# 6.1. Layers and Modules¶ Open the notebook in SageMaker Studio Lab

When we first introduced neural networks, we focused on linear models with a single output. Here, the entire model consists of just a single neuron. Note that a single neuron (i) takes some set of inputs; (ii) generates a corresponding scalar output; and (iii) has a set of associated parameters that can be updated to optimize some objective function of interest. Then, once we started thinking about networks with multiple outputs, we leveraged vectorized arithmetic to characterize an entire layer of neurons. Just like individual neurons, layers (i) take a set of inputs, (ii) generate corresponding outputs, and (iii) are described by a set of tunable parameters. When we worked through softmax regression, a single layer was itself the model. However, even when we subsequently introduced MLPs, we could still think of the model as retaining this same basic structure.

Interestingly, for MLPs, both the entire model and its constituent layers share this structure. The entire model takes in raw inputs (the features), generates outputs (the predictions), and possesses parameters (the combined parameters from all constituent layers). Likewise, each individual layer ingests inputs (supplied by the previous layer) generates outputs (the inputs to the subsequent layer), and possesses a set of tunable parameters that are updated according to the signal that flows backwards from the subsequent layer.

While you might think that neurons, layers, and models give us enough
abstractions to go about our business, it turns out that we often find
it convenient to speak about components that are larger than an
individual layer but smaller than the entire model. For example, the
ResNet-152 architecture, which is wildly popular in computer vision,
possesses hundreds of layers. These layers consist of repeating patterns
of *groups of layers*. Implementing such a network one layer at a time
can grow tedious. This concern is not just hypothetical—such design
patterns are common in practice. The ResNet architecture mentioned above
won the 2015 ImageNet and COCO computer vision competitions for both
recognition and detection (He *et al.*, 2016) and remains a
go-to architecture for many vision tasks. Similar architectures in which
layers are arranged in various repeating patterns are now ubiquitous in
other domains, including natural language processing and speech.

To implement these complex networks, we introduce the concept of a
neural network *module*. A module could describe a single layer, a
component consisting of multiple layers, or the entire model itself! One
benefit of working with the module abstraction is that they can be
combined into larger artifacts, often recursively. This is illustrated
in Fig. 6.1.1. By defining code to generate modules of
arbitrary complexity on demand, we can write surprisingly compact code
and still implement complex neural networks.

From a programming standpoint, a module is represented by a *class*. Any
subclass of it must define a forward propagation method that transforms
its input into output and must store any necessary parameters. Note that
some modules do not require any parameters at all. Finally a module must
possess a backpropagation method, for purposes of calculating gradients.
Fortunately, due to some behind-the-scenes magic supplied by the auto
differentiation (introduced in Section 2.5) when defining
our own module, we only need to worry about parameters and the forward
propagation method.

```
import torch
from torch import nn
from torch.nn import functional as F
```

```
from mxnet import np, npx
from mxnet.gluon import nn
npx.set_np()
```

```
from typing import List
import jax
from flax import linen as nn
from jax import numpy as jnp
from d2l import jax as d2l
```

```
No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)
```

```
import tensorflow as tf
```

To begin, we revisit the code that we used to implement MLPs (Section 5.1). The following code generates a network with one fully connected hidden layer with 256 units and ReLU activation, followed by a fully connected output layer with ten units (no activation function).

```
net = nn.Sequential(nn.LazyLinear(256), nn.ReLU(), nn.LazyLinear(10))
X = torch.rand(2, 20)
net(X).shape
```

```
torch.Size([2, 10])
```

In this example, we constructed our model by instantiating an
`nn.Sequential`

, with layers in the order that they should be executed
passed as arguments. In short, `nn.Sequential`

defines a special kind
of `Module`

, the class that presents a module in PyTorch. It maintains
an ordered list of constituent `Module`

s. Note that each of the two
fully connected layers is an instance of the `Linear`

class which is
itself a subclass of `Module`

. The forward propagation (`forward`

)
method is also remarkably simple: it chains each module in the list
together, passing the output of each as input to the next. Note that
until now, we have been invoking our models via the construction
`net(X)`

to obtain their outputs. This is actually just shorthand for
`net.__call__(X)`

.

```
net = nn.Sequential()
net.add(nn.Dense(256, activation='relu'))
net.add(nn.Dense(10))
net.initialize()
X = np.random.uniform(size=(2, 20))
net(X).shape
```

```
[21:53:59] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU
```

```
(2, 10)
```

In this example, we constructed our model by instantiating an
`nn.Sequential`

, assigning the returned object to the `net`

variable. Next, we repeatedly call its `add`

method, appending layers
in the order that they should be executed. In short, `nn.Sequential`

defines a special kind of `Block`

, the class that presents a *module*
in Gluon. It maintains an ordered list of constituent `Block`

s. The
`add`

method simply facilitates the addition of each successive
`Block`

to the list. Note that each layer is an instance of the
`Dense`

class which is itself a subclass of `Block`

. The forward
propagation (`forward`

) method is also remarkably simple: it chains
each `Block`

in the list together, passing the output of each as input
to the next. Note that until now, we have been invoking our models via
the construction `net(X)`

to obtain their outputs. This is actually
just shorthand for `net.forward(X)`

, a slick Python trick achieved via
the `Block`

class’s `__call__`

method.

```
net = nn.Sequential([nn.Dense(256), nn.relu, nn.Dense(10)])
# get_key is a d2l saved function returning jax.random.PRNGKey(random_seed)
X = jax.random.uniform(d2l.get_key(), (2, 20))
params = net.init(d2l.get_key(), X)
net.apply(params, X).shape
```

```
(2, 10)
```

```
net = tf.keras.models.Sequential([
tf.keras.layers.Dense(256, activation=tf.nn.relu),
tf.keras.layers.Dense(10),
])
X = tf.random.uniform((2, 20))
net(X).shape
```

```
TensorShape([2, 10])
```

In this example, we constructed our model by instantiating an
`keras.models.Sequential`

, with layers in the order that they should
be executed passed as arguments. In short, `Sequential`

defines a
special kind of `keras.Model`

, the class that presents a module in
Keras. It maintains an ordered list of constituent `Model`

s. Note
that each of the two fully connected layers is an instance of the
`Dense`

class which is itself a subclass of `Model`

. The forward
propagation (`call`

) method is also remarkably simple: it chains each
module in the list together, passing the output of each as input to the
next. Note that until now, we have been invoking our models via the
construction `net(X)`

to obtain their outputs. This is actually just
shorthand for `net.call(X)`

, a slick Python trick achieved via the
module class’s `__call__`

method.

## 6.1.1. A Custom Module¶

Perhaps the easiest way to develop intuition about how a module works is to implement one ourselves. Before we do that, we briefly summarize the basic functionality that each module must provide:

Ingest input data as arguments to its forward propagation method.

Generate an output by having the forward propagation method return a value. Note that the output may have a different shape from the input. For example, the first fully connected layer in our model above ingests an input of arbitrary dimension but returns an output of dimension 256.

Calculate the gradient of its output with respect to its input, which can be accessed via its backpropagation method. Typically this happens automatically.

Store and provide access to those parameters necessary for executing the forward propagation computation.

Initialize model parameters as needed.

In the following snippet, we code up a module from scratch corresponding
to an MLP with one hidden layer with 256 hidden units, and a
10-dimensional output layer. Note that the `MLP`

class below inherits
the class that represents a module. We will heavily rely on the parent
class’s methods, supplying only our own constructor (the `__init__`

method in Python) and the forward propagation method.

```
class MLP(nn.Module):
def __init__(self):
# Call the constructor of the parent class nn.Module to perform
# the necessary initialization
super().__init__()
self.hidden = nn.LazyLinear(256)
self.out = nn.LazyLinear(10)
# Define the forward propagation of the model, that is, how to return the
# required model output based on the input X
def forward(self, X):
return self.out(F.relu(self.hidden(X)))
```

```
class MLP(nn.Block):
def __init__(self):
# Call the constructor of the MLP parent class nn.Block to perform
# the necessary initialization
super().__init__()
self.hidden = nn.Dense(256, activation='relu')
self.out = nn.Dense(10)
# Define the forward propagation of the model, that is, how to return the
# required model output based on the input X
def forward(self, X):
return self.out(self.hidden(X))
```

```
class MLP(nn.Module):
def setup(self):
# Define the layers
self.hidden = nn.Dense(256)
self.out = nn.Dense(10)
# Define the forward propagation of the model, that is, how to return the
# required model output based on the input X
def __call__(self, X):
return self.out(nn.relu(self.hidden(X)))
```

```
class MLP(tf.keras.Model):
def __init__(self):
# Call the constructor of the parent class tf.keras.Model to perform
# the necessary initialization
super().__init__()
self.hidden = tf.keras.layers.Dense(units=256, activation=tf.nn.relu)
self.out = tf.keras.layers.Dense(units=10)
# Define the forward propagation of the model, that is, how to return the
# required model output based on the input X
def call(self, X):
return self.out(self.hidden((X)))
```

Let’s first focus on the forward propagation method. Note that it takes
`X`

as input, calculates the hidden representation with the activation
function applied, and outputs its logits. In this `MLP`

implementation, both layers are instance variables. To see why this is
reasonable, imagine instantiating two MLPs, `net1`

and `net2`

, and
training them on different data. Naturally, we would expect them to
represent two different learned models.

We instantiate the MLP’s layers in the constructor and subsequently
invoke these layers on each call to the forward propagation method. Note
a few key details. First, our customized `__init__`

method invokes the
parent class’s `__init__`

method via `super().__init__()`

sparing us
the pain of restating boilerplate code applicable to most modules. We
then instantiate our two fully connected layers, assigning them to
`self.hidden`

and `self.out`

. Note that unless we implement a new
layer, we need not worry about the backpropagation method or parameter
initialization. The system will generate these methods automatically.
Let’s try this out.

```
net = MLP()
net(X).shape
```

```
torch.Size([2, 10])
```

```
net = MLP()
net.initialize()
net(X).shape
```

```
(2, 10)
```

```
net = MLP()
params = net.init(d2l.get_key(), X)
net.apply(params, X).shape
```

```
(2, 10)
```

```
net = MLP()
net(X).shape
```

```
TensorShape([2, 10])
```

A key virtue of the module abstraction is its versatility. We can
subclass a module to create layers (such as the fully connected layer
class), entire models (such as the `MLP`

class above), or various
components of intermediate complexity. We exploit this versatility
throughout the coming chapters, such as when addressing convolutional
neural networks.

## 6.1.2. The Sequential Module¶

We can now take a closer look at how the `Sequential`

class works.
Recall that `Sequential`

was designed to daisy-chain other modules
together. To build our own simplified `MySequential`

, we just need to
define two key methods:

A method for appending modules one by one to a list.

A forward propagation method for passing an input through the chain of modules, in the same order as they were appended.

The following `MySequential`

class delivers the same functionality of
the default `Sequential`

class.

```
class MySequential(nn.Module):
def __init__(self, *args):
super().__init__()
for idx, module in enumerate(args):
self.add_module(str(idx), module)
def forward(self, X):
for module in self.children():
X = module(X)
return X
```

In the `__init__`

method, we add every module by calling the
`add_modules`

method. These modules can be accessed by the
`children`

method at a later date. In this way the system knows the
added modules, and it will properly initialize each module’s parameters.

```
class MySequential(nn.Block):
def add(self, block):
# Here, block is an instance of a Block subclass, and we assume that
# it has a unique name. We save it in the member variable _children of
# the Block class, and its type is OrderedDict. When the MySequential
# instance calls the initialize method, the system automatically
# initializes all members of _children
self._children[block.name] = block
def forward(self, X):
# OrderedDict guarantees that members will be traversed in the order
# they were added
for block in self._children.values():
X = block(X)
return X
```

The `add`

method adds a single block to the ordered dictionary
`_children`

. You might wonder why every Gluon `Block`

possesses a
`_children`

attribute and why we used it rather than just define a
Python list ourselves. In short the chief advantage of `_children`

is
that during our block’s parameter initialization, Gluon knows to look
inside the `_children`

dictionary to find sub-blocks whose parameters
also need to be initialized.

```
class MySequential(nn.Module):
modules: List
def __call__(self, X):
for module in self.modules:
X = module(X)
return X
```

```
class MySequential(tf.keras.Model):
def __init__(self, *args):
super().__init__()
self.modules = args
def call(self, X):
for module in self.modules:
X = module(X)
return X
```

When our `MySequential`

’s forward propagation method is invoked,
each added module is executed in the order in which they were added. We
can now reimplement an MLP using our `MySequential`

class.

```
net = MySequential(nn.LazyLinear(256), nn.ReLU(), nn.LazyLinear(10))
net(X).shape
```

```
torch.Size([2, 10])
```

```
net = MySequential()
net.add(nn.Dense(256, activation='relu'))
net.add(nn.Dense(10))
net.initialize()
net(X).shape
```

```
(2, 10)
```

```
net = MySequential([nn.Dense(256), nn.relu, nn.Dense(10)])
params = net.init(d2l.get_key(), X)
net.apply(params, X).shape
```

```
(2, 10)
```

```
net = MySequential(
tf.keras.layers.Dense(units=256, activation=tf.nn.relu),
tf.keras.layers.Dense(10))
net(X).shape
```

```
TensorShape([2, 10])
```

Note that this use of `MySequential`

is identical to the code we
previously wrote for the `Sequential`

class (as described in
Section 5.1).

## 6.1.3. Executing Code in the Forward Propagation Method¶

The `Sequential`

class makes model construction easy, allowing us to
assemble new architectures without having to define our own class.
However, not all architectures are simple daisy chains. When greater
flexibility is required, we will want to define our own blocks. For
example, we might want to execute Python’s control flow within the
forward propagation method. Moreover, we might want to perform arbitrary
mathematical operations, not simply relying on predefined neural network
layers.

You may have noticed that until now, all of the operations in our
networks have acted upon our network’s activations and its parameters.
Sometimes, however, we might want to incorporate terms that are neither
the result of previous layers nor updatable parameters. We call these
*constant parameters*. Say for example that we want a layer that
calculates the function
\(f(\mathbf{x},\mathbf{w}) = c \cdot \mathbf{w}^\top \mathbf{x}\),
where \(\mathbf{x}\) is the input, \(\mathbf{w}\) is our
parameter, and \(c\) is some specified constant that is not updated
during optimization. So we implement a `FixedHiddenMLP`

class as
follows.

```
class FixedHiddenMLP(nn.Module):
def __init__(self):
super().__init__()
# Random weight parameters that will not compute gradients and
# therefore keep constant during training
self.rand_weight = torch.rand((20, 20))
self.linear = nn.LazyLinear(20)
def forward(self, X):
X = self.linear(X)
X = F.relu(X @ self.rand_weight + 1)
# Reuse the fully connected layer. This is equivalent to sharing
# parameters with two fully connected layers
X = self.linear(X)
# Control flow
while X.abs().sum() > 1:
X /= 2
return X.sum()
```

```
class FixedHiddenMLP(nn.Block):
def __init__(self):
super().__init__()
# Random weight parameters created with the get_constant method
# are not updated during training (i.e., constant parameters)
self.rand_weight = self.params.get_constant(
'rand_weight', np.random.uniform(size=(20, 20)))
self.dense = nn.Dense(20, activation='relu')
def forward(self, X):
X = self.dense(X)
# Use the created constant parameters, as well as the relu and dot
# functions
X = npx.relu(np.dot(X, self.rand_weight.data()) + 1)
# Reuse the fully connected layer. This is equivalent to sharing
# parameters with two fully connected layers
X = self.dense(X)
# Control flow
while np.abs(X).sum() > 1:
X /= 2
return X.sum()
```

```
class FixedHiddenMLP(nn.Module):
# Random weight parameters that will not compute gradients and
# therefore keep constant during training
rand_weight: jnp.array = jax.random.uniform(d2l.get_key(), (20, 20))
def setup(self):
self.dense = nn.Dense(20)
def __call__(self, X):
X = self.dense(X)
X = nn.relu(X @ self.rand_weight + 1)
# Reuse the fully connected layer. This is equivalent to sharing
# parameters with two fully connected layers
X = self.dense(X)
# Control flow
while jnp.abs(X).sum() > 1:
X /= 2
return X.sum()
```

```
class FixedHiddenMLP(tf.keras.Model):
def __init__(self):
super().__init__()
self.flatten = tf.keras.layers.Flatten()
# Random weight parameters created with tf.constant are not updated
# during training (i.e., constant parameters)
self.rand_weight = tf.constant(tf.random.uniform((20, 20)))
self.dense = tf.keras.layers.Dense(20, activation=tf.nn.relu)
def call(self, inputs):
X = self.flatten(inputs)
# Use the created constant parameters, as well as the relu and
# matmul functions
X = tf.nn.relu(tf.matmul(X, self.rand_weight) + 1)
# Reuse the fully connected layer. This is equivalent to sharing
# parameters with two fully connected layers
X = self.dense(X)
# Control flow
while tf.reduce_sum(tf.math.abs(X)) > 1:
X /= 2
return tf.reduce_sum(X)
```

In this model, we implement a hidden layer whose weights
(`self.rand_weight`

) are initialized randomly at instantiation and are
thereafter constant. This weight is not a model parameter and thus it is
never updated by backpropagation. The network then passes the output of
this “fixed” layer through a fully connected layer.

Note that before returning the output, our model did something unusual.
We ran a while-loop, testing on the condition its \(\ell_1\) norm is
larger than \(1\), and dividing our output vector by \(2\) until
it satisfied the condition. Finally, we returned the sum of the entries
in `X`

. To our knowledge, no standard neural network performs this
operation. Note that this particular operation may not be useful in any
real-world task. Our point is only to show you how to integrate
arbitrary code into the flow of your neural network computations.

```
net = FixedHiddenMLP()
net(X)
```

```
tensor(-0.3836, grad_fn=<SumBackward0>)
```

```
net = FixedHiddenMLP()
net.initialize()
net(X)
```

```
array(0.52637565)
```

```
net = FixedHiddenMLP()
params = net.init(d2l.get_key(), X)
net.apply(params, X)
```

```
Array(0.32849464, dtype=float32)
```

```
net = FixedHiddenMLP()
net(X)
```

```
<tf.Tensor: shape=(), dtype=float32, numpy=0.6186229>
```

We can mix and match various ways of assembling modules together. In the following example, we nest modules in some creative ways.

```
class NestMLP(nn.Module):
def __init__(self):
super().__init__()
self.net = nn.Sequential(nn.LazyLinear(64), nn.ReLU(),
nn.LazyLinear(32), nn.ReLU())
self.linear = nn.LazyLinear(16)
def forward(self, X):
return self.linear(self.net(X))
chimera = nn.Sequential(NestMLP(), nn.LazyLinear(20), FixedHiddenMLP())
chimera(X)
```

```
tensor(0.0679, grad_fn=<SumBackward0>)
```

```
class NestMLP(nn.Block):
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.net = nn.Sequential()
self.net.add(nn.Dense(64, activation='relu'),
nn.Dense(32, activation='relu'))
self.dense = nn.Dense(16, activation='relu')
def forward(self, X):
return self.dense(self.net(X))
chimera = nn.Sequential()
chimera.add(NestMLP(), nn.Dense(20), FixedHiddenMLP())
chimera.initialize()
chimera(X)
```

```
array(0.97720534)
```

```
class NestMLP(nn.Module):
def setup(self):
self.net = nn.Sequential([nn.Dense(64), nn.relu,
nn.Dense(32), nn.relu])
self.dense = nn.Dense(16)
def __call__(self, X):
return self.dense(self.net(X))
chimera = nn.Sequential([NestMLP(), nn.Dense(20), FixedHiddenMLP()])
params = chimera.init(d2l.get_key(), X)
chimera.apply(params, X)
```

```
Array(-0.1306174, dtype=float32)
```

```
class NestMLP(tf.keras.Model):
def __init__(self):
super().__init__()
self.net = tf.keras.Sequential()
self.net.add(tf.keras.layers.Dense(64, activation=tf.nn.relu))
self.net.add(tf.keras.layers.Dense(32, activation=tf.nn.relu))
self.dense = tf.keras.layers.Dense(16, activation=tf.nn.relu)
def call(self, inputs):
return self.dense(self.net(inputs))
chimera = tf.keras.Sequential()
chimera.add(NestMLP())
chimera.add(tf.keras.layers.Dense(20))
chimera.add(FixedHiddenMLP())
chimera(X)
```

```
<tf.Tensor: shape=(), dtype=float32, numpy=0.59283525>
```

## 6.1.4. Summary¶

Individual layers can be modules. Many layers can comprise a module. Many modules can comprise a module.

A module can contain code. Modules take care of lots of housekeeping,
including parameter initialization and backpropagation. Sequential
concatenations of layers and modules are handled by the `Sequential`

module.

## 6.1.5. Exercises¶

What kinds of problems will occur if you change

`MySequential`

to store modules in a Python list?Implement a module that takes two modules as an argument, say

`net1`

and`net2`

and returns the concatenated output of both networks in the forward propagation. This is also called a*parallel module*.Assume that you want to concatenate multiple instances of the same network. Implement a factory function that generates multiple instances of the same module and build a larger network from it.