.. _sec_linear_scratch:
线性回归的从零开始实现
======================
在了解线性回归的关键思想之后，我们可以开始通过代码来动手实现线性回归了。
在这一节中，我们将从零开始实现整个方法，
包括数据流水线、模型、损失函数和小批量随机梯度下降优化器。
虽然现代的深度学习框架几乎可以自动化地进行所有这些工作，但从零开始实现可以确保我们真正知道自己在做什么。
同时，了解更细致的工作原理将方便我们自定义模型、自定义层或自定义损失函数。
在这一节中，我们将只使用张量和自动求导。
在之后的章节中，我们会充分利用深度学习框架的优势，介绍更简洁的实现方式。
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\diilbookstyleinputcell
.. code:: python
%matplotlib inline
import random
from mxnet import autograd, np, npx
from d2l import mxnet as d2l
npx.set_np()
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\diilbookstyleinputcell
.. code:: python
%matplotlib inline
import random
import torch
from d2l import torch as d2l
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\diilbookstyleinputcell
.. code:: python
%matplotlib inline
import random
import tensorflow as tf
from d2l import tensorflow as d2l
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\diilbookstyleinputcell
.. code:: python
%matplotlib inline
import warnings
from d2l import paddle as d2l
warnings.filterwarnings("ignore")
import random
import paddle
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\diilbookstyleinputcell
.. code:: python
def synthetic_data(w, b, num_examples): #@save
"""生成y=Xw+b+噪声"""
X = np.random.normal(0, 1, (num_examples, len(w)))
y = np.dot(X, w) + b
y += np.random.normal(0, 0.01, y.shape)
return X, y.reshape((-1, 1))
true_w = np.array([2, -3.4])
true_b = 4.2
features, labels = synthetic_data(true_w, true_b, 1000)
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
[07:03:27] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU
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\diilbookstyleinputcell
.. code:: python
def synthetic_data(w, b, num_examples): #@save
"""生成y=Xw+b+噪声"""
X = torch.normal(0, 1, (num_examples, len(w)))
y = torch.matmul(X, w) + b
y += torch.normal(0, 0.01, y.shape)
return X, y.reshape((-1, 1))
true_w = torch.tensor([2, -3.4])
true_b = 4.2
features, labels = synthetic_data(true_w, true_b, 1000)
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\diilbookstyleinputcell
.. code:: python
def synthetic_data(w, b, num_examples): #@save
"""生成y=Xw+b+噪声"""
X = tf.zeros((num_examples, w.shape[0]))
X += tf.random.normal(shape=X.shape)
y = tf.matmul(X, tf.reshape(w, (-1, 1))) + b
y += tf.random.normal(shape=y.shape, stddev=0.01)
y = tf.reshape(y, (-1, 1))
return X, y
true_w = tf.constant([2, -3.4])
true_b = 4.2
features, labels = synthetic_data(true_w, true_b, 1000)
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\diilbookstyleinputcell
.. code:: python
def synthetic_data(w, b, num_examples): #@save
"""生成y=Xw+b+噪声"""
X = paddle.normal(0, 1, (num_examples, len(w)))
y = paddle.matmul(X, w) + b
y += paddle.normal(0, 0.01, y.shape)
return X, y.reshape((-1, 1))
true_w = paddle.to_tensor([2, -3.4])
true_b = 4.2
features, labels = synthetic_data(true_w, true_b, 1000)
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\diilbookstyleinputcell
.. code:: python
print('features:', features[0],'\nlabel:', labels[0])
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
features: [2.2122064 1.1630787]
label: [4.662078]
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\diilbookstyleinputcell
.. code:: python
print('features:', features[0],'\nlabel:', labels[0])
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
features: tensor([1.4632, 0.5511])
label: tensor([5.2498])
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\diilbookstyleinputcell
.. code:: python
print('features:', features[0],'\nlabel:', labels[0])
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
features: tf.Tensor([-0.07650175 0.6898775 ], shape=(2,), dtype=float32)
label: tf.Tensor([1.6964029], shape=(1,), dtype=float32)
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\diilbookstyleinputcell
.. code:: python
print('features:', features[0],'\nlabel:', labels[0])
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
features: Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.17713280, 0.61215985])
label: Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
[2.46172976])
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\diilbookstyleinputcell
.. code:: python
d2l.set_figsize()
d2l.plt.scatter(features[:, (1)].asnumpy(), labels.asnumpy(), 1);
.. figure:: output_linear-regression-scratch_58de05_48_0.svg
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\diilbookstyleinputcell
.. code:: python
d2l.set_figsize()
d2l.plt.scatter(features[:, (1)].detach().numpy(), labels.detach().numpy(), 1);
.. figure:: output_linear-regression-scratch_58de05_51_0.svg
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\diilbookstyleinputcell
.. code:: python
d2l.set_figsize()
d2l.plt.scatter(features[:, (1)].numpy(), labels.numpy(), 1);
.. figure:: output_linear-regression-scratch_58de05_54_0.svg
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\diilbookstyleinputcell
.. code:: python
d2l.set_figsize()
d2l.plt.scatter(features[:, (1)].detach().numpy(), labels.detach().numpy(), 1);
.. figure:: output_linear-regression-scratch_58de05_57_0.svg
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\diilbookstyleinputcell
.. code:: python
def data_iter(batch_size, features, labels):
num_examples = len(features)
indices = list(range(num_examples))
# 这些样本是随机读取的，没有特定的顺序
random.shuffle(indices)
for i in range(0, num_examples, batch_size):
batch_indices = np.array(
indices[i: min(i + batch_size, num_examples)])
yield features[batch_indices], labels[batch_indices]
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\diilbookstyleinputcell
.. code:: python
def data_iter(batch_size, features, labels):
num_examples = len(features)
indices = list(range(num_examples))
# 这些样本是随机读取的，没有特定的顺序
random.shuffle(indices)
for i in range(0, num_examples, batch_size):
batch_indices = torch.tensor(
indices[i: min(i + batch_size, num_examples)])
yield features[batch_indices], labels[batch_indices]
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\diilbookstyleinputcell
.. code:: python
def data_iter(batch_size, features, labels):
num_examples = len(features)
indices = list(range(num_examples))
# 这些样本是随机读取的，没有特定的顺序
random.shuffle(indices)
for i in range(0, num_examples, batch_size):
j = tf.constant(indices[i: min(i + batch_size, num_examples)])
yield tf.gather(features, j), tf.gather(labels, j)
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\diilbookstyleinputcell
.. code:: python
def data_iter(batch_size, features, labels):
num_examples = len(features)
indices = list(range(num_examples))
# 这些样本是随机读取的，没有特定的顺序
random.shuffle(indices)
for i in range(0, num_examples, batch_size):
batch_indices = paddle.to_tensor(
indices[i: min(i + batch_size, num_examples)])
yield features[batch_indices], labels[batch_indices]
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\diilbookstyleinputcell
.. code:: python
batch_size = 10
for X, y in data_iter(batch_size, features, labels):
print(X, '\n', y)
break
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
[[-0.594405 0.7975923 ]
[ 0.5477517 0.15074243]
[-0.34835348 0.929739 ]
[-1.5249145 0.701587 ]
[-2.298264 0.10911477]
[ 1.6356094 0.14286116]
[-0.19882555 -0.85171705]
[ 1.2024101 -1.7029836 ]
[-0.60534513 -0.39319903]
[-1.771029 -0.5459446 ]]
[[ 0.30207413]
[ 4.786745 ]
[ 0.33858034]
[-1.2297847 ]
[-0.75900215]
[ 6.979927 ]
[ 6.7001796 ]
[12.39533 ]
[ 4.3220677 ]
[ 2.517848 ]]
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\diilbookstyleinputcell
.. code:: python
batch_size = 10
for X, y in data_iter(batch_size, features, labels):
print(X, '\n', y)
break
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tensor([[ 0.3934, 2.5705],
[ 0.5849, -0.7124],
[ 0.1008, 0.6947],
[-0.4493, -0.9037],
[ 2.3104, -0.2798],
[-0.0173, -0.2552],
[ 0.1963, -0.5445],
[-1.0580, -0.5180],
[ 0.8417, -1.5547],
[-0.6316, 0.9732]])
tensor([[-3.7623],
[ 7.7852],
[ 2.0443],
[ 6.3767],
[ 9.7776],
[ 5.0301],
[ 6.4541],
[ 3.8407],
[11.1396],
[-0.3836]])
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
batch_size = 10
for X, y in data_iter(batch_size, features, labels):
print(X, '\n', y)
break
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tf.Tensor(
[[ 0.15857808 1.0263954 ]
[-0.2139615 0.41953972]
[ 0.82911646 -0.43722913]
[-1.4051182 -0.31091216]
[ 0.4279184 0.6564329 ]
[-1.6576649 -0.01954047]
[ 1.4908514 1.115273 ]
[-0.4216003 1.4159069 ]
[ 1.6962595 -0.6179955 ]
[-0.7782186 -0.07165246]], shape=(10, 2), dtype=float32)
tf.Tensor(
[[ 1.030544 ]
[ 2.330862 ]
[ 7.3499002]
[ 2.4424326]
[ 2.8114896]
[ 0.9454506]
[ 3.3814855]
[-1.4531627]
[ 9.692594 ]
[ 2.8894465]], shape=(10, 1), dtype=float32)
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\diilbookstyleinputcell
.. code:: python
batch_size = 10
for X, y in data_iter(batch_size, features, labels):
print(X, '\n', y)
break
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Tensor(shape=[10, 2], dtype=float32, place=Place(cpu), stop_gradient=True,
[[ 0.11829354, 0.57692450],
[-0.57479954, -0.09607600],
[-1.07854998, -0.25145546],
[-0.84905797, 0.67849380],
[-0.27850190, 0.56761360],
[-1.61706865, -0.68674469],
[ 1.58607817, 0.16652523],
[ 0.73623341, -0.98004287],
[ 1.20183730, 0.04473243],
[-0.38806343, -0.88529688]])
Tensor(shape=[10, 1], dtype=float32, place=Place(cpu), stop_gradient=True,
[[2.46672344],
[3.37174177],
[2.89900732],
[0.18522391],
[1.71305954],
[3.29967785],
[6.81773567],
[9.01074409],
[6.46295214],
[6.43930912]])
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\diilbookstyleinputcell
.. code:: python
w = np.random.normal(0, 0.01, (2, 1))
b = np.zeros(1)
w.attach_grad()
b.attach_grad()
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\diilbookstyleinputcell
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w = torch.normal(0, 0.01, size=(2,1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)
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\diilbookstyleinputcell
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w = tf.Variable(tf.random.normal(shape=(2, 1), mean=0, stddev=0.01),
trainable=True)
b = tf.Variable(tf.zeros(1), trainable=True)
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\diilbookstyleinputcell
.. code:: python
w = paddle.normal(0, 0.01, shape=(2,1))
b = paddle.zeros(shape=[1])
# w和b为创建的模型参数，stop_gradient默认为True，即梯度不更新，因此需要指定为False已更新梯度
w.stop_gradient = False
b.stop_gradient = False
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\diilbookstyleinputcell
.. code:: python
def linreg(X, w, b): #@save
"""线性回归模型"""
return np.dot(X, w) + b
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\diilbookstyleinputcell
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def linreg(X, w, b): #@save
"""线性回归模型"""
return torch.matmul(X, w) + b
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\diilbookstyleinputcell
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def linreg(X, w, b): #@save
"""线性回归模型"""
return tf.matmul(X, w) + b
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\diilbookstyleinputcell
.. code:: python
def linreg(X, w, b): #@save
"""线性回归模型"""
return paddle.matmul(X, w) + b
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\diilbookstyleinputcell
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def squared_loss(y_hat, y): #@save
"""均方损失"""
return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2
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\diilbookstyleinputcell
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def squared_loss(y_hat, y): #@save
"""均方损失"""
return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2
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\diilbookstyleinputcell
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def squared_loss(y_hat, y): #@save
"""均方损失"""
return (y_hat - tf.reshape(y, y_hat.shape)) ** 2 / 2
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\diilbookstyleinputcell
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def squared_loss(y_hat, y): #@save
"""均方损失"""
return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2
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\diilbookstyleinputcell
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def sgd(params, lr, batch_size): #@save
"""小批量随机梯度下降"""
for param in params:
param[:] = param - lr * param.grad / batch_size
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\diilbookstyleinputcell
.. code:: python
def sgd(params, lr, batch_size): #@save
"""小批量随机梯度下降"""
with torch.no_grad():
for param in params:
param -= lr * param.grad / batch_size
param.grad.zero_()
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\diilbookstyleinputcell
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def sgd(params, grads, lr, batch_size): #@save
"""小批量随机梯度下降"""
for param, grad in zip(params, grads):
param.assign_sub(lr*grad/batch_size)
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\diilbookstyleinputcell
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#@save
def sgd(params, lr, batch_size):
"""小批量随机梯度下降"""
with paddle.no_grad():
for i, param in enumerate(params):
param -= lr * params[i].grad / batch_size
params[i].set_value(param)
params[i].clear_gradient()
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\diilbookstyleinputcell
.. code:: python
lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss
for epoch in range(num_epochs):
for X, y in data_iter(batch_size, features, labels):
with autograd.record():
l = loss(net(X, w, b), y) # X和y的小批量损失
# 计算l关于[w,b]的梯度
l.backward()
sgd([w, b], lr, batch_size) # 使用参数的梯度更新参数
train_l = loss(net(features, w, b), labels)
print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
[07:03:28] ../src/base.cc:48: GPU context requested, but no GPUs found.
epoch 1, loss 0.024969
epoch 2, loss 0.000089
epoch 3, loss 0.000051
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\diilbookstyleinputcell
.. code:: python
lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss
for epoch in range(num_epochs):
for X, y in data_iter(batch_size, features, labels):
l = loss(net(X, w, b), y) # X和y的小批量损失
# 因为l形状是(batch_size,1)，而不是一个标量。l中的所有元素被加到一起，
# 并以此计算关于[w,b]的梯度
l.sum().backward()
sgd([w, b], lr, batch_size) # 使用参数的梯度更新参数
with torch.no_grad():
train_l = loss(net(features, w, b), labels)
print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
epoch 1, loss 0.042790
epoch 2, loss 0.000162
epoch 3, loss 0.000051
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\diilbookstyleinputcell
.. code:: python
lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss
for epoch in range(num_epochs):
for X, y in data_iter(batch_size, features, labels):
with tf.GradientTape() as g:
l = loss(net(X, w, b), y) # X和y的小批量损失
# 计算l关于[w,b]的梯度
dw, db = g.gradient(l, [w, b])
# 使用参数的梯度更新参数
sgd([w, b], [dw, db], lr, batch_size)
train_l = loss(net(features, w, b), labels)
print(f'epoch {epoch + 1}, loss {float(tf.reduce_mean(train_l)):f}')
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
epoch 1, loss 0.045469
epoch 2, loss 0.000220
epoch 3, loss 0.000050
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss
for epoch in range(num_epochs):
for X, y in data_iter(batch_size, features, labels):
l = loss(net(X, w, b), y) # X和y的小批量损失
# 因为l形状是(batch_size,1)，而不是一个标量。l中的所有元素被加到一起，
# 并以此计算关于[w,b]的梯度
l.sum().backward()
sgd([w, b], lr, batch_size) # 使用参数的梯度更新参数
with paddle.no_grad():
train_l = loss(net(features, w, b), labels)
print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
epoch 1, loss 0.037368
epoch 2, loss 0.000136
epoch 3, loss 0.000048
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
print(f'w的估计误差: {true_w - w.reshape(true_w.shape)}')
print(f'b的估计误差: {true_b - b}')
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
w的估计误差: [ 4.171133e-04 -7.724762e-05]
b的估计误差: [0.00061893]
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
print(f'w的估计误差: {true_w - w.reshape(true_w.shape)}')
print(f'b的估计误差: {true_b - b}')
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
w的估计误差: tensor([-1.3804e-04, 5.7936e-05], grad_fn=)
b的估计误差: tensor([0.0006], grad_fn=)
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
print(f'w的估计误差: {true_w - tf.reshape(w, true_w.shape)}')
print(f'b的估计误差: {true_b - b}')
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
w的估计误差: [0.00169134 0.00011921]
b的估计误差: [0.00055313]
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
print(f'w的估计误差: {true_w - w.reshape(true_w.shape)}')
print(f'b的估计误差: {true_b - b}')
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
w的估计误差: Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=False,
[ 0.00068414, -0.00082040])
b的估计误差: Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=False,
[0.00085068])
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`Discussions `__
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`Discussions `__
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`Discussions `__
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`Discussions `__
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