🏷️chapter_softmax_scratch
Just as we implemented linear regression from scratch,
we believe that multiclass logistic (softmax) regression
is similarly fundamental and you ought to know
the gory details of how to implement it from scratch.
As with linear regression, after doing things by hand
we will breeze through an implementation in Gluon for comparison.
To begin, let's import our packages
(only autograd, nd are needed here
because we will be doing the heavy lifting ourselves.)
import sys
sys.path.insert(0, '..')
%matplotlib inline
import d2l
from mxnet import autograd, nd
We will work with the Fashion-MNIST dataset just introduced, cuing up an iterator with batch size 256.
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
Just as in linear regression, we represent each example as a vector.
Since each example is a
Recall that in softmax regression,
we have as many outputs as there are categories.
Because our dataset has
num_inputs = 784
num_outputs = 10
W = nd.random.normal(scale=0.01, shape=(num_inputs, num_outputs))
b = nd.zeros(num_outputs)
Recall that we need to attach gradients to the model parameters. More literally, we are allocating memory for future gradients to be stored and notifiying MXNet that we want gradients to be calculated with respect to these parameters in the first place.
W.attach_grad()
b.attach_grad()
Before implementing the softmax regression model,
let's briefly review how operators such as sum work
along specific dimensions in an NDArray.
Given a matrix X we can sum over all elements (default) or only
over elements in the same column (axis=0) or the same row (axis=1).
Note that if X is an array with shape (2, 3)
and we sum over the columns (X.sum(axis=0),
the result will be a (1D) vector with shape (3,).
If we want to keep the number of axes in the original array
(resulting in a 2D array with shape (1,3)),
rather than collapsing out the dimension that we summed over
we can specify keepdims=True when invoking sum.
X = nd.array([[1, 2, 3], [4, 5, 6]])
X.sum(axis=0, keepdims=True), X.sum(axis=1, keepdims=True)
We are now ready to implement the softmax function.
Recall that softmax consists of two steps:
First, we exponentiate each term (using exp).
Then, we sum over each row (we have one row per example in the batch)
to get the normalization constants for each example.
Finally, we divide each row by its normalization constant,
ensuring that the result sums to
$$ \mathrm{softmax}(\mathbf{X}){ij} = \frac{\exp(X{ij})}{\sum_k \exp(X_{ik})} $$
The denominator, or normalization constant, is also sometimes called the partition function (and its logarithm the log-partition function). The origins of that name are in statistical physics where a related equation models the distribution over an ensemble of particles).
def softmax(X):
X_exp = X.exp()
partition = X_exp.sum(axis=1, keepdims=True)
return X_exp / partition # The broadcast mechanism is applied here
As you can see, for any random input, we turn each element into a non-negative number. Moreover, each row sums up to 1, as is required for a probability.
Note that while this looks correct mathematically,
we were a bit sloppy in our implementation
because failed to take precautions against numerical overflow or underflow
due to large (or very small) elements of the matrix,
as we did in
:numref:chapter_naive_bayes.
X = nd.random.normal(shape=(2, 5))
X_prob = softmax(X)
X_prob, X_prob.sum(axis=1)
Now that we have defined the softmax operation,
we can implement the softmax regression model.
The below code defines the forward pass through the network.
Note that we flatten each original image in the batch
into a vector with length num_inputs with the reshape function
before passing the data through our model.
def net(X):
return softmax(nd.dot(X.reshape((-1, num_inputs)), W) + b)
Next, we need to implement the cross entropy loss function,
introduced in :numref:chapter_softmax.
This may be the most common loss function
in all of deep learning because, at the moment,
classification problems far outnumber regression problems.
Recall that cross entropy takes the negative log likelihood
of the predicted probability assigned to the true label for loop
(which tends to be inefficient), we can use the pick function
which allows us to select the appropriate terms
from the matrix of softmax entries easily.
Below, we illustrate the pick function on a toy example,
with 3 categories and 2 examples.
y_hat = nd.array([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]])
y = nd.array([0, 2], dtype='int32')
nd.pick(y_hat, y)
Now we can implement the cross-entropy loss function efficiently with just one line of code.
def cross_entropy(y_hat, y):
return -nd.pick(y_hat, y).log()
Given the predicted probability distribution y_hat,
we typically choose the class with highest predicted probability
whenever we must output a hard prediction. Indeed, many applications require that we make a choice. Gmail must catetegorize an email into Primary, Social, Updates, or Forums. It might estimate probabilities internally, but at the end of the day it has to choose one among the categories.
When predictions are consistent with the actual category y, they are coorect. The classification accuracy is the fraction of all predictions that are correct. Although we cannot optimize accuracy directly (it is not differentiable), it's often the performance metric that we care most about, and we will nearly always report it when training classifiers.
To compute accuracy we do the following:
First, we execute y_hat.argmax(axis=1)
to gather the predicted classes
(given by the indices for the largest entires each row).
The result has the same shape as the variable y.
Now we just need to check how frequently the two match.
Since the equality operator == is datatype-sensitive
(e.g. an int and a float32 are never equal),
we also need to convert both to the same type (we pick float32).
The result is an NDArray containing entries of 0 (false) and 1 (true).
Taking the mean yields the desired result.
def accuracy(y_hat, y):
return (y_hat.argmax(axis=1) == y.astype('float32')).mean().asscalar()
We will continue to use the variables y_hat and y
defined in the pick function,
as the predicted probability distribution and label, respectively.
We can see that the first example's prediction category is 2
(the largest element of the row is 0.6 with an index of 2),
which is inconsistent with the actual label, 0.
The second example's prediction category is 2
(the largest element of the row is 0.5 with an index of 2),
which is consistent with the actual label, 2.
Therefore, the classification accuracy rate for these two examples is 0.5.
accuracy(y_hat, y)
Similarly, we can evaluate the accuracy for model net on the data set
(accessed via data_iter).
# The function will be gradually improved: the complete implementation will be
# discussed in the "Image Augmentation" section
def evaluate_accuracy(data_iter, net):
acc_sum, n = 0.0, 0
for X, y in data_iter:
y = y.astype('float32')
acc_sum += (net(X).argmax(axis=1) == y).sum().asscalar()
n += y.size
return acc_sum / n
Because we initialized the net model with random weights,
the accuracy of this model should be close to random guessing,
i.e. 0.1 for 10 classes.
evaluate_accuracy(test_iter, net)
The training loop for softmax regression should look strikingly familiar
if you read through our implementation
of linear regression earlier in this chapter.
Again, we use the mini-batch stochastic gradient descent
to optimize the loss function of the model.
Note that the number of epochs (num_epochs),
and learning rate (lr) are both adjustable hyper-parameters.
By changing their values, we may be able to increase the classification accuracy of the model. In practice we'll want to split our data three ways
into training, validation, and test data, using the validation data to choose the best values of our hyperparameters.
num_epochs, lr = 5, 0.1
# This function has been saved in the d2l package for future use
def train_ch3(net, train_iter, test_iter, loss, num_epochs, batch_size,
params=None, lr=None, trainer=None):
for epoch in range(num_epochs):
train_l_sum, train_acc_sum, n = 0.0, 0.0, 0
for X, y in train_iter:
with autograd.record():
y_hat = net(X)
l = loss(y_hat, y).sum()
l.backward()
if trainer is None:
d2l.sgd(params, lr, batch_size)
else:
# This will be illustrated in the next section
trainer.step(batch_size)
y = y.astype('float32')
train_l_sum += l.asscalar()
train_acc_sum += (y_hat.argmax(axis=1) == y).sum().asscalar()
n += y.size
test_acc = evaluate_accuracy(test_iter, net)
print('epoch %d, loss %.4f, train acc %.3f, test acc %.3f'
% (epoch + 1, train_l_sum / n, train_acc_sum / n, test_acc))
train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs,
batch_size, [W, b], lr)
Now that training is complete, our model is ready to classify some images. Given a series of images, we will compare their actual labels (first line of text output) and the model predictions (second line of text output).
for X, y in test_iter:
break
true_labels = d2l.get_fashion_mnist_labels(y.asnumpy())
pred_labels = d2l.get_fashion_mnist_labels(net(X).argmax(axis=1).asnumpy())
titles = [truelabel + '\n' + predlabel
for truelabel, predlabel in zip(true_labels, pred_labels)]
d2l.show_fashion_mnist(X[0:9], titles[0:9])
With softmax regression, we can train models for multi-category classification. The training loop is very similar to that in linear regression: retrieve and read data, define models and loss functions, then train models using optimization algorithms. As you'll soon find out, most common deep learning models have similar training procedures.
- In this section, we directly implemented the softmax function based on the mathematical definition of the softmax operation. What problems might this cause (hint - try to calculate the size of $\exp(50)$)?
- The function
cross_entropyin this section is implemented according to the definition of the cross-entropy loss function. What could be the problem with this implementation (hint - consider the domain of the logarithm)? - What solutions you can think of to fix the two problems above?
- Is it always a good idea to return the most likely label. E.g. would you do this for medical diagnosis?
- Assume that we want to use softmax regression to predict the next word based on some features. What are some problems that might arise from a large vocabulary?
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