4.2. Implementation of Multilayer Perceptrons from Scratch
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Now that we have characterized multilayer perceptrons (MLPs) mathematically, let us try to implement one ourselves. To compare against our previous results achieved with softmax regression (Section 3.6), we will continue to work with the Fashion-MNIST image classification dataset (Section 3.5).

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


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

4.2.1. Initializing Model Parameters

Recall that Fashion-MNIST contains 10 classes, and that each image consists of a \(28 \times 28 = 784\) grid of grayscale pixel values. Again, we will disregard the spatial structure among the pixels for now, so we can think of this as simply a classification dataset with 784 input features and 10 classes. To begin, we will implement an MLP with one hidden layer and 256 hidden units. Note that we can regard both of these quantities as hyperparameters. Typically, we choose layer widths in powers of 2, which tend to be computationally efficient because of how memory is allocated and addressed in hardware.

Again, we will represent our parameters with several tensors. Note that for every layer, we must keep track of one weight matrix and one bias vector. As always, we allocate memory for the gradients of the loss with respect to these parameters.

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:
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.random.normal([num_outputs], stddev=.01))

params = [W1, b1, W2, b2]

4.2.2. Activation Function

To make sure we know how everything works, we will implement the ReLU activation ourselves using the maximum function rather than invoking the built-in relu function directly.

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)

4.2.3. Model

Because we are disregarding spatial structure, we reshape each two-dimensional image into a flat vector of length num_inputs. Finally, we implement our model with just a few lines of code.

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)  # Here '@' stands for matrix multiplication
    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

4.2.4. Loss Function

To ensure numerical stability, and because we already implemented the softmax function from scratch (Section 3.6), we leverage the integrated function from high-level APIs for calculating the softmax and cross-entropy loss. Recall our earlier discussion of these intricacies in Section 3.7.2. We encourage the interested reader to examine the source code for the loss function to deepen their knowledge of implementation details.

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)

4.2.5. Training

Fortunately, the training loop for MLPs is exactly the same as for softmax regression. Leveraging the d2l package again, we call the train_ch3 function (see Section 3.6), setting the number of epochs to 10 and the learning rate to 0.1.

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))
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)
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)

To evaluate the learned model, we apply it on some test data.

d2l.predict_ch3(net, test_iter)
d2l.predict_ch3(net, test_iter)
d2l.predict_ch3(net, test_iter)

4.2.6. Summary

  • We saw that implementing a simple MLP is easy, even when done manually.

  • However, with a large number of layers, implementing MLPs from scratch can still get messy (e.g., naming and keeping track of our model’s parameters).

4.2.7. Exercises

  1. Change the value of the hyperparameter num_hiddens and see how this hyperparameter influences your results. Determine the best value of this hyperparameter, keeping all others constant.

  2. Try adding an additional hidden layer to see how it affects the results.

  3. How does changing the learning rate alter your results? Fixing the model architecture and other hyperparameters (including number of epochs), what learning rate gives you the best results?

  4. What is the best result you can get by optimizing over all the hyperparameters (learning rate, number of epochs, number of hidden layers, number of hidden units per layer) jointly?

  5. Describe why it is much more challenging to deal with multiple hyperparameters.

  6. What is the smartest strategy you can think of for structuring a search over multiple hyperparameters?