4.2. Implementation of Multilayer Perceptrons from Scratch¶
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 npx.set_np() 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: 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.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
relu function directly.
Because we are disregarding spatial structure, we
two-dimensional image into a flat vector of length
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.
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.
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).
Change the value of the hyperparameter
num_hiddensand see how this hyperparameter influences your results. Determine the best value of this hyperparameter, keeping all others constant.
Try adding an additional hidden layer to see how it affects the results.
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?
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?
Describe why it is much more challenging to deal with multiple hyperparameters.
What is the smartest strategy you can think of for structuring a search over multiple hyperparameters?