4.2. 多层感知机的从零开始实现¶

Open the notebook in Colab

Open the notebook in Colab

Open the notebook in Colab

Open the notebook in Colab

Open the notebook in SageMaker Studio Lab

我们已经在 4.1节中描述了多层感知机（MLP），现在让我们尝试自己实现一个多层感知机。为了与之前softmax回归（ 3.6节）获得的结果进行比较，我们将继续使用Fashion-MNIST图像分类数据集（ 3.5节）。

from mxnet import gluon, np, npx
from d2l import mxnet as d2l

npx.set_np()

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

[07:19:40] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU

import torch
from torch import nn
from d2l import torch as d2l

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

import tensorflow as tf
from d2l import tensorflow as d2l

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

import warnings
from d2l import paddle as d2l

warnings.filterwarnings("ignore")
import paddle
from paddle import nn

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

4.2.1. 初始化模型参数¶

回想一下，Fashion-MNIST中的每个图像由 \(28 \times 28 = 784\)个灰度像素值组成。所有图像共分为10个类别。忽略像素之间的空间结构，我们可以将每个图像视为具有784个输入特征和10个类的简单分类数据集。首先，我们将实现一个具有单隐藏层的多层感知机，它包含256个隐藏单元。注意，我们可以将这两个变量都视为超参数。通常，我们选择2的若干次幂作为层的宽度。因为内存在硬件中的分配和寻址方式，这么做往往可以在计算上更高效。

我们用几个张量来表示我们的参数。注意，对于每一层我们都要记录一个权重矩阵和一个偏置向量。跟以前一样，我们要为损失关于这些参数的梯度分配内存。

mxnet pytorch tensorflow paddle

num_inputs, num_outputs, num_hiddens = 784, 10, 256

W1 = np.random.normal(scale=0.01, size=(num_inputs, num_hiddens))
b1 = np.zeros(num_hiddens)
W2 = np.random.normal(scale=0.01, size=(num_hiddens, num_outputs))
b2 = np.zeros(num_outputs)
params = [W1, b1, W2, b2]

for param in params:
    param.attach_grad()

num_inputs, num_outputs, num_hiddens = 784, 10, 256

W1 = nn.Parameter(torch.randn(
    num_inputs, num_hiddens, requires_grad=True) * 0.01)
b1 = nn.Parameter(torch.zeros(num_hiddens, requires_grad=True))
W2 = nn.Parameter(torch.randn(
    num_hiddens, num_outputs, requires_grad=True) * 0.01)
b2 = nn.Parameter(torch.zeros(num_outputs, requires_grad=True))

params = [W1, b1, W2, b2]

num_inputs, num_outputs, num_hiddens = 784, 10, 256

W1 = tf.Variable(tf.random.normal(
    shape=(num_inputs, num_hiddens), mean=0, stddev=0.01))
b1 = tf.Variable(tf.zeros(num_hiddens))
W2 = tf.Variable(tf.random.normal(
    shape=(num_hiddens, num_outputs), mean=0, stddev=0.01))
b2 = tf.Variable(tf.zeros(num_outputs))

params = [W1, b1, W2, b2]

num_inputs, num_outputs, num_hiddens = 784, 10, 256

W1 = paddle.randn([num_inputs, num_hiddens]) * 0.01
W1.stop_gradient = False
b1 = paddle.zeros([num_hiddens])
b1.stop_gradient = False
W2 = paddle.randn([num_hiddens, num_outputs]) * 0.01
W2.stop_gradient = False
b2 = paddle.zeros([num_outputs])
b2.stop_gradient = False

params = [W1, b1, W2, b2]

4.2.2. 激活函数¶

为了确保我们对模型的细节了如指掌，我们将实现ReLU激活函数，而不是直接调用内置的relu函数。

mxnet pytorch tensorflow paddle

def relu(X):
    return np.maximum(X, 0)

def relu(X):
    a = torch.zeros_like(X)
    return torch.max(X, a)

def relu(X):
    return tf.math.maximum(X, 0)

def relu(X):
    a = paddle.zeros_like(X)
    return paddle.maximum(X, a)

4.2.3. 模型¶

因为我们忽略了空间结构，所以我们使用reshape将每个二维图像转换为一个长度为num_inputs的向量。只需几行代码就可以实现我们的模型。

mxnet pytorch tensorflow paddle

def net(X):
    X = X.reshape((-1, num_inputs))
    H = relu(np.dot(X, W1) + b1)
    return np.dot(H, W2) + b2

def net(X):
    X = X.reshape((-1, num_inputs))
    H = relu(X@W1 + b1)  # 这里“@”代表矩阵乘法
    return (H@W2 + b2)

def net(X):
    X = tf.reshape(X, (-1, num_inputs))
    H = relu(tf.matmul(X, W1) + b1)
    return tf.matmul(H, W2) + b2

def net(X):
    X = X.reshape((-1, num_inputs))
    H = relu(X@W1 + b1)  # 这里“@”代表矩阵乘法
    return (H@W2 + b2)

4.2.4. 损失函数¶

由于我们已经从零实现过softmax函数（ 3.6节），因此在这里我们直接使用高级API中的内置函数来计算softmax和交叉熵损失。回想一下我们之前在 3.7.2节中对这些复杂问题的讨论。我们鼓励感兴趣的读者查看损失函数的源代码，以加深对实现细节的了解。

mxnet pytorch tensorflow paddle

loss = gluon.loss.SoftmaxCrossEntropyLoss()

loss = nn.CrossEntropyLoss(reduction='none')

def loss(y_hat, y):
    return tf.losses.sparse_categorical_crossentropy(
        y, y_hat, from_logits=True)

loss = nn.CrossEntropyLoss(reduction='none')

4.2.5. 训练¶

幸运的是，多层感知机的训练过程与softmax回归的训练过程完全相同。可以直接调用d2l包的train_ch3函数（参见 3.6节），将迭代周期数设置为10，并将学习率设置为0.1.

mxnet pytorch tensorflow paddle

num_epochs, lr = 10, 0.1
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs,
              lambda batch_size: d2l.sgd(params, lr, batch_size))

../_images/output_mlp-scratch_106d07_78_0.svg

num_epochs, lr = 10, 0.1
updater = torch.optim.SGD(params, lr=lr)
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, updater)

../_images/output_mlp-scratch_106d07_81_0.svg

num_epochs, lr = 10, 0.1
updater = d2l.Updater([W1, W2, b1, b2], lr)
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, updater)

../_images/output_mlp-scratch_106d07_84_0.svg

num_epochs, lr = 10, 0.1
updater = paddle.optimizer.SGD(learning_rate=lr, parameters=params)
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, updater)