims(attention_weights, 0), 0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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d2l.show_heatmaps(attention_weights.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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X = np.ones((2, 1, 4))
Y = np.ones((2, 4, 6))
npx.batch_dot(X, Y).shape
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(2, 1, 6)
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X = torch.ones((2, 1, 4))
Y = torch.ones((2, 4, 6))
torch.bmm(X, Y).shape
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torch.Size([2, 1, 6])
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weights = np.ones((2, 10)) * 0.1
values = np.arange(20).reshape((2, 10))
npx.batch_dot(np.expand_dims(weights, 1), np.expand_dims(values, -1))
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array([[[ 4.5]],
[[14.5]]])
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weights = torch.ones((2, 10)) * 0.1
values = torch.arange(20.0).reshape((2, 10))
torch.bmm(weights.unsqueeze(1), values.unsqueeze(-1))
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tensor([[[ 4.5000]],
[[14.5000]]])
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class NWKernelRegression(nn.Block):
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.w = self.params.get('w', shape=(1,))
def forward(self, queries, keys, values):
# Shape of the output `queries` and `attention_weights`:
# (no. of queries, no. of key-value pairs)
queries = queries.repeat(keys.shape[1]).reshape((-1, keys.shape[1]))
self.attention_weights = npx.softmax(
-((queries - keys) * self.w.data())**2 / 2)
# Shape of `values`: (no. of queries, no. of key-value pairs)
return npx.batch_dot(np.expand_dims(self.attention_weights, 1),
np.expand_dims(values, -1)).reshape(-1)
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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):
# Shape of the output `queries` and `attention_weights`:
# (no. of queries, no. of key-value pairs)
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)
# Shape of `values`: (no. of queries, no. of key-value pairs)
return torch.bmm(self.attention_weights.unsqueeze(1),
values.unsqueeze(-1)).reshape(-1)
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# Shape of `X_tile`: (`n_train`, `n_train`), where each column contains the
# same training inputs
X_tile = np.tile(x_train, (n_train, 1))
# Shape of `Y_tile`: (`n_train`, `n_train`), where each column contains the
# same training outputs
Y_tile = np.tile(y_train, (n_train, 1))
# Shape of `keys`: ('n_train', 'n_train' - 1)
keys = X_tile[(1 - np.eye(n_train)).astype('bool')].reshape((n_train, -1))
# Shape of `values`: ('n_train', 'n_train' - 1)
values = Y_tile[(1 - np.eye(n_train)).astype('bool')].reshape((n_train, -1))
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# Shape of `X_tile`: (`n_train`, `n_train`), where each column contains the
# same training inputs
X_tile = x_train.repeat((n_train, 1))
# Shape of `Y_tile`: (`n_train`, `n_train`), where each column contains the
# same training outputs
Y_tile = y_train.repeat((n_train, 1))
# Shape of `keys`: ('n_train', 'n_train' - 1)
keys = X_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape((n_train, -1))
# Shape of `values`: ('n_train', 'n_train' - 1)
values = Y_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape((n_train, -1))
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net = NWKernelRegression()
net.initialize()
loss = gluon.loss.L2Loss()
trainer = gluon.Trainer(net.collect_params(), 'sgd', {'learning_rate': 0.5})
animator = d2l.Animator(xlabel='epoch', ylabel='loss', xlim=[1, 5])
for epoch in range(5):
with autograd.record():
l = loss(net(x_train, keys, values), y_train)
l.backward()
trainer.step(1)
print(f'epoch {epoch + 1}, loss {float(l.sum()):.6f}')
animator.add(epoch + 1, float(l.sum()))
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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()
# Note: L2 Loss = 1/2 * MSE Loss. PyTorch has MSE Loss which is slightly
# different from MXNet's L2Loss by a factor of 2. Hence we halve the loss
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()))
.. figure:: output_nadaraya-waston_736177_96_0.svg
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# Shape of `keys`: (`n_test`, `n_train`), where each column contains the same
# training inputs (i.e., same keys)
keys = np.tile(x_train, (n_test, 1))
# Shape of `value`: (`n_test`, `n_train`)
values = np.tile(y_train, (n_test, 1))
y_hat = net(x_test, keys, values)
plot_kernel_reg(y_hat)
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# Shape of `keys`: (`n_test`, `n_train`), where each column contains the same
# training inputs (i.e., same keys)
keys = x_train.repeat((n_test, 1))
# Shape of `value`: (`n_test`, `n_train`)
values = y_train.repeat((n_test, 1))
y_hat = net(x_test, keys, values).unsqueeze(1).detach()
plot_kernel_reg(y_hat)
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d2l.show_heatmaps(np.expand_dims(np.expand_dims(net.attention_weights, 0), 0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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d2l.show_heatmaps(net.attention_weights.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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`Discussions `__
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`Discussions `__
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