6.6  Convolutional Neural Networks (LeNet)

We now have all the ingredients required to assemble a fully-functional CNN. In our earlier encounter with image data, we applied a linear model with softmax regression (Section 3.4) and an MLP (Section 4.2) to pictures of clothing in the Fashion-MNIST dataset. To make such data amenable we first flattened each image from a \(28\times28\) matrix into a fixed-length \(784\)-dimensional vector, and thereafter processed them in fully connected layers. Now that we have a handle on convolutional layers, we can retain the spatial structure in our images. As an additional benefit of replacing fully connected layers with convolutional layers, we will enjoy more parsimonious models that require far fewer parameters.

In this section, we will introduce LeNet, among the first published CNNs to capture wide attention for its performance on computer vision tasks. The model was introduced by (and named for) Yann LeCun, then a researcher at AT&T Bell Labs, for the purpose of recognizing handwritten digits in images (LeCun et al. 1998). This work represented the culmination of a decade of research developing the technology; LeCun’s team published the first study to successfully train CNNs via backpropagation (LeCun et al. 1989).

At the time LeNet achieved outstanding results matching the performance of support vector machines, then a dominant approach in supervised learning, achieving an error rate of less than 1% per digit. LeNet was eventually adapted to recognize digits for processing deposits in ATM machines. To this day, some ATMs still run the code that Yann LeCun and his colleague Leon Bottou wrote in the 1990s!

from d2l import torch as d2l
import torch
from torch import nn
import tensorflow as tf
from d2l import tensorflow as d2l
from d2l import jax as d2l
from flax import nnx
from jax import numpy as jnp
from d2l import mxnet as d2l
from mxnet import autograd, gluon, init, np, npx
from mxnet.gluon import nn
npx.set_np()

6.6.1 LeNet

At a high level, LeNet (LeNet-5) consists of two parts: (i) a convolutional encoder consisting of two convolutional layers; and (ii) a dense block consisting of three fully connected layers. The architecture is summarized in Figure 6.6.1.

Figure 6.6.1: Data flow in LeNet. The input is a handwritten digit, the output is a probability over 10 possible outcomes.

The basic units in each convolutional block are a convolutional layer, a sigmoid activation function, and a subsequent average pooling operation. ReLUs and max-pooling were not part of LeNet-5 and had not yet become the standard choices for trained CNNs. Each convolutional layer uses a \(5\times 5\) kernel and a sigmoid activation function. These layers map spatially arranged inputs to a number of two-dimensional feature maps, typically increasing the number of channels. The first convolutional layer has 6 output channels, while the second has 16. Each \(2\times2\) pooling operation (stride 2) reduces dimensionality by a factor of \(4\) via spatial downsampling. The convolutional block emits an output with shape (batch size, number of channels, height, width) in the PyTorch/MXNet convention; TensorFlow and JAX use the channels-last layout (batch, height, width, channels).

In order to pass output from the convolutional block to the dense block, we must flatten each example in the minibatch. In other words, we take this four-dimensional input and transform it into the two-dimensional input expected by fully connected layers: as a reminder, the two-dimensional representation that we desire uses the first dimension to index examples in the minibatch and the second to give the flat vector representation of each example. LeNet’s dense block has three fully connected layers, with 120, 84, and 10 outputs, respectively. Because we are still performing classification, the 10-dimensional output layer corresponds to the number of possible output classes.

While getting to the point where you truly understand what is going on inside LeNet may have taken a bit of work, we hope that the following code snippet will convince you that implementing such models with modern deep learning frameworks is remarkably simple. We need only to instantiate a Sequential block and chain together the appropriate layers, using Xavier initialization as introduced in Section 4.4.2.2.

def init_cnn(module):
    """Initialize weights for CNNs."""
    if type(module) == nn.Linear or type(module) == nn.Conv2d:
        nn.init.xavier_uniform_(module.weight)
class LeNet(d2l.Classifier):
    """The LeNet-5 model."""
    def __init__(self, lr=0.1, num_classes=10):
        super().__init__()
        self.save_hyperparameters()
        self.net = nn.Sequential(
            nn.LazyConv2d(6, kernel_size=5, padding=2), nn.Sigmoid(),
            nn.AvgPool2d(kernel_size=2, stride=2),
            nn.LazyConv2d(16, kernel_size=5), nn.Sigmoid(),
            nn.AvgPool2d(kernel_size=2, stride=2),
            nn.Flatten(),
            nn.LazyLinear(120), nn.Sigmoid(),
            nn.LazyLinear(84), nn.Sigmoid(),
            nn.LazyLinear(num_classes))
class LeNet(d2l.Classifier):
    """The LeNet-5 model."""
    def __init__(self, lr=0.1, num_classes=10):
        super().__init__()
        self.save_hyperparameters()
        self.net = tf.keras.models.Sequential([
            tf.keras.layers.Conv2D(filters=6, kernel_size=5,
                                   activation='sigmoid', padding='same'),
            tf.keras.layers.AvgPool2D(pool_size=2, strides=2),
            tf.keras.layers.Conv2D(filters=16, kernel_size=5,
                                   activation='sigmoid'),
            tf.keras.layers.AvgPool2D(pool_size=2, strides=2),
            tf.keras.layers.Flatten(),
            tf.keras.layers.Dense(120, activation='sigmoid'),
            tf.keras.layers.Dense(84, activation='sigmoid'),
            tf.keras.layers.Dense(num_classes)])
class LeNet(d2l.Classifier):
    """The LeNet-5 model."""
    def __init__(self, lr=0.1, num_classes=10, kernel_init=None, rngs=None):
        super().__init__()
        self.save_hyperparameters(ignore=['rngs', 'kernel_init'])
        rngs = nnx.Rngs(d2l.get_key()) if rngs is None else rngs
        kernel_init = (nnx.initializers.xavier_uniform() if kernel_init is None
                       else kernel_init)
        self.net = nnx.Sequential(
            nnx.Conv(1, 6, kernel_size=(5, 5), padding='SAME',
                     kernel_init=kernel_init, rngs=rngs),
            nnx.sigmoid,
            lambda x: nnx.avg_pool(x, window_shape=(2, 2), strides=(2, 2)),
            nnx.Conv(6, 16, kernel_size=(5, 5), padding='VALID',
                     kernel_init=kernel_init, rngs=rngs),
            nnx.sigmoid,
            lambda x: nnx.avg_pool(x, window_shape=(2, 2), strides=(2, 2)),
            lambda x: x.reshape((x.shape[0], -1)),  # flatten
            nnx.Linear(400, 120, kernel_init=kernel_init, rngs=rngs),
            nnx.sigmoid,
            nnx.Linear(120, 84, kernel_init=kernel_init, rngs=rngs),
            nnx.sigmoid,
            nnx.Linear(84, num_classes, kernel_init=kernel_init, rngs=rngs))
class LeNet(d2l.Classifier):
    """The LeNet-5 model."""
    def __init__(self, lr=0.1, num_classes=10):
        super().__init__()
        self.save_hyperparameters()
        self.net = nn.Sequential()
        self.net.add(
            nn.Conv2D(channels=6, kernel_size=5, padding=2,
                      activation='sigmoid'),
            nn.AvgPool2D(pool_size=2, strides=2),
            nn.Conv2D(channels=16, kernel_size=5, activation='sigmoid'),
            nn.AvgPool2D(pool_size=2, strides=2),
            nn.Dense(120, activation='sigmoid'),
            nn.Dense(84, activation='sigmoid'),
            nn.Dense(num_classes))
        self.net.initialize(init.Xavier())

This is a teaching variant of LeNet-5 rather than an exact historical reproduction. We use \(28\times28\) inputs with padding instead of the original \(32\times32\) inputs, ordinary average pooling instead of trainable subsampling, full connectivity between convolutional channels instead of LeNet-5’s partial C3 connections, logistic sigmoid instead of scaled hyperbolic tangent, and a linear logit head trained with cross-entropy instead of radial-basis output units. These changes keep the alternating convolution–pooling structure and the 6–16–120–84 channel/hidden dimensions while making every component recognizable in a modern library.

Let’s see what happens inside the network. By passing a single-channel (black and white) \(28 \times 28\) image through the network and printing the output shape at each layer, we can inspect the model to ensure that its operations line up with what we expect from Figure 6.6.2.

Let’s see what happens inside the network. By passing a single-channel (black and white) \(28 \times 28\) image through the network and printing the output shape at each layer, we can inspect the model to ensure that its operations line up with what we expect from Figure 6.6.2.

Let’s see what happens inside the network. By passing a single-channel (black and white) \(28 \times 28\) image through the network and printing the output shape at each layer, we can inspect the model to ensure that its operations line up with what we expect from Figure 6.6.2. Because an NNX model already owns its initialized layers, we can pass an array through the callables stored in Sequential and print each intermediate shape directly.

Let’s see what happens inside the network. By passing a single-channel (black and white) \(28 \times 28\) image through the network and printing the output shape at each layer, we can inspect the model to ensure that its operations line up with what we expect from Figure 6.6.2.

Figure 6.6.2: Compressed notation for LeNet-5. 
@d2l.add_to_class(d2l.Classifier)
def layer_summary(self, X_shape):
    X = d2l.randn(*X_shape)
    for layer in self.net:
        X = layer(X)
        print(layer.__class__.__name__, 'output shape:\t', X.shape)
        
model = LeNet()
model.layer_summary((1, 1, 28, 28))
Conv2d output shape:     torch.Size([1, 6, 28, 28])
Sigmoid output shape:    torch.Size([1, 6, 28, 28])
AvgPool2d output shape:  torch.Size([1, 6, 14, 14])
Conv2d output shape:     torch.Size([1, 16, 10, 10])
Sigmoid output shape:    torch.Size([1, 16, 10, 10])
AvgPool2d output shape:  torch.Size([1, 16, 5, 5])
Flatten output shape:    torch.Size([1, 400])
Linear output shape:     torch.Size([1, 120])
Sigmoid output shape:    torch.Size([1, 120])
Linear output shape:     torch.Size([1, 84])
Sigmoid output shape:    torch.Size([1, 84])
Linear output shape:     torch.Size([1, 10])
@d2l.add_to_class(d2l.Classifier)
def layer_summary(self, X_shape):
    X = d2l.normal(X_shape)
    for layer in self.net.layers:
        X = layer(X)
        print(layer.__class__.__name__, 'output shape:\t', X.shape)

model = LeNet()
model.layer_summary((1, 28, 28, 1))
Conv2D output shape:     (1, 28, 28, 6)
AveragePooling2D output shape:   (1, 14, 14, 6)
Conv2D output shape:     (1, 10, 10, 16)
AveragePooling2D output shape:   (1, 5, 5, 16)
Flatten output shape:    (1, 400)
Dense output shape:  (1, 120)
Dense output shape:  (1, 84)
Dense output shape:  (1, 10)
@d2l.add_to_class(d2l.Classifier)
def layer_summary(self, X_shape):
    X = jnp.zeros(X_shape)
    for layer in self.net.layers:
        X = layer(X)
        print(layer.__class__.__name__, 'output shape:\t', X.shape)

model = LeNet()
model.layer_summary((1, 28, 28, 1))
Conv output shape:   (1, 28, 28, 6)
PjitFunction output shape:   (1, 28, 28, 6)
function output shape:   (1, 14, 14, 6)
Conv output shape:   (1, 10, 10, 16)
PjitFunction output shape:   (1, 10, 10, 16)
function output shape:   (1, 5, 5, 16)
function output shape:   (1, 400)
Linear output shape:     (1, 120)
PjitFunction output shape:   (1, 120)
Linear output shape:     (1, 84)
PjitFunction output shape:   (1, 84)
Linear output shape:     (1, 10)
@d2l.add_to_class(d2l.Classifier)
def layer_summary(self, X_shape):
    X = d2l.randn(*X_shape)
    for layer in self.net:
        X = layer(X)
        print(layer.__class__.__name__, 'output shape:\t', X.shape)
        
model = LeNet()
model.layer_summary((1, 1, 28, 28))
Conv2D output shape:     (1, 6, 28, 28)
AvgPool2D output shape:  (1, 6, 14, 14)
Conv2D output shape:     (1, 16, 10, 10)
AvgPool2D output shape:  (1, 16, 5, 5)
Dense output shape:  (1, 120)
Dense output shape:  (1, 84)
Dense output shape:  (1, 10)

Note that the height and width of the representation at each layer throughout the convolutional block is reduced (compared with the previous layer). The first convolutional layer uses two pixels of padding to compensate for the reduction in height and width that would otherwise result from using a \(5 \times 5\) kernel. As an aside, the image size of \(28 \times 28\) pixels in the original MNIST OCR dataset is a result of trimming two pixel rows (and columns) from the original scans that measured \(32 \times 32\) pixels. This was done primarily to save space (a 30% reduction) at a time when megabytes mattered.

In contrast, the second convolutional layer forgoes padding, and thus the height and width are both reduced by four pixels. As we go up the stack of layers, the number of channels increases layer-over-layer from 1 in the input to 6 after the first convolutional layer and 16 after the second convolutional layer. However, each pooling layer halves the height and width. Finally, each fully connected layer reduces dimensionality, finally emitting an output whose dimension matches the number of classes.

6.6.2 Training

Now that we have implemented the model, let’s run an experiment to see how the LeNet-5 model fares on Fashion-MNIST.

While CNNs have fewer parameters, they can still be more expensive to compute than similarly deep MLPs because each parameter participates in many more multiplications. If you have access to a GPU, this might be a good time to put it into action to speed up training. Note that the d2l.Trainer class takes care of all details. By default, it initializes the model parameters on the available devices. Just as with MLPs, our loss function is cross-entropy, and we minimize it via minibatch stochastic gradient descent.

trainer = d2l.Trainer(max_epochs=10, num_gpus=1)
data = d2l.FashionMNIST(batch_size=128)
model = LeNet(lr=0.1)
model.apply_init([next(iter(data.get_dataloader(True)))[0]], init_cnn)
trainer.fit(model, data)

trainer = d2l.Trainer(max_epochs=10)
data = d2l.FashionMNIST(batch_size=128)
with d2l.try_gpu():
    model = LeNet(lr=0.1)
    trainer.fit(model, data)

trainer = d2l.Trainer(max_epochs=10, num_gpus=1)
data = d2l.FashionMNIST(batch_size=128)
model = LeNet(lr=0.1)
trainer.fit(model, data)

trainer = d2l.Trainer(max_epochs=10, num_gpus=1)
data = d2l.FashionMNIST(batch_size=128)
model = LeNet(lr=0.1)
trainer.fit(model, data)

6.6.3 Summary

We have made significant progress in this chapter. We moved from the MLPs of the 1980s to the CNNs of the 1990s and early 2000s. The architectures proposed, e.g., in the form of LeNet-5 remain meaningful, even to this day. Compare the error rates on Fashion-MNIST achievable with LeNet-5 both to the very best possible with MLPs (Section 4.2) and to those of significantly more advanced architectures such as ResNet (Section 7.4). LeNet is much more similar to the latter than to the former. One of the primary differences, as we shall see, is that greater amounts of computation enabled significantly more complex architectures.

A second difference is the relative ease with which we were able to implement LeNet. What used to be an engineering challenge worth months of C++ and assembly code, engineering to improve SN (Simulateur Neuristique), an early Lisp-based deep learning tool (Bottou and Le Cun 1988), and finally experimentation with models can now be accomplished in minutes. It is this incredible productivity boost that has democratized deep learning model development tremendously.

A third way to read LeNet is as a checklist of what thirty years of progress replaced. The skeleton, a convolutional encoder feeding a small classification head, is the one part that has not changed. Every component inside it has:

LeNet (1998) Modern (2020s) What the change buys
sigmoid activation ReLU or GELU gradients that survive depth instead of saturating
average pooling max-pooling or strided convolution keeps the strongest local response rather than diluting it
no normalization batch or layer normalization stable activation scales, so much deeper stacks train
dense head on flattened features global average pooling plus one linear layer removes most of the parameters (the \(400 \times 120\) block here)
Xavier initialization He initialization variance matched to ReLU rather than to sigmoid

Each row except the last is a section of Chapter 7: ReLU and max-pooling arrive with AlexNet (Section 7.1), the global-average-pooling head with NiN (Section 7.2.2), batch normalization in Section 7.3, and strided-convolution downsampling with ResNet (Section 7.4). He initialization you have already met in Section 5.3; it exists because of the switch to ReLU. The next chapter walks through this table row by row while keeping LeNet’s skeleton fixed.

6.6.4 Exercises

  1. Let’s modernize LeNet. Implement and test the following changes:
    1. Replace average pooling with max-pooling.
    2. Replace the sigmoid activations with ReLU.
  2. Try to change the size of the LeNet style network to improve its accuracy in addition to max-pooling and ReLU.
    1. Adjust the convolution window size.
    2. Adjust the number of output channels.
    3. Adjust the number of convolution layers.
    4. Adjust the number of fully connected layers.
    5. Adjust the learning rates and other training details (e.g., initialization and number of epochs).
  3. Try out the improved network on the original MNIST dataset.
  4. Display the activations of the first and second layer of LeNet for different inputs (e.g., sweaters and coats).
  5. What happens to the activations when you feed significantly different images into the network (e.g., cats, cars, or even random noise)?