跟着示例学习 PyTorch

Author: Justin Johnson

本教程通过自包含示例介绍了 PyTorch 的基本概念。

在其核心，PyTorch提供了两个主要功能：

• 一个n维张量，类似于numpy，但可以在GPU上运行

• 用于构建和训练神经网络的自动微分

我们将使用拟合 \(y=\sin(x)\) 的问题作为示例。网络将具有四个参数，并将使用梯度下降法训练，通过最小化网络输出与真实输出之间的欧几里得距离(Euclidean distance)来拟合随机数据。

Note

可在 本文末尾处 查看示例。

张量

热身：numpy

在介绍 PyTorch 之前，我们将先使用 numpy 来实现网络。

Numpy提供了一个n维数组对象，并提供了许多用于操作这些数组的函数。Numpy是一个通用的科学计算框架；它不知道任何关于计算图、深度学习或梯度的信息。 然而，我们可以很容易地使用 numpy 提供的方法，手动实现前向和后向传播过程，来拟合一个三次多项式到正弦函数：

# -*- coding: utf-8 -*-
import numpy as np
import math

# Create random input and output data
x = np.linspace(-math.pi, math.pi, 2000)
y = np.sin(x)

# Randomly initialize weights
a = np.random.randn()
b = np.random.randn()
c = np.random.randn()
d = np.random.randn()

learning_rate = 1e-6
for t in range(2000):
    # Forward pass: compute predicted y
    # y = a + b x + c x^2 + d x^3
    y_pred = a + b * x + c * x ** 2 + d * x ** 3

    # Compute and print loss
    loss = np.square(y_pred - y).sum()
    if t % 100 == 99:
        print(t, loss)

    # Backprop to compute gradients of a, b, c, d with respect to loss
    grad_y_pred = 2.0 * (y_pred - y)
    grad_a = grad_y_pred.sum()
    grad_b = (grad_y_pred * x).sum()
    grad_c = (grad_y_pred * x ** 2).sum()
    grad_d = (grad_y_pred * x ** 3).sum()

    # Update weights
    a -= learning_rate * grad_a
    b -= learning_rate * grad_b
    c -= learning_rate * grad_c
    d -= learning_rate * grad_d

print(f'Result: y = {a} + {b} x + {c} x^2 + {d} x^3')

PyTorch：张量

Numpy是一个很棒的框架，但它不能利用GPU来加速其数值计算。对于现代深度神经网络，GPU通常提供 50倍或更大的加速， 因此，numpy对于现代深度学习来说还是不够的。

在这里，我们介绍了PyTorch最基本的概念： 张量。 一个PyTorch张量在概念上与numpy数组相同：一个n维数组， PyTorch提供了许多操作这些张量的函数，可以自动跟踪计算图和梯度，它们也作为科学计算的通用工具非常有用。

与 numpy 不同，PyTorch 张量可以利用GPU来加速它们的数值计算。要在GPU上运行 PyTorch 张量，您只需要指定正确的设备。

在这里，我们使用 PyTorch 张量来拟合一个三次多项式到正弦函数中。 与上面的numpy示例类似，我们需要手动实现网络的前向和后向传递：

# -*- coding: utf-8 -*-

import torch
import math


dtype = torch.float
device = torch.device("cpu")
# device = torch.device("cuda:0") # Uncomment this to run on GPU

# Create random input and output data
x = torch.linspace(-math.pi, math.pi, 2000, device=device, dtype=dtype)
y = torch.sin(x)

# Randomly initialize weights
a = torch.randn((), device=device, dtype=dtype)
b = torch.randn((), device=device, dtype=dtype)
c = torch.randn((), device=device, dtype=dtype)
d = torch.randn((), device=device, dtype=dtype)

learning_rate = 1e-6
for t in range(2000):
    # Forward pass: compute predicted y
    y_pred = a + b * x + c * x ** 2 + d * x ** 3

    # Compute and print loss
    loss = (y_pred - y).pow(2).sum().item()
    if t % 100 == 99:
        print(t, loss)

    # Backprop to compute gradients of a, b, c, d with respect to loss
    grad_y_pred = 2.0 * (y_pred - y)
    grad_a = grad_y_pred.sum()
    grad_b = (grad_y_pred * x).sum()
    grad_c = (grad_y_pred * x ** 2).sum()
    grad_d = (grad_y_pred * x ** 3).sum()

    # Update weights using gradient descent
    a -= learning_rate * grad_a
    b -= learning_rate * grad_b
    c -= learning_rate * grad_c
    d -= learning_rate * grad_d


print(f'Result: y = {a.item()} + {b.item()} x + {c.item()} x^2 + {d.item()} x^3')

自动求导

PyTorch：张量和自动求导

在上面的示例中，我们必须手动实现神经网络的前向和后向传递。对于一个小型的两层网络来说，手动实现后向传递并不是什么大问题，但对于大型复杂的网络来说，很快就会变得非常麻烦。

幸运的是，我们可以使用 自动微分 来自动计算神经网络中的后向传递。 PyTorch中的 autograd 包正是提供了这种功能。当使用自动求导时，网络的前向传递将定义一个 计算图 ；图中的节点是张量，边是从输入张量生成输出张量的函数。通过这个图进行反向传播，然后可以轻松计算梯度。

这听起来很复杂，但在实际使用中非常简单。每个张量代表计算图中的一个节点。如果 x 是一个设置了 x.requires_grad=True 的张量，那么 x.grad 将是另一个张量，它包含了 x 相对于某个标量值的梯度。

在这里，我们使用PyTorch张量和自动求导来实现我们用三次多项式拟合正弦波的示例；现在我们不再需要手动实现网络的后向传递：

# -*- coding: utf-8 -*-
import torch
import math

dtype = torch.float
device = "cuda" if torch.cuda.is_available() else "cpu"
torch.set_default_device(device)

# Create Tensors to hold input and outputs.
# By default, requires_grad=False, which indicates that we do not need to
# compute gradients with respect to these Tensors during the backward pass.
x = torch.linspace(-math.pi, math.pi, 2000, dtype=dtype)
y = torch.sin(x)

# Create random Tensors for weights. For a third order polynomial, we need
# 4 weights: y = a + b x + c x^2 + d x^3
# Setting requires_grad=True indicates that we want to compute gradients with
# respect to these Tensors during the backward pass.
a = torch.randn((), dtype=dtype, requires_grad=True)
b = torch.randn((), dtype=dtype, requires_grad=True)
c = torch.randn((), dtype=dtype, requires_grad=True)
d = torch.randn((), dtype=dtype, requires_grad=True)

learning_rate = 1e-6
for t in range(2000):
    # Forward pass: compute predicted y using operations on Tensors.
    y_pred = a + b * x + c * x ** 2 + d * x ** 3

    # Compute and print loss using operations on Tensors.
    # Now loss is a Tensor of shape (1,)
    # loss.item() gets the scalar value held in the loss.
    loss = (y_pred - y).pow(2).sum()
    if t % 100 == 99:
        print(t, loss.item())

    # Use autograd to compute the backward pass. This call will compute the
    # gradient of loss with respect to all Tensors with requires_grad=True.
    # After this call a.grad, b.grad. c.grad and d.grad will be Tensors holding
    # the gradient of the loss with respect to a, b, c, d respectively.
    loss.backward()

    # Manually update weights using gradient descent. Wrap in torch.no_grad()
    # because weights have requires_grad=True, but we don't need to track this
    # in autograd.
    with torch.no_grad():
        a -= learning_rate * a.grad
        b -= learning_rate * b.grad
        c -= learning_rate * c.grad
        d -= learning_rate * d.grad

        # Manually zero the gradients after updating weights
        a.grad = None
        b.grad = None
        c.grad = None
        d.grad = None

print(f'Result: y = {a.item()} + {b.item()} x + {c.item()} x^2 + {d.item()} x^3')

PyTorch：定义新的自动求导函数

在底层，每个原始的自动求导操作实际上是对张量进行操作的两个函数。**前向**函数从输入张量计算输出张量。**后向**函数接收输出张量相对于某个标量值的梯度，并计算输入张量相对于同一标量值的梯度。

在PyTorch中，我们可以通过定义一个``torch.autograd.Function``的子类并实现``forward``和``backward``函数，轻松定义自己的自动求导操作符。然后，我们可以通过构造实例并像函数一样调用它，传递包含输入数据的张量，来使用我们新的自动求导操作符。

在这个示例中，我们将模型定义为:math:y=a+b P_3(c+dx)`而不是:math:`y=a+bx+cx^2+dx^3，其中:math:`P_3(x)=\frac{1}{2}\left(5x^3-3x\right)`是三阶的`勒让德多项式`_。我们编写了自己的自定义自动求导函数来计算:math:`P_3`的前向和后向传递，并使用它来实现我们的模型：

# -*- coding: utf-8 -*-
import torch
import math


class LegendrePolynomial3(torch.autograd.Function):
    """
    We can implement our own custom autograd Functions by subclassing
    torch.autograd.Function and implementing the forward and backward passes
    which operate on Tensors.
    """

    @staticmethod
    def forward(ctx, input):
        """
        In the forward pass we receive a Tensor containing the input and return
        a Tensor containing the output. ctx is a context object that can be used
        to stash information for backward computation. You can cache arbitrary
        objects for use in the backward pass using the ctx.save_for_backward method.
        """
        ctx.save_for_backward(input)
        return 0.5 * (5 * input ** 3 - 3 * input)

    @staticmethod
    def backward(ctx, grad_output):
        """
        In the backward pass we receive a Tensor containing the gradient of the loss
        with respect to the output, and we need to compute the gradient of the loss
        with respect to the input.
        """
        input, = ctx.saved_tensors
        return grad_output * 1.5 * (5 * input ** 2 - 1)


dtype = torch.float
device = torch.device("cpu")
# device = torch.device("cuda:0")  # Uncomment this to run on GPU

# Create Tensors to hold input and outputs.
# By default, requires_grad=False, which indicates that we do not need to
# compute gradients with respect to these Tensors during the backward pass.
x = torch.linspace(-math.pi, math.pi, 2000, device=device, dtype=dtype)
y = torch.sin(x)

# Create random Tensors for weights. For this example, we need
# 4 weights: y = a + b * P3(c + d * x), these weights need to be initialized
# not too far from the correct result to ensure convergence.
# Setting requires_grad=True indicates that we want to compute gradients with
# respect to these Tensors during the backward pass.
a = torch.full((), 0.0, device=device, dtype=dtype, requires_grad=True)
b = torch.full((), -1.0, device=device, dtype=dtype, requires_grad=True)
c = torch.full((), 0.0, device=device, dtype=dtype, requires_grad=True)
d = torch.full((), 0.3, device=device, dtype=dtype, requires_grad=True)

learning_rate = 5e-6
for t in range(2000):
    # To apply our Function, we use Function.apply method. We alias this as 'P3'.
    P3 = LegendrePolynomial3.apply

    # Forward pass: compute predicted y using operations; we compute
    # P3 using our custom autograd operation.
    y_pred = a + b * P3(c + d * x)

    # Compute and print loss
    loss = (y_pred - y).pow(2).sum()
    if t % 100 == 99:
        print(t, loss.item())

    # Use autograd to compute the backward pass.
    loss.backward()

    # Update weights using gradient descent
    with torch.no_grad():
        a -= learning_rate * a.grad
        b -= learning_rate * b.grad
        c -= learning_rate * c.grad
        d -= learning_rate * d.grad

        # Manually zero the gradients after updating weights
        a.grad = None
        b.grad = None
        c.grad = None
        d.grad = None

print(f'Result: y = {a.item()} + {b.item()} * P3({c.item()} + {d.item()} x)')

nn module

PyTorch: nn

计算图(Computational graphs) 和自动求导(autograd) 是定义复杂操作非常强大的功能；然而，原始的自动求导还是不足以支持实现大型神经网络。

在构建神经网络时，我们通常会考虑将计算安排成 层(layers)，其中一些层具有 可学习的参数(learnable parameters)，这些参数将在学习过程中进行优化。

在TensorFlow中，像 Keras、 TensorFlow-Slim 和 TFLearn 提供了相较于原始计算图的更高层次的抽象，这些抽象对于构建神经网络非常有用。

在PyTorch中， ``nn``包起到了同样的作用。 ``nn``包定义了一组 模块，这些模块相当于神经网络层。一个模块接收输入张量并计算输出张量， 但也可以包含内部状态，例如包含可学习参数的张量。``nn``包还定义了一组常用于训练神经网络的有用的损失函数(loss functions)。

在这个示例中，我们使用 nn 包来实现我们的多项式模型网络：

# -*- coding: utf-8 -*-
import torch
import math


# Create Tensors to hold input and outputs.
x = torch.linspace(-math.pi, math.pi, 2000)
y = torch.sin(x)

# For this example, the output y is a linear function of (x, x^2, x^3), so
# we can consider it as a linear layer neural network. Let's prepare the
# tensor (x, x^2, x^3).
p = torch.tensor([1, 2, 3])
xx = x.unsqueeze(-1).pow(p)

# In the above code, x.unsqueeze(-1) has shape (2000, 1), and p has shape
# (3,), for this case, broadcasting semantics will apply to obtain a tensor
# of shape (2000, 3) 

# Use the nn package to define our model as a sequence of layers. nn.Sequential
# is a Module which contains other Modules, and applies them in sequence to
# produce its output. The Linear Module computes output from input using a
# linear function, and holds internal Tensors for its weight and bias.
# The Flatten layer flatens the output of the linear layer to a 1D tensor,
# to match the shape of `y`.
model = torch.nn.Sequential(
    torch.nn.Linear(3, 1),
    torch.nn.Flatten(0, 1)
)

# The nn package also contains definitions of popular loss functions; in this
# case we will use Mean Squared Error (MSE) as our loss function.
loss_fn = torch.nn.MSELoss(reduction='sum')

learning_rate = 1e-6
for t in range(2000):

    # Forward pass: compute predicted y by passing x to the model. Module objects
    # override the __call__ operator so you can call them like functions. When
    # doing so you pass a Tensor of input data to the Module and it produces
    # a Tensor of output data.
    y_pred = model(xx)

    # Compute and print loss. We pass Tensors containing the predicted and true
    # values of y, and the loss function returns a Tensor containing the
    # loss.
    loss = loss_fn(y_pred, y)
    if t % 100 == 99:
        print(t, loss.item())

    # Zero the gradients before running the backward pass.
    model.zero_grad()

    # Backward pass: compute gradient of the loss with respect to all the learnable
    # parameters of the model. Internally, the parameters of each Module are stored
    # in Tensors with requires_grad=True, so this call will compute gradients for
    # all learnable parameters in the model.
    loss.backward()

    # Update the weights using gradient descent. Each parameter is a Tensor, so
    # we can access its gradients like we did before.
    with torch.no_grad():
        for param in model.parameters():
            param -= learning_rate * param.grad

# You can access the first layer of `model` like accessing the first item of a list
linear_layer = model[0]

# For linear layer, its parameters are stored as `weight` and `bias`.
print(f'Result: y = {linear_layer.bias.item()} + {linear_layer.weight[:, 0].item()} x + {linear_layer.weight[:, 1].item()} x^2 + {linear_layer.weight[:, 2].item()} x^3')

PyTorch: optim

我们通过使用 torch.no_grad() 手动更改张量的可学习参数，来更新模型的权重。 对于像随机梯度下降这样的优化算法来说，这并不是一个很大的负担，但在实践中，我们经常使用更复杂的优化器来训练神经网络， 比如 AdaGrad、 RMSProp、 Adam 等。

PyTorch中的 optim 包抽象了优化算法的定义，并提供了常用优化算法的实现。

在这个示例中，我们将像以前一样使用 nn``包来定义我们的模型，但我们将使用 ``optim 包提供的 RMSprop 算法来优化模型：

# -*- coding: utf-8 -*-
import torch
import math


# Create Tensors to hold input and outputs.
x = torch.linspace(-math.pi, math.pi, 2000)
y = torch.sin(x)

# Prepare the input tensor (x, x^2, x^3).
p = torch.tensor([1, 2, 3])
xx = x.unsqueeze(-1).pow(p)

# Use the nn package to define our model and loss function.
model = torch.nn.Sequential(
    torch.nn.Linear(3, 1),
    torch.nn.Flatten(0, 1)
)
loss_fn = torch.nn.MSELoss(reduction='sum')

# Use the optim package to define an Optimizer that will update the weights of
# the model for us. Here we will use RMSprop; the optim package contains many other
# optimization algorithms. The first argument to the RMSprop constructor tells the
# optimizer which Tensors it should update.
learning_rate = 1e-3
optimizer = torch.optim.RMSprop(model.parameters(), lr=learning_rate)
for t in range(2000):
    # Forward pass: compute predicted y by passing x to the model.
    y_pred = model(xx)

    # Compute and print loss.
    loss = loss_fn(y_pred, y)
    if t % 100 == 99:
        print(t, loss.item())

    # Before the backward pass, use the optimizer object to zero all of the
    # gradients for the variables it will update (which are the learnable
    # weights of the model). This is because by default, gradients are
    # accumulated in buffers( i.e, not overwritten) whenever .backward()
    # is called. Checkout docs of torch.autograd.backward for more details.
    optimizer.zero_grad()

    # Backward pass: compute gradient of the loss with respect to model
    # parameters
    loss.backward()

    # Calling the step function on an Optimizer makes an update to its
    # parameters
    optimizer.step()


linear_layer = model[0]
print(f'Result: y = {linear_layer.bias.item()} + {linear_layer.weight[:, 0].item()} x + {linear_layer.weight[:, 1].item()} x^2 + {linear_layer.weight[:, 2].item()} x^3')

PyTorch: 自定义 nn 模块

有时你可能会希望自定义比现有模块集更复杂的模型；在这些情况下，你可以通过继承 nn.Module 并定义一个 forward 方法来自定义模块， 该方法接收输入张量，并使用其他模块或在张量上自动求导等操作生成新的输出张量。

在这个示例中，我们将实现一个三次多项式作为自定义模块的子类：

# -*- coding: utf-8 -*-
import torch
import math


class Polynomial3(torch.nn.Module):
    def __init__(self):
        """
        In the constructor we instantiate four parameters and assign them as
        member parameters.
        """
        super().__init__()
        self.a = torch.nn.Parameter(torch.randn(()))
        self.b = torch.nn.Parameter(torch.randn(()))
        self.c = torch.nn.Parameter(torch.randn(()))
        self.d = torch.nn.Parameter(torch.randn(()))

    def forward(self, x):
        """
        In the forward function we accept a Tensor of input data and we must return
        a Tensor of output data. We can use Modules defined in the constructor as
        well as arbitrary operators on Tensors.
        """
        return self.a + self.b * x + self.c * x ** 2 + self.d * x ** 3

    def string(self):
        """
        Just like any class in Python, you can also define custom method on PyTorch modules
        """
        return f'y = {self.a.item()} + {self.b.item()} x + {self.c.item()} x^2 + {self.d.item()} x^3'


# Create Tensors to hold input and outputs.
x = torch.linspace(-math.pi, math.pi, 2000)
y = torch.sin(x)

# Construct our model by instantiating the class defined above
model = Polynomial3()

# Construct our loss function and an Optimizer. The call to model.parameters()
# in the SGD constructor will contain the learnable parameters (defined 
# with torch.nn.Parameter) which are members of the model.
criterion = torch.nn.MSELoss(reduction='sum')
optimizer = torch.optim.SGD(model.parameters(), lr=1e-6)
for t in range(2000):
    # Forward pass: Compute predicted y by passing x to the model
    y_pred = model(x)

    # Compute and print loss
    loss = criterion(y_pred, y)
    if t % 100 == 99:
        print(t, loss.item())

    # Zero gradients, perform a backward pass, and update the weights.
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

print(f'Result: {model.string()}')

PyTorch: 控制流 + 权重共享

作为动态计算图和权重共享的一个示例，我们实现了一个非常奇特的模型：一个三至五阶的多项式，在每次前向传递时随机选择一个3到5之间的数字， 并使用该阶数多项式来计算，重复使用相同的权重多次以计算四阶和五阶多项式。

对于这个模型，我们可以使用Python流控制来实现循环，并且可以通过在定义前向传递时，多次重复使用相同的参数，来实现权重共享。

我们可以很容易地将这个模型实现为一个模块的子类：

# -*- coding: utf-8 -*-
import random
import torch
import math


class DynamicNet(torch.nn.Module):
    def __init__(self):
        """
        In the constructor we instantiate five parameters and assign them as members.
        """
        super().__init__()
        self.a = torch.nn.Parameter(torch.randn(()))
        self.b = torch.nn.Parameter(torch.randn(()))
        self.c = torch.nn.Parameter(torch.randn(()))
        self.d = torch.nn.Parameter(torch.randn(()))
        self.e = torch.nn.Parameter(torch.randn(()))

    def forward(self, x):
        """
        For the forward pass of the model, we randomly choose either 4, 5
        and reuse the e parameter to compute the contribution of these orders.

        Since each forward pass builds a dynamic computation graph, we can use normal
        Python control-flow operators like loops or conditional statements when
        defining the forward pass of the model.

        Here we also see that it is perfectly safe to reuse the same parameter many
        times when defining a computational graph.
        """
        y = self.a + self.b * x + self.c * x ** 2 + self.d * x ** 3
        for exp in range(4, random.randint(4, 6)):
            y = y + self.e * x ** exp
        return y

    def string(self):
        """
        Just like any class in Python, you can also define custom method on PyTorch modules
        """
        return f'y = {self.a.item()} + {self.b.item()} x + {self.c.item()} x^2 + {self.d.item()} x^3 + {self.e.item()} x^4 ? + {self.e.item()} x^5 ?'


# Create Tensors to hold input and outputs.
x = torch.linspace(-math.pi, math.pi, 2000)
y = torch.sin(x)

# Construct our model by instantiating the class defined above
model = DynamicNet()

# Construct our loss function and an Optimizer. Training this strange model with
# vanilla stochastic gradient descent is tough, so we use momentum
criterion = torch.nn.MSELoss(reduction='sum')
optimizer = torch.optim.SGD(model.parameters(), lr=1e-8, momentum=0.9)
for t in range(30000):
    # Forward pass: Compute predicted y by passing x to the model
    y_pred = model(x)

    # Compute and print loss
    loss = criterion(y_pred, y)
    if t % 2000 == 1999:
        print(t, loss.item())

    # Zero gradients, perform a backward pass, and update the weights.
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

print(f'Result: {model.string()}')

示例

具体示例如下

Tensors

Autograd

nn module