# 4.5. Weight Decay¶

Now that we have characterized the problem of overfitting, we can introduce some standard techniques for regularizing models. Recall that we can always mitigate overfitting by going out and collecting more training data. That can be costly, time consuming, or entirely out of our control, making it impossible in the short run. For now, we can assume that we already have as much high-quality data as our resources permit and focus on regularization techniques.

Recall that in our polynomial regression example
(Section 4.4) we could limit our model’s capacity
simply by tweaking the degree of the fitted polynomial. Indeed, limiting
the number of features is a popular technique to mitigate overfitting.
However, simply tossing aside features can be too blunt an instrument
for the job. Sticking with the polynomial regression example, consider
what might happen with high-dimensional inputs. The natural extensions
of polynomials to multivariate data are called *monomials*, which are
simply products of powers of variables. The degree of a monomial is the
sum of the powers. For example, \(x_1^2 x_2\), and \(x_3 x_5^2\)
are both monomials of degree 3.

Note that the number of terms with degree \(d\) blows up rapidly as \(d\) grows larger. Given \(k\) variables, the number of monomials of degree \(d\) (i.e., \(k\) multichoose \(d\)) is \({k - 1 + d} \choose {k - 1}\). Even small changes in degree, say from \(2\) to \(3\), dramatically increase the complexity of our model. Thus we often need a more fine-grained tool for adjusting function complexity.

## 4.5.1. Norms and Weight Decay¶

We have described both the \(L_2\) norm and the \(L_1\) norm,
which are special cases of the more general \(L_p\) norm in
Section 2.3.10. *Weight decay* (commonly called
\(L_2\) regularization), might be the most widely-used technique for
regularizing parametric machine learning models. The technique is
motivated by the basic intuition that among all functions \(f\), the
function \(f = 0\) (assigning the value \(0\) to all inputs) is
in some sense the *simplest*, and that we can measure the complexity of
a function by its distance from zero. But how precisely should we
measure the distance between a function and zero? There is no single
right answer. In fact, entire branches of mathematics, including parts
of functional analysis and the theory of Banach spaces, are devoted to
answering this issue.

One simple interpretation might be to measure the complexity of a linear
function \(f(\mathbf{x}) = \mathbf{w}^\top \mathbf{x}\) by some norm
of its weight vector, e.g., \(\| \mathbf{w} \|^2\). The most common
method for ensuring a small weight vector is to add its norm as a
penalty term to the problem of minimizing the loss. Thus we replace our
original objective, *minimizing the prediction loss on the training
labels*, with new objective, *minimizing the sum of the prediction loss
and the penalty term*. Now, if our weight vector grows too large, our
learning algorithm might focus on minimizing the weight norm
\(\| \mathbf{w} \|^2\) vs. minimizing the training error. That is
exactly what we want. To illustrate things in code, let us revive our
previous example from Section 3.1 for linear
regression. There, our loss was given by

Recall that \(\mathbf{x}^{(i)}\) are the features, \(y^{(i)}\)
are labels for all data examples \(i\), and \((\mathbf{w}, b)\)
are the weight and bias parameters, respectively. To penalize the size
of the weight vector, we must somehow add \(\| \mathbf{w} \|^2\) to
the loss function, but how should the model trade off the standard loss
for this new additive penalty? In practice, we characterize this
tradeoff via the *regularization constant* \(\lambda\), a
non-negative hyperparameter that we fit using validation data:

For \(\lambda = 0\), we recover our original loss function. For \(\lambda > 0\), we restrict the size of \(\| \mathbf{w} \|\). We divide by \(2\) by convention: when we take the derivative of a quadratic function, the \(2\) and \(1/2\) cancel out, ensuring that the expression for the update looks nice and simple. The astute reader might wonder why we work with the squared norm and not the standard norm (i.e., the Euclidean distance). We do this for computational convenience. By squaring the \(L_2\) norm, we remove the square root, leaving the sum of squares of each component of the weight vector. This makes the derivative of the penalty easy to compute: the sum of derivatives equals the derivative of the sum.

Moreover, you might ask why we work with the \(L_2\) norm in the
first place and not, say, the \(L_1\) norm. In fact, other choices
are valid and popular throughout statistics. While
\(L_2\)-regularized linear models constitute the classic *ridge
regression* algorithm, \(L_1\)-regularized linear regression is a
similarly fundamental model in statistics, which is popularly known as
*lasso regression*.

One reason to work with the \(L_2\) norm is that it places an
outsize penalty on large components of the weight vector. This biases
our learning algorithm towards models that distribute weight evenly
across a larger number of features. In practice, this might make them
more robust to measurement error in a single variable. By contrast,
\(L_1\) penalties lead to models that concentrate weights on a small
set of features by clearing the other weights to zero. This is called
*feature selection*, which may be desirable for other reasons.

Using the same notation in (3.1.10), the minibatch stochastic gradient descent updates for \(L_2\)-regularized regression follow:

As before, we update \(\mathbf{w}\) based on the amount by which our
estimate differs from the observation. However, we also shrink the size
of \(\mathbf{w}\) towards zero. That is why the method is sometimes
called “weight decay”: given the penalty term alone, our optimization
algorithm *decays* the weight at each step of training. In contrast to
feature selection, weight decay offers us a continuous mechanism for
adjusting the complexity of a function. Smaller values of
\(\lambda\) correspond to less constrained \(\mathbf{w}\),
whereas larger values of \(\lambda\) constrain \(\mathbf{w}\)
more considerably.

Whether we include a corresponding bias penalty \(b^2\) can vary across implementations, and may vary across layers of a neural network. Often, we do not regularize the bias term of a network’s output layer.

## 4.5.2. High-Dimensional Linear Regression¶

We can illustrate the benefits of weight decay through a simple synthetic example.

```
%matplotlib inline
from d2l import mxnet as d2l
from mxnet import autograd, gluon, init, np, npx
from mxnet.gluon import nn
npx.set_np()
```

```
%matplotlib inline
from d2l import torch as d2l
import torch
import torch.nn as nn
```

```
%matplotlib inline
from d2l import tensorflow as d2l
import tensorflow as tf
```

First, we generate some data as before

We choose our label to be a linear function of our inputs, corrupted by Gaussian noise with zero mean and standard deviation 0.01. To make the effects of overfitting pronounced, we can increase the dimensionality of our problem to \(d = 200\) and work with a small training set containing only 20 examples.

```
n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5
true_w, true_b = np.ones((num_inputs, 1)) * 0.01, 0.05
train_data = d2l.synthetic_data(true_w, true_b, n_train)
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)
```

```
n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5
true_w, true_b = torch.ones((num_inputs, 1)) * 0.01, 0.05
train_data = d2l.synthetic_data(true_w, true_b, n_train)
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)
```

```
n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5
true_w, true_b = tf.ones((num_inputs, 1)) * 0.01, 0.05
train_data = d2l.synthetic_data(true_w, true_b, n_train)
train_iter = d2l.load_array(train_data, batch_size)
test_data = d2l.synthetic_data(true_w, true_b, n_test)
test_iter = d2l.load_array(test_data, batch_size, is_train=False)
```

## 4.5.3. Implementation from Scratch¶

In the following, we will implement weight decay from scratch, simply by adding the squared \(L_2\) penalty to the original target function.

### 4.5.3.1. Initializing Model Parameters¶

First, we will define a function to randomly initialize our model parameters.

```
def init_params():
w = np.random.normal(scale=1, size=(num_inputs, 1))
b = np.zeros(1)
w.attach_grad()
b.attach_grad()
return [w, b]
```

```
def init_params():
w = torch.normal(0, 1, size=(num_inputs, 1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)
return [w, b]
```

```
def init_params():
w = tf.Variable(tf.random.normal(mean=1, shape=(num_inputs, 1)))
b = tf.Variable(tf.zeros(shape=(1, )))
return [w, b]
```

### 4.5.3.2. Defining \(L_2\) Norm Penalty¶

Perhaps the most convenient way to implement this penalty is to square all terms in place and sum them up.

```
def l2_penalty(w):
return (w**2).sum() / 2
```

```
def l2_penalty(w):
return torch.sum(w.pow(2)) / 2
```

```
def l2_penalty(w):
return tf.reduce_sum(tf.pow(w, 2)) / 2
```

### 4.5.3.3. Defining the Training Loop¶

The following code fits a model on the training set and evaluates it on
the test set. The linear network and the squared loss have not changed
since Section 3, so we will just import them via
`d2l.linreg`

and `d2l.squared_loss`

. The only change here is that
our loss now includes the penalty term.

```
def train(lambd):
w, b = init_params()
net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss
num_epochs, lr = 100, 0.003
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test'])
for epoch in range(num_epochs):
for X, y in train_iter:
with autograd.record():
# The L2 norm penalty term has been added, and broadcasting
# makes `l2_penalty(w)` a vector whose length is `batch_size`
l = loss(net(X), y) + lambd * l2_penalty(w)
l.backward()
d2l.sgd([w, b], lr, batch_size)
if (epoch + 1) % 5 == 0:
animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),
d2l.evaluate_loss(net, test_iter, loss)))
print('L2 norm of w:', np.linalg.norm(w))
```

```
def train(lambd):
w, b = init_params()
net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss
num_epochs, lr = 100, 0.003
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test'])
for epoch in range(num_epochs):
for X, y in train_iter:
with torch.enable_grad():
# The L2 norm penalty term has been added, and broadcasting
# makes `l2_penalty(w)` a vector whose length is `batch_size`
l = loss(net(X), y) + lambd * l2_penalty(w)
l.sum().backward()
d2l.sgd([w, b], lr, batch_size)
if (epoch + 1) % 5 == 0:
animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),
d2l.evaluate_loss(net, test_iter, loss)))
print('L2 norm of w:', torch.norm(w).item())
```

```
def train(lambd):
w, b = init_params()
net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss
num_epochs, lr = 100, 0.003
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test'])
for epoch in range(num_epochs):
for X, y in train_iter:
with tf.GradientTape() as tape:
# The L2 norm penalty term has been added, and broadcasting
# makes `l2_penalty(w)` a vector whose length is `batch_size`
l = loss(net(X), y) + lambd * l2_penalty(w)
grads = tape.gradient(l, [w, b])
d2l.sgd([w, b], grads, lr, batch_size)
if (epoch + 1) % 5 == 0:
animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),
d2l.evaluate_loss(net, test_iter, loss)))
print('L2 norm of w:', tf.norm(w).numpy())
```

### 4.5.3.4. Training without Regularization¶

We now run this code with `lambd = 0`

, disabling weight decay. Note
that we overfit badly, decreasing the training error but not the test
error—a textook case of overfitting.

```
train(lambd=0)
```

```
L2 norm of w: 13.259393
```

```
train(lambd=0)
```

```
L2 norm of w: 14.27925968170166
```

```
train(lambd=0)
```

```
L2 norm of w: 19.222454
```

### 4.5.3.5. Using Weight Decay¶

Below, we run with substantial weight decay. Note that the training error increases but the test error decreases. This is precisely the effect we expect from regularization.

```
train(lambd=3)
```

```
L2 norm of w: 0.38251483
```

```
train(lambd=3)
```

```
L2 norm of w: 0.3773614466190338
```

```
train(lambd=3)
```

```
L2 norm of w: 0.56827617
```

## 4.5.4. Concise Implementation¶

Because weight decay is ubiquitous in neural network optimization, the deep learning framework makes it especially convenient, integrating weight decay into the optimization algorithm itself for easy use in combination with any loss function. Moreover, this integration serves a computational benefit, allowing implementation tricks to add weight decay to the algorithm, without any additional computational overhead. Since the weight decay portion of the update depends only on the current value of each parameter, the optimizer must touch each parameter once anyway.

In the following code, we specify the weight decay hyperparameter
directly through `wd`

when instantiating our `Trainer`

. By default,
Gluon decays both weights and biases simultaneously. Note that the
hyperparameter `wd`

will be multiplied by `wd_mult`

when updating
model parameters. Thus, if we set `wd_mult`

to zero, the bias
parameter \(b\) will not decay.

```
def train_concise(wd):
net = nn.Sequential()
net.add(nn.Dense(1))
net.initialize(init.Normal(sigma=1))
loss = gluon.loss.L2Loss()
num_epochs, lr = 100, 0.003
trainer = gluon.Trainer(net.collect_params(), 'sgd',
{'learning_rate': lr, 'wd': wd})
# The bias parameter has not decayed. Bias names generally end with "bias"
net.collect_params('.*bias').setattr('wd_mult', 0)
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test'])
for epoch in range(num_epochs):
for X, y in train_iter:
with autograd.record():
l = loss(net(X), y)
l.backward()
trainer.step(batch_size)
if (epoch + 1) % 5 == 0:
animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),
d2l.evaluate_loss(net, test_iter, loss)))
print('L2 norm of w:', np.linalg.norm(net[0].weight.data()))
```

In the following code, we specify the weight decay hyperparameter
directly through `weight_decay`

when instantiating our optimizer. By
default, PyTorch decays both weights and biases simultaneously. Here we
only set `weight_decay`

for the weight, so the bias parameter
\(b\) will not decay.

```
def train_concise(wd):
net = nn.Sequential(nn.Linear(num_inputs, 1))
for param in net.parameters():
param.data.normal_()
loss = nn.MSELoss()
num_epochs, lr = 100, 0.003
# The bias parameter has not decayed
trainer = torch.optim.SGD([
{"params":net[0].weight,'weight_decay': wd},
{"params":net[0].bias}], lr=lr)
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test'])
for epoch in range(num_epochs):
for X, y in train_iter:
with torch.enable_grad():
trainer.zero_grad()
l = loss(net(X), y)
l.backward()
trainer.step()
if (epoch + 1) % 5 == 0:
animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),
d2l.evaluate_loss(net, test_iter, loss)))
print('L2 norm of w:', net[0].weight.norm().item())
```

In the following code, we create an \(L_2\) regularizer with the
weight decay hyperparameter `wd`

and apply it to the layer through the
`kernel_regularizer`

argument.

```
def train_concise(wd):
net = tf.keras.models.Sequential()
net.add(tf.keras.layers.Dense(
1, kernel_regularizer=tf.keras.regularizers.l2(wd)))
net.build(input_shape=(1, num_inputs))
w, b = net.trainable_variables
loss = tf.keras.losses.MeanSquaredError()
num_epochs, lr = 100, 0.003
trainer = tf.keras.optimizers.SGD(learning_rate=lr)
animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',
xlim=[5, num_epochs], legend=['train', 'test'])
for epoch in range(num_epochs):
for X, y in train_iter:
with tf.GradientTape() as tape:
# `tf.keras` requires retrieving and adding the losses from
# layers manually for custom training loop.
l = loss(net(X), y) + net.losses
grads = tape.gradient(l, net.trainable_variables)
trainer.apply_gradients(zip(grads, net.trainable_variables))
if (epoch + 1) % 5 == 0:
animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),
d2l.evaluate_loss(net, test_iter, loss)))
print('L2 norm of w:', tf.norm(net.get_weights()[0]).numpy())
```

The plots look identical to those when we implemented weight decay from scratch. However, they run appreciably faster and are easier to implement, a benefit that will become more pronounced for larger problems.

```
train_concise(0)
```

```
L2 norm of w: 15.014067
```

```
train_concise(3)
```

```
L2 norm of w: 0.33991632
```

```
train_concise(0)
```

```
L2 norm of w: 13.397233009338379
```

```
train_concise(3)
```

```
L2 norm of w: 0.3846113085746765
```

```
train_concise(0)
```

```
L2 norm of w: 1.29725
```

```
train_concise(3)
```

```
L2 norm of w: 0.03512145
```

So far, we only touched upon one notion of what constitutes a simple linear function. Moreover, what constitutes a simple nonlinear function can be an even more complex question. For instance, reproducing kernel Hilbert space (RKHS) allows one to apply tools introduced for linear functions in a nonlinear context. Unfortunately, RKHS-based algorithms tend to scale poorly to large, high-dimensional data. In this book we will default to the simple heuristic of applying weight decay on all layers of a deep network.

## 4.5.5. Summary¶

Regularization is a common method for dealing with overfitting. It adds a penalty term to the loss function on the training set to reduce the complexity of the learned model.

One particular choice for keeping the model simple is weight decay using an \(L_2\) penalty. This leads to weight decay in the update steps of the learning algorithm.

The weight decay functionality is provided in optimizers from deep learning frameworks.

Different sets of parameters can have different update behaviors within the same training loop.

## 4.5.6. Exercises¶

Experiment with the value of \(\lambda\) in the estimation problem in this section. Plot training and test accuracy as a function of \(\lambda\). What do you observe?

Use a validation set to find the optimal value of \(\lambda\). Is it really the optimal value? Does this matter?

What would the update equations look like if instead of \(\|\mathbf{w}\|^2\) we used \(\sum_i |w_i|\) as our penalty of choice (\(L_1\) regularization)?

We know that \(\|\mathbf{w}\|^2 = \mathbf{w}^\top \mathbf{w}\). Can you find a similar equation for matrices (see the Frobenius norm in Section 2.3.10)?

Review the relationship between training error and generalization error. In addition to weight decay, increased training, and the use of a model of suitable complexity, what other ways can you think of to deal with overfitting?

In Bayesian statistics we use the product of prior and likelihood to arrive at a posterior via \(P(w \mid x) \propto P(x \mid w) P(w)\). How can you identify \(P(w)\) with regularization?