4.5. Weight Decay
Open the notebook in Colab

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 curve-fitting 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 avoid overfitting. However, simply tossing aside features can be too blunt a hammer for the job. Sticking with the polynomial curve-fitting 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 \(d\) 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. Squared Norm Regularization

Weight decay (commonly called L2 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, minimize the prediction loss on the training labels, with new objective, minimize 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 \(|| \mathbf{w} ||^2\) versus 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

(4.5.1)\[l(\mathbf{w}, b) = \frac{1}{n}\sum_{i=1}^n \frac{1}{2}\left(\mathbf{w}^\top \mathbf{x}^{(i)} + b - y^{(i)}\right)^2.\]

Recall that \(\mathbf{x}^{(i)}\) are the observations, \(y^{(i)}\) are labels, 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 \(\lambda > 0\), a non-negative hyperparameter that we fit using validation data:

(4.5.2)\[l(\mathbf{w}, b) + \frac{\lambda}{2} \|\mathbf{w}\|^2.\]

For \(\lambda = 0\), we recover our original loss function. For \(\lambda > 0\), we restrict the size of \(|| \mathbf{w} ||\). 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 L2 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 L2 norm in the first place and not, say, the L1 norm.

In fact, other choices are valid and popular throughout statistics. While L2-regularized linear models constitute the classic ridge regression algorithm, L1-regularized linear regression is a similarly fundamental model in statistics (popularly known as lasso regression).

More generally, the \(\ell_2\) is just one among an infinite class of norms call p-norms, many of which you might encounter in the future. In general, for some number \(p\), the \(\ell_p\) norm is defined as

(4.5.3)\[\|\mathbf{w}\|_p^p := \sum_{i=1}^d |w_i|^p.\]

One reason to work with the L2 norm is that it places and 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, L1 penalties lead to models that concentrate weight on a small set of features, which may be desirable for other reasons.

The stochastic gradient descent updates for L2-regularized regression follow:

(4.5.4)\[\begin{aligned} \mathbf{w} & \leftarrow \left(1- \eta\lambda \right) \mathbf{w} - \frac{\eta}{|\mathcal{B}|} \sum_{i \in \mathcal{B}} \mathbf{x}^{(i)} \left(\mathbf{w}^\top \mathbf{x}^{(i)} + b - y^{(i)}\right), \end{aligned}\]

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 \(0\). 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 \(f\). Small values of \(\lambda\) correspond to unconstrained \(\mathbf{w}\), whereas large values of \(\lambda\) constrain \(\mathbf{w}\) 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 over feature selection through a simple synthetic example. First, we generate some data as before

(4.5.5)\[y = 0.05 + \sum_{i = 1}^d 0.01 x_i + \epsilon \text{ where } \epsilon \sim \mathcal{N}(0, 0.01).\]

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

%matplotlib inline
import d2l
from mxnet import autograd, gluon, init, np, npx
from mxnet.gluon import nn

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)

4.5.3. Implementation from Scratch

Next, we will implement weight decay from scratch, simply by adding the squared \(\ell_2\) penalty to the original target function. Initializing Model Parameters

First, we will define a function to randomly initialize our model parameters and run attach_grad on each to allocate memory for the gradients we will calculate.

def init_params():
    w = np.random.normal(scale=1, size=(num_inputs, 1))
    b = np.zeros(1)
    return [w, b] Defining \(\ell_2\) Norm Penalty

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

def l2_penalty(w):
    return (w**2).sum() / 2 Defining the Train and Test Functions

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 the previous chapter, 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=[1, num_epochs], legend=['train', 'test'])
    for epoch in range(1, num_epochs + 1):
        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)
            d2l.sgd([w, b], lr, batch_size)
        if epoch % 5 == 0:
            animator.add(epoch, (d2l.evaluate_loss(net, train_iter, loss),
                                 d2l.evaluate_loss(net, test_iter, loss)))
    print('l1 norm of w:', np.abs(w).sum()) 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.

l1 norm of w: 152.89601
../_images/output_weight-decay_035474_9_1.svg 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. As an exercise, you might want to check that the \(\ell_2\) norm of the weights \(\mathbf{w}\) has actually decreased.

l1 norm of w: 4.249245

4.5.4. Concise Implementation

Because weight decay is ubiquitous in neural network optimization, Gluon 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, and the optimizer must to 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 \(0\), the bias parameter \(b\) will not decay.

def train_gluon(wd):
    net = nn.Sequential()
    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=[1, num_epochs], legend=['train', 'test'])
    for epoch in range(1, num_epochs+1):
        for X, y in train_iter:
            with autograd.record():
                l = loss(net(X), y)
        if epoch % 5 == 0:
            animator.add(epoch, (d2l.evaluate_loss(net, train_iter, loss),
                                 d2l.evaluate_loss(net, test_iter, loss)))
    print('L1 norm of w:', np.abs(net[0].weight.data()).sum())

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 large problems.

L1 norm of w: 163.57933
L1 norm of w: 3.8908558

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 Spaces (RKHS) allow one to apply tools introduced for linear functions in a nonlinear context. Unfortunately, RKHS-based algorithms tend to scale purely 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 \(\ell_2\) penalty. This leads to weight decay in the update steps of the learning algorithm.

  • Gluon provides automatic weight decay functionality in the optimizer by setting the hyperparameter wd.

  • You can have different optimizers within the same training loop, e.g., for different sets of parameters.

4.5.6. Exercises

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

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

  3. What would the update equations look like if instead of \(\|\mathbf{w}\|^2\) we used \(\sum_i |w_i|\) as our penalty of choice (this is called \(\ell_1\) regularization).

  4. We know that \(\|\mathbf{w}\|^2 = \mathbf{w}^\top \mathbf{w}\). Can you find a similar equation for matrices (mathematicians call this the Frobenius norm)?

  5. 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?

  6. 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?

4.5.7. Discussions