3.6. Implementation of Softmax Regression from Scratch¶ Open the notebook in SageMaker Studio Lab
Just as we implemented linear regression from scratch, we believe that softmax regression is similarly fundamental and you ought to know the gory details of and how to implement it yourself. We will work with the Fashion-MNIST dataset, just introduced in Section 3.5, setting up a data iterator with batch size 256.
from IPython import display
from mxnet import autograd, gluon, np, npx
from d2l import mxnet as d2l
npx.set_np()
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
import torch
from IPython import display
from d2l import torch as d2l
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
import tensorflow as tf
from IPython import display
from d2l import tensorflow as d2l
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
3.6.1. Initializing Model Parameters¶
As in our linear regression example, each example here will be represented by a fixed-length vector. Each example in the raw dataset is a \(28 \times 28\) image. In this section, we will flatten each image, treating them as vectors of length 784. In the future, we will talk about more sophisticated strategies for exploiting the spatial structure in images, but for now we treat each pixel location as just another feature.
Recall that in softmax regression, we have as many outputs as there are
classes. Because our dataset has 10 classes, our network will have an
output dimension of 10. Consequently, our weights will constitute a
\(784 \times 10\) matrix and the biases will constitute a
\(1 \times 10\) row vector. As with linear regression, we will
initialize our weights W with Gaussian noise and our biases to take
the initial value 0.
num_inputs = 784
num_outputs = 10
W = np.random.normal(0, 0.01, (num_inputs, num_outputs))
b = np.zeros(num_outputs)
W.attach_grad()
b.attach_grad()
num_inputs = 784
num_outputs = 10
W = torch.normal(0, 0.01, size=(num_inputs, num_outputs), requires_grad=True)
b = torch.zeros(num_outputs, requires_grad=True)
num_inputs = 784
num_outputs = 10
W = tf.Variable(tf.random.normal(shape=(num_inputs, num_outputs),
mean=0, stddev=0.01))
b = tf.Variable(tf.zeros(num_outputs))
3.6.2. Defining the Softmax Operation¶
Before implementing the softmax regression model, let us briefly review
how the sum operator works along specific dimensions in a tensor, as
discussed in Section 2.3.6 and
Section 2.3.6.1. Given a matrix X we can
sum over all elements (by default) or only over elements in the same
axis, i.e., the same column (axis 0) or the same row (axis 1). Note that
if X is a tensor with shape (2, 3) and we sum over the columns, the
result will be a vector with shape (3,). When invoking the sum operator,
we can specify to keep the number of axes in the original tensor, rather
than collapsing out the dimension that we summed over. This will result
in a two-dimensional tensor with shape (1, 3).
X = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
X.sum(0, keepdims=True), X.sum(1, keepdims=True)
(array([[5., 7., 9.]]),
array([[ 6.],
[15.]]))
X = torch.tensor([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
X.sum(0, keepdim=True), X.sum(1, keepdim=True)
(tensor([[5., 7., 9.]]),
tensor([[ 6.],
[15.]]))
X = tf.constant([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
tf.reduce_sum(X, 0, keepdims=True), tf.reduce_sum(X, 1, keepdims=True)
(<tf.Tensor: shape=(1, 3), dtype=float32, numpy=array([[5., 7., 9.]], dtype=float32)>,
<tf.Tensor: shape=(2, 1), dtype=float32, numpy=
array([[ 6.],
[15.]], dtype=float32)>)
We are now ready to implement the softmax operation. Recall that softmax
consists of three steps: (i) we exponentiate each term (using exp);
(ii) we sum over each row (we have one row per example in the batch) to
get the normalization constant for each example; (iii) we divide each
row by its normalization constant, ensuring that the result sums to 1.
Before looking at the code, let us recall how this looks expressed as an
equation:
The denominator, or normalization constant, is also sometimes called the partition function (and its logarithm is called 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 = np.exp(X)
partition = X_exp.sum(1, keepdims=True)
return X_exp / partition # The broadcasting mechanism is applied here
def softmax(X):
X_exp = torch.exp(X)
partition = X_exp.sum(1, keepdim=True)
return X_exp / partition # The broadcasting mechanism is applied here
def softmax(X):
X_exp = tf.exp(X)
partition = tf.reduce_sum(X_exp, 1, keepdims=True)
return X_exp / partition # The broadcasting 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.
X = np.random.normal(0, 1, (2, 5))
X_prob = softmax(X)
X_prob, X_prob.sum(1)
(array([[0.22376052, 0.06659239, 0.06583703, 0.29964197, 0.3441681 ],
[0.63209665, 0.03179282, 0.194987 , 0.09209415, 0.04902935]]),
array([1. , 0.99999994]))
X = torch.normal(0, 1, (2, 5))
X_prob = softmax(X)
X_prob, X_prob.sum(1)
(tensor([[0.4805, 0.0865, 0.1355, 0.2927, 0.0047],
[0.7267, 0.1369, 0.0596, 0.0387, 0.0381]]),
tensor([1., 1.]))
X = tf.random.normal((2, 5), 0, 1)
X_prob = softmax(X)
X_prob, tf.reduce_sum(X_prob, 1)
(<tf.Tensor: shape=(2, 5), dtype=float32, numpy=
array([[0.2237906 , 0.38737074, 0.05478511, 0.2320722 , 0.10198135],
[0.02762193, 0.02948399, 0.17606923, 0.3539647 , 0.4128601 ]],
dtype=float32)>,
<tf.Tensor: shape=(2,), dtype=float32, numpy=array([1.0000001 , 0.99999994], dtype=float32)>)
Note that while this looks correct mathematically, we were a bit sloppy in our implementation because we failed to take precautions against numerical overflow or underflow due to large or very small elements of the matrix.
3.6.3. Defining the Model¶
Now that we have defined the softmax operation, we can implement the
softmax regression model. The below code defines how the input is mapped
to the output through the network. Note that we flatten each original
image in the batch into a vector using the reshape function before
passing the data through our model.
def net(X):
return softmax(np.dot(X.reshape((-1, W.shape[0])), W) + b)
def net(X):
return softmax(torch.matmul(X.reshape((-1, W.shape[0])), W) + b)
def net(X):
return softmax(tf.matmul(tf.reshape(X, (-1, W.shape[0])), W) + b)
3.6.4. Defining the Loss Function¶
Next, we need to implement the cross-entropy loss function, as introduced in Section 3.4. 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. Rather than iterating
over the predictions with a Python for-loop (which tends to be
inefficient), we can pick all elements by a single operator. Below, we
create sample data y_hat with 2 examples of predicted probabilities
over 3 classes and their corresponding labels y. With y we know
that in the first example the first class is the correct prediction and
in the second example the third class is the ground-truth. Using y
as the indices of the probabilities in y_hat, we pick the
probability of the first class in the first example and the probability
of the third class in the second example.
y = np.array([0, 2])
y_hat = np.array([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]])
y_hat[[0, 1], y]
array([0.1, 0.5])
y = torch.tensor([0, 2])
y_hat = torch.tensor([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]])
y_hat[[0, 1], y]
tensor([0.1000, 0.5000])
y_hat = tf.constant([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]])
y = tf.constant([0, 2])
tf.boolean_mask(y_hat, tf.one_hot(y, depth=y_hat.shape[-1]))
<tf.Tensor: shape=(2,), dtype=float32, numpy=array([0.1, 0.5], dtype=float32)>
Now we can implement the cross-entropy loss function efficiently with just one line of code.
def cross_entropy(y_hat, y):
return - np.log(y_hat[range(len(y_hat)), y])
cross_entropy(y_hat, y)
array([2.3025851, 0.6931472])
def cross_entropy(y_hat, y):
return - torch.log(y_hat[range(len(y_hat)), y])
cross_entropy(y_hat, y)
tensor([2.3026, 0.6931])
def cross_entropy(y_hat, y):
return -tf.math.log(tf.boolean_mask(
y_hat, tf.one_hot(y, depth=y_hat.shape[-1])))
cross_entropy(y_hat, y)
<tf.Tensor: shape=(2,), dtype=float32, numpy=array([2.3025851, 0.6931472], dtype=float32)>
3.6.5. Classification Accuracy¶
Given the predicted probability distribution y_hat, we typically
choose the class with the highest predicted probability whenever we must
output a hard prediction. Indeed, many applications require that we make
a choice. Gmail must categorize 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 classes.
When predictions are consistent with the label class y, they are
correct. The classification accuracy is the fraction of all predictions
that are correct. Although it can be difficult to optimize accuracy
directly (it is not differentiable), it is often the performance measure
that we care most about, and we will nearly always report it when
training classifiers.
To compute accuracy we do the following. First, if y_hat is a
matrix, we assume that the second dimension stores prediction scores for
each class. We use argmax to obtain the predicted class by the index
for the largest entry in each row. Then we compare the predicted class
with the ground-truth y elementwise. Since the equality operator
== is sensitive to data types, we convert y_hat’s data type to
match that of y. The result is a tensor containing entries of 0
(false) and 1 (true). Taking the sum yields the number of correct
predictions.
def accuracy(y_hat, y): #@save
"""Compute the number of correct predictions."""
if len(y_hat.shape) > 1 and y_hat.shape[1] > 1:
y_hat = y_hat.argmax(axis=1)
cmp = y_hat.astype(y.dtype) == y
return float(cmp.astype(y.dtype).sum())
def accuracy(y_hat, y): #@save
"""Compute the number of correct predictions."""
if len(y_hat.shape) > 1 and y_hat.shape[1] > 1:
y_hat = y_hat.argmax(axis=1)
cmp = y_hat.type(y.dtype) == y
return float(cmp.type(y.dtype).sum())
def accuracy(y_hat, y): #@save
"""Compute the number of correct predictions."""
if len(y_hat.shape) > 1 and y_hat.shape[1] > 1:
y_hat = tf.argmax(y_hat, axis=1)
cmp = tf.cast(y_hat, y.dtype) == y
return float(tf.reduce_sum(tf.cast(cmp, y.dtype)))
We will continue to use the variables y_hat and y defined before
as the predicted probability distributions and labels, respectively. We
can see that the first example’s prediction class is 2 (the largest
element of the row is 0.6 with the index 2), which is inconsistent with
the actual label, 0. The second example’s prediction class is 2 (the
largest element of the row is 0.5 with the 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) / len(y)
0.5
accuracy(y_hat, y) / len(y)
0.5
accuracy(y_hat, y) / len(y)
0.5
Similarly, we can evaluate the accuracy for any model net on a
dataset that is accessed via the data iterator data_iter.
def evaluate_accuracy(net, data_iter): #@save
"""Compute the accuracy for a model on a dataset."""
metric = Accumulator(2) # No. of correct predictions, no. of predictions
for X, y in data_iter:
metric.add(accuracy(net(X), y), d2l.size(y))
return metric[0] / metric[1]
def evaluate_accuracy(net, data_iter): #@save
"""Compute the accuracy for a model on a dataset."""
if isinstance(net, torch.nn.Module):
net.eval() # Set the model to evaluation mode
metric = Accumulator(2) # No. of correct predictions, no. of predictions
with torch.no_grad():
for X, y in data_iter:
metric.add(accuracy(net(X), y), y.numel())
return metric[0] / metric[1]
def evaluate_accuracy(net, data_iter): #@save
"""Compute the accuracy for a model on a dataset."""
metric = Accumulator(2) # No. of correct predictions, no. of predictions
for X, y in data_iter:
metric.add(accuracy(net(X), y), d2l.size(y))
return metric[0] / metric[1]
Here Accumulator is a utility class to accumulate sums over multiple
variables. In the above evaluate_accuracy function, we create 2
variables in the Accumulator instance for storing both the number of
correct predictions and the number of predictions, respectively. Both
will be accumulated over time as we iterate over the dataset.
class Accumulator: #@save
"""For accumulating sums over `n` variables."""
def __init__(self, n):
self.data = [0.0] * n
def add(self, *args):
self.data = [a + float(b) for a, b in zip(self.data, args)]
def reset(self):
self.data = [0.0] * len(self.data)
def __getitem__(self, idx):
return self.data[idx]
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(net, test_iter)
0.0811
evaluate_accuracy(net, test_iter)
0.1248
evaluate_accuracy(net, test_iter)
0.1623
3.6.6. Training¶
The training loop for softmax regression should look strikingly familiar
if you read through our implementation of linear regression in
Section 3.2. Here we refactor the implementation to
make it reusable. First, we define a function to train for one epoch.
Note that updater is a general function to update the model
parameters, which accepts the batch size as an argument. It can be
either a wrapper of the d2l.sgd function or a framework’s built-in
optimization function.
def train_epoch_ch3(net, train_iter, loss, updater): #@save
"""Train a model within one epoch (defined in Chapter 3)."""
# Sum of training loss, sum of training accuracy, no. of examples
metric = Accumulator(3)
if isinstance(updater, gluon.Trainer):
updater = updater.step
for X, y in train_iter:
# Compute gradients and update parameters
with autograd.record():
y_hat = net(X)
l = loss(y_hat, y)
l.backward()
updater(X.shape[0])
metric.add(float(l.sum()), accuracy(y_hat, y), y.size)
# Return training loss and training accuracy
return metric[0] / metric[2], metric[1] / metric[2]
def train_epoch_ch3(net, train_iter, loss, updater): #@save
"""The training loop defined in Chapter 3."""
# Set the model to training mode
if isinstance(net, torch.nn.Module):
net.train()
# Sum of training loss, sum of training accuracy, no. of examples
metric = Accumulator(3)
for X, y in train_iter:
# Compute gradients and update parameters
y_hat = net(X)
l = loss(y_hat, y)
if isinstance(updater, torch.optim.Optimizer):
# Using PyTorch in-built optimizer & loss criterion
updater.zero_grad()
l.mean().backward()
updater.step()
else:
# Using custom built optimizer & loss criterion
l.sum().backward()
updater(X.shape[0])
metric.add(float(l.sum()), accuracy(y_hat, y), y.numel())
# Return training loss and training accuracy
return metric[0] / metric[2], metric[1] / metric[2]
def train_epoch_ch3(net, train_iter, loss, updater): #@save
"""The training loop defined in Chapter 3."""
# Sum of training loss, sum of training accuracy, no. of examples
metric = Accumulator(3)
for X, y in train_iter:
# Compute gradients and update parameters
with tf.GradientTape() as tape:
y_hat = net(X)
# Keras implementations for loss takes (labels, predictions)
# instead of (predictions, labels) that users might implement
# in this book, e.g. `cross_entropy` that we implemented above
if isinstance(loss, tf.keras.losses.Loss):
l = loss(y, y_hat)
else:
l = loss(y_hat, y)
if isinstance(updater, tf.keras.optimizers.Optimizer):
params = net.trainable_variables
grads = tape.gradient(l, params)
updater.apply_gradients(zip(grads, params))
else:
updater(X.shape[0], tape.gradient(l, updater.params))
# Keras loss by default returns the average loss in a batch
l_sum = l * float(tf.size(y)) if isinstance(
loss, tf.keras.losses.Loss) else tf.reduce_sum(l)
metric.add(l_sum, accuracy(y_hat, y), tf.size(y))
# Return training loss and training accuracy
return metric[0] / metric[2], metric[1] / metric[2]
Before showing the implementation of the training function, we define a utility class that plot data in animation. Again, it aims to simplify code in the rest of the book.
class Animator: #@save
"""For plotting data in animation."""
def __init__(self, xlabel=None, ylabel=None, legend=None, xlim=None,
ylim=None, xscale='linear', yscale='linear',
fmts=('-', 'm--', 'g-.', 'r:'), nrows=1, ncols=1,
figsize=(3.5, 2.5)):
# Incrementally plot multiple lines
if legend is None:
legend = []
d2l.use_svg_display()
self.fig, self.axes = d2l.plt.subplots(nrows, ncols, figsize=figsize)
if nrows * ncols == 1:
self.axes = [self.axes, ]
# Use a lambda function to capture arguments
self.config_axes = lambda: d2l.set_axes(
self.axes[0], xlabel, ylabel, xlim, ylim, xscale, yscale, legend)
self.X, self.Y, self.fmts = None, None, fmts
def add(self, x, y):
# Add multiple data points into the figure
if not hasattr(y, "__len__"):
y = [y]
n = len(y)
if not hasattr(x, "__len__"):
x = [x] * n
if not self.X:
self.X = [[] for _ in range(n)]
if not self.Y:
self.Y = [[] for _ in range(n)]
for i, (a, b) in enumerate(zip(x, y)):
if a is not None and b is not None:
self.X[i].append(a)
self.Y[i].append(b)
self.axes[0].cla()
for x, y, fmt in zip(self.X, self.Y, self.fmts):
self.axes[0].plot(x, y, fmt)
self.config_axes()
display.display(self.fig)
display.clear_output(wait=True)
The following training function then trains a model net on a
training dataset accessed via train_iter for multiple epochs, which
is specified by num_epochs. At the end of each epoch, the model is
evaluated on a testing dataset accessed via test_iter. We will
leverage the Animator class to visualize the training progress.
def train_ch3(net, train_iter, test_iter, loss, num_epochs, updater): #@save
"""Train a model (defined in Chapter 3)."""
animator = Animator(xlabel='epoch', xlim=[1, num_epochs], ylim=[0.3, 0.9],
legend=['train loss', 'train acc', 'test acc'])
for epoch in range(num_epochs):
train_metrics = train_epoch_ch3(net, train_iter, loss, updater)
test_acc = evaluate_accuracy(net, test_iter)
animator.add(epoch + 1, train_metrics + (test_acc,))
train_loss, train_acc = train_metrics
assert train_loss < 0.5, train_loss
assert train_acc <= 1 and train_acc > 0.7, train_acc
assert test_acc <= 1 and test_acc > 0.7, test_acc
As an implementation from scratch, we use the minibatch stochastic gradient descent defined in Section 3.2 to optimize the loss function of the model with a learning rate 0.1.
lr = 0.1
def updater(batch_size):
return d2l.sgd([W, b], lr, batch_size)
lr = 0.1
def updater(batch_size):
return d2l.sgd([W, b], lr, batch_size)
class Updater(): #@save
"""For updating parameters using minibatch stochastic gradient descent."""
def __init__(self, params, lr):
self.params = params
self.lr = lr
def __call__(self, batch_size, grads):
d2l.sgd(self.params, grads, self.lr, batch_size)
updater = Updater([W, b], lr=0.1)
Now we train the model with 10 epochs. Note that both the number of
epochs (num_epochs), and learning rate (lr) are adjustable
hyperparameters. By changing their values, we may be able to increase
the classification accuracy of the model.
num_epochs = 10
train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)
num_epochs = 10
train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)
num_epochs = 10
train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)
3.6.7. Prediction¶
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 predictions from the model (second line of text output).
def predict_ch3(net, test_iter, n=6): #@save
"""Predict labels (defined in Chapter 3)."""
for X, y in test_iter:
break
trues = d2l.get_fashion_mnist_labels(y)
preds = d2l.get_fashion_mnist_labels(net(X).argmax(axis=1))
titles = [true +'\n' + pred for true, pred in zip(trues, preds)]
d2l.show_images(
X[0:n].reshape((n, 28, 28)), 1, n, titles=titles[0:n])
predict_ch3(net, test_iter)
def predict_ch3(net, test_iter, n=6): #@save
"""Predict labels (defined in Chapter 3)."""
for X, y in test_iter:
break
trues = d2l.get_fashion_mnist_labels(y)
preds = d2l.get_fashion_mnist_labels(net(X).argmax(axis=1))
titles = [true +'\n' + pred for true, pred in zip(trues, preds)]
d2l.show_images(
X[0:n].reshape((n, 28, 28)), 1, n, titles=titles[0:n])
predict_ch3(net, test_iter)
def predict_ch3(net, test_iter, n=6): #@save
"""Predict labels (defined in Chapter 3)."""
for X, y in test_iter:
break
trues = d2l.get_fashion_mnist_labels(y)
preds = d2l.get_fashion_mnist_labels(tf.argmax(net(X), axis=1))
titles = [true +'\n' + pred for true, pred in zip(trues, preds)]
d2l.show_images(
tf.reshape(X[0:n], (n, 28, 28)), 1, n, titles=titles[0:n])
predict_ch3(net, test_iter)
3.6.8. Summary¶
With softmax regression, we can train models for multiclass classification.
The training loop of softmax regression 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 will soon find out, most common deep learning models have similar training procedures.
3.6.9. Exercises¶
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 was 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? For example, 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?