4.5. Concise Implementation of Softmax Regression
Open the notebook in Colab
Open the notebook in Colab
Open the notebook in Colab
Open the notebook in SageMaker Studio Lab

Just as high-level deep learning frameworks made it easier to implement linear regression (see Section 3.5), they are similarly convenient here.

import torch
from torch import nn
from torch.nn import functional as F
from d2l import torch as d2l
from mxnet import gluon, init, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l

npx.set_np()
import tensorflow as tf
from d2l import tensorflow as d2l

4.5.1. Defining the Model

As in Section 3.5, we construct our fully connected layer using the built-in layer. The built-in __call__ method then invokes forward whenever we need to apply the network to some input.

We use a Flatten layer to convert the 4th order tensor X to 2nd order by keeping the dimensionality along the first axis unchanged.

class SoftmaxRegression(d2l.Classifier):
    def __init__(self, num_outputs, lr):
        super().__init__()
        self.save_hyperparameters()
        self.net = nn.Sequential(nn.Flatten(),
                                 nn.LazyLinear(num_outputs))
    def forward(self, X):
        return self.net(X)

Even though the input X is a 4th order tensor, the built-in Dense layer will automatically convert X into a 2nd order tensor by keeping the dimensionality along the first axis unchanged.

class SoftmaxRegression(d2l.Classifier):
    def __init__(self, num_outputs, lr):
        super().__init__()
        self.save_hyperparameters()
        self.net = nn.Dense(num_outputs)
        self.net.initialize()
    def forward(self, X):
        return self.net(X)

We use a Flatten layer to convert the 4th order tensor X by keeping the dimension along the first axis unchanged.

class SoftmaxRegression(d2l.Classifier):
    def __init__(self, num_outputs, lr):
        super().__init__()
        self.save_hyperparameters()
        self.net = tf.keras.models.Sequential()
        self.net.add(tf.keras.layers.Flatten())
        self.net.add(tf.keras.layers.Dense(num_outputs))

    def forward(self, X):
        return self.net(X)

4.5.2. Softmax Revisited

In Section 4.4 we calculated our model’s output and applied the cross-entropy loss. While this is perfectly reasonable mathematically, it is risky computationally, due to numerical underflow and overflow in the exponentiation.

Recall that the softmax function computes probabilities via \(\hat y_j = \frac{\exp(o_j)}{\sum_k \exp(o_k)}\). If some of the \(o_k\) are very large, i.e., very positive, then \(\exp(o_k)\) might be larger than the largest number we can have for certain data types. This is called overflow. Likewise, if all arguments are very negative, we will get underflow. For instance, single precision floating point numbers approximately cover the range of \(10^{-38}\) to \(10^{38}\). As such, if the largest term in \(\mathbf{o}\) lies outside the interval \([-90, 90]\), the result will not be stable. A solution to this problem is to subtract \(\bar{o} \stackrel{\mathrm{def}}{=} \max_k o_k\) from all entries:

(4.5.1)\[\hat y_j = \frac{\exp o_j}{\sum_k \exp o_k} = \frac{\exp(o_j - \bar{o}) \exp \bar{o}}{\sum_k \exp (o_k - \bar{o}) \exp \bar{o}} = \frac{\exp(o_j - \bar{o})}{\sum_k \exp (o_k - \bar{o})}.\]

By construction we know that \(o_j - \bar{o} \leq 0\) for all \(j\). As such, for a \(q\)-class classification problem, the denominator is contained in the interval \([1, q]\). Moreover, the numerator never exceeds \(1\), thus preventing numerical overflow. Numerical underflow only occurs when \(\exp(o_j - \bar{o})\) numerically evaluates as \(0\). Nonetheless, a few steps down the road we might find ourselves in trouble when we want to compute \(\log \hat{y}_j\) as \(\log 0\). In particular, in backpropagation, we might find ourselves faced with a screenful of the dreaded NaN (Not a Number) results.

Fortunately, we are saved by the fact that even though we are computing exponential functions, we ultimately intend to take their log (when calculating the cross-entropy loss). By combining softmax and cross-entropy, we can escape the numerical stability issues altogether. We have:

(4.5.2)\[\log \hat{y}_j = \log \frac{\exp(o_j - \bar{o})}{\sum_k \exp (o_k - \bar{o})} = o_j - \bar{o} - \log \sum_k \exp (o_k - \bar{o}).\]

This avoids both overflow and underflow. We will want to keep the conventional softmax function handy in case we ever want to evaluate the output probabilities by our model. But instead of passing softmax probabilities into our new loss function, we just pass the logits and compute the softmax and its log all at once inside the cross-entropy loss function, which does smart things like the “LogSumExp trick”.

@d2l.add_to_class(d2l.Classifier)  #@save
def loss(self, Y_hat, Y, averaged=True):
    Y_hat = Y_hat.reshape((-1, Y_hat.shape[-1]))
    Y = Y.reshape((-1,))
    return F.cross_entropy(
        Y_hat, Y, reduction='mean' if averaged else 'none')
@d2l.add_to_class(d2l.Classifier)  #@save
def loss(self, Y_hat, Y, averaged=True):
    Y_hat = Y_hat.reshape((-1, Y_hat.shape[-1]))
    Y = Y.reshape((-1,))
    fn = gluon.loss.SoftmaxCrossEntropyLoss()
    l = fn(Y_hat, Y)
    return l.mean() if averaged else l
@d2l.add_to_class(d2l.Classifier)  #@save
def loss(self, Y_hat, Y, averaged=True):
    Y_hat = tf.reshape(Y_hat, (-1, Y_hat.shape[-1]))
    Y = tf.reshape(Y, (-1,))
    fn = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)
    return fn(Y, Y_hat)

4.5.3. Training

Next we train our model. As before, we use Fashion-MNIST images, flattened to 784-dimensional feature vectors.

data = d2l.FashionMNIST(batch_size=256)
model = SoftmaxRegression(num_outputs=10, lr=0.1)
trainer = d2l.Trainer(max_epochs=10)
trainer.fit(model, data)
../_images/output_softmax-regression-concise_0b22ca_42_0.svg
data = d2l.FashionMNIST(batch_size=256)
model = SoftmaxRegression(num_outputs=10, lr=0.1)
trainer = d2l.Trainer(max_epochs=10)
trainer.fit(model, data)
../_images/output_softmax-regression-concise_0b22ca_45_0.svg
data = d2l.FashionMNIST(batch_size=256)
model = SoftmaxRegression(num_outputs=10, lr=0.1)
trainer = d2l.Trainer(max_epochs=10)
trainer.fit(model, data)
../_images/output_softmax-regression-concise_0b22ca_48_0.svg

As before, this algorithm converges to a solution that achieves a decent accuracy, albeit this time with fewer lines of code than before.

4.5.4. Summary

High-level APIs are very convenient at hiding potentially dangerous aspects from their user, such as numerical stability. Moreover, they allow users to design models concisely with very few lines of code. This is both a blessing and a curse. The obvious benefit is that it makes things highly accessible, even to engineers who never took a single class of statistics in their life (in fact, this is one of the target audiences of the book). But hiding the sharp edges also comes with a price: a disincentive to add new and different components on your own, since there’s little muscle memory for doing it. Moreover, it makes it more difficult to fix things whenever the protective padding of a framework fails to cover all the corner cases entirely. Again, this is due to lack of familiarity.

As such, we strongly urge you to review both the bare bones and the elegant versions of many of the implementations that follow. While we emphasize ease of understanding, the implementations are nonetheless usually quite performant (convolutions are the big exception here). It is our intention to allow you to build on these when you invent something new that no framework can give you.

4.5.5. Exercises

  1. Deep learning uses many different number formats, including FP64 double precision (used extremely rarely), FP32 single precision, BFLOAT16 (good for compressed representations), FP16 (very unstable), TF32 (a new format from NVIDIA), and INT8. Compute the smallest and largest argument of the exponential function for which the result does not lead to a numerical underflow or overflow.

  2. INT8 is a very limited format with nonzero numbers from \(1\) to \(255\). How could you extend its dynamic range without using more bits? Do standard multiplication and addition still work?

  3. Increase the number of epochs for training. Why might the validation accuracy decrease after a while? How could we fix this?

  4. What happens as you increase the learning rate? Compare the loss curves for several learning rates. Which one works better? When?