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Binary Cross-Entropy
Kajal Pawar
2 years ago
Table of Content
- What is Binary Cross Entropy?
- Loss Function: Binary Cross-Entropy / Log Loss
- Computation of Binary cross-entropy in visual way.
- Implementation of Binary Cross-entropy / log loss using Python
Cross-entropy is one of the most used as a loss function when we try to
optimize any classification models.
Classification is a type of supervised learning problem,
which involves the prediction of a class label using one or more input
variables.
Basically, Classification problems that have just two labels
for the target, the variable is referred to as binary classification problems
and the problems with more than two labels/classes are referred to as categorical
or multi-class classification problems.
What is Binary Cross Entropy?
Binary
cross-entropy is
a loss function that is used in binary classification problems. The main aim of
these tasks is to answer a question with only two choices.
For example, we are
classifying
- Will, it rains today? Yes or No
- Is the email Spam or Not Spam?
Let me take
a simple classification example and explain to you how it actually works.
Let’s take some 10 random data points as
shown below and call it as x feature:
x = [-2.2, -1.4, -0.8, 0.2,
0.4, 0.8, 1.2, 2.2, 2.9, 4.6]
Now, let’s
we represent these data points on a number line as shown below.
Let’s assign
some different color to different data points which represent different classes
or labels as shown below.
We can see
that our classification task here is quite simple and straight-forward.
Task:
Given the feature x we have to classify the data points into red or green
labels or classes.
Now from the
above task, we can see that it is a binary classification task, we can
also take this task as: “Is the data point green” or in a better way,
“what is the probability of the point being green”?
Basically,
green points would have a probability equal to 1 of being green and
the red points would have a probability equal to 0 of being
green.
In the
above-mentioned scenario, green points belong to the positive class, i.e., Yes,
they are green, while the red points belong to the negative class, i.e., No,
they are not green.
Now, our
task is to build a model to perform this classification task, it will predict a
probability of being green for each of the data points. Given what we know
about the color of the points, how can we evaluate and know how good or bad are
the predictions made by our model.
This is
where the loss function comes into the picture, which will help us to
check whether our model is performing good or bad. It will return high
values for bad predictions and low values for good
predictions.
In our case,
for a binary classification, the typical loss functions are called as the
binary cross-entropy / log loss.
Loss Function:
Binary Cross-Entropy / Log Loss
The Binary cross-entropy loss function
actually
calculates the average cross entropy across all examples.
The formula of this loss function can be given by:
- Here, y represents the label / class (1 for the green points and 0 for the red points)
- p(y) represents the predicted probability of the data point being green for all N data points.
Let me explain you, what the formula given above actually
tells you.
For each green point (y=1), it adds log(p(y)) to the loss,
which means that the log probability of it being green. On the other hand, it
will add log(1-p(y)), which means that the log probability of it being red, for
each red point (y=0).
Computation of Binary cross-entropy
in visual way.
Let’s see how we can compute Binary cross-entropy in a visual
way first and then I will take you through how we can implement it using
python.
Let’s consider the above example only.
First, let’s
split these data points according to their respective classes, i.e., positive
and negative as shown below.
Next, let’s
build a Logistic Regression model to classify the given data points. The
fitted logistic regression model is a sigmoid curve which is
representing the probability of a point being green for any given x. It can be
given by:
Now, you
might be thinking what are the predicted probabilities of all the points
belonging to the positive class (green) or negative class (red) by our model.
Let’s me
show you how it will look actually.
These are
the green bars under the sigmoid curve, at the x coordinates corresponding to
the points as shown in the below figure.
And for the
negative class (red) it looks like as shown below:
Now
combining both the above figure we will get:
Since as of
now we have the predicted probabilities and let’s calculate the binary
cross-entropy / log loss.
Now we will
only take the bars graph which gives us the probabilities which all we need. It
can be shown as below:
Rearranging
the bars of the above figure, we will get:
As we are
trying to compute a loss, we need to penalize the bad predictions. But how?
Let’s see
If
the probability associated
with the true
class is 1.0, we need
its loss to
be zero.
On the other hand, if that probability is low,
say, 0.02,
we need its loss to be large.
So,
taking the (negative) log of the probability suits us well enough for this
purpose as the log of values between 0 and 1 is negative, we take the negative
log to obtain a positive value for the loss. To get more better understanding
we have to understand the math behind. It’s actually comes from the
cross-entropy.
From the below plot, we can see the predicted
probability of the true class gets closer to zero, the loss increases
exponentially
So, now let’s take the
(negative) log of the probabilities — these are the
corresponding losses of each and every point.
Finally, then we compute the mean of all
these losses as shown below
So, the calculated binary cross-entropy / log loss of the taken example here comes out to
be 0.3329.
Implementation of Binary Cross-entropy / log loss
using Python
Let’s
us implement it on python
# import libraries
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import log_loss
import numpy as np
x = np.array([-2.2, -1.4, -.8, .2, .4, .8, 1.2, 2.2, 2.9, 4.6])
y = np.array([0.0, 0.0, 1.0, 0.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])
logr = LogisticRegression(solver='lbfgs')
logr.fit(x.reshape(-1, 1), y)
y_pred = logr.predict_proba(x.reshape(-1, 1))[:, 1].ravel()
loss = log_loss(y, y_pred)
print('x = {}'.format(x))
print('y = {}'.format(y))
print('p(y) = {}'.format(np.round(y_pred, 2)))
print('Log Loss / Cross Entropy = {:.4f}'.format(loss))
Output:
x = [-2.2 -1.4 -0.8 0.2 0.4 0.8 1.2 2.2 2.9 4.6]
y = [0. 0. 1. 0. 1. 1. 1. 1. 1. 1.]
p(y) = [0.19 0.33 0.47 0.7 0.74 0.81 0.86 0.94 0.97 0.99]
Log Loss / Cross Entropy = 0.3329
Let’s take
an and write a full python code and calculate the binary cross-entropy / log
loss.
# mlp for the circles problem with cross entropy loss
from sklearn.datasets import make_circles
from keras.models import Sequential
from keras.layers import Dense
from keras.optimizers import SGD
from matplotlib import pyplot
# generate 2d classification dataset
X, y = make_circles(n_samples=1000, noise=0.1, random_state=1)
# split into train and test
n_train = 500
trainX, testX = X[:n_train, :], X[n_train:, :]
trainy, testy = y[:n_train], y[n_train:]
# define model
model = Sequential()
model.add(Dense(50, input_dim=2, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(1, activation='sigmoid'))
opt = SGD(lr=0.01, momentum=0.9)
model.compile(loss='binary_crossentropy', optimizer=opt, metrics=['accuracy'])
# fit model
history = model.fit(trainX, trainy, validation_data=(testX, testy), epochs=200, verbose=0)
# evaluate the model
_, train_acc = model.evaluate(trainX, trainy, verbose=0)
_, test_acc = model.evaluate(testX, testy, verbose=0)
print('Train: %.3f, Test: %.3f' % (train_acc, test_acc))
# plot loss during training
pyplot.subplot(211)
pyplot.title('Loss')
pyplot.plot(history.history['loss'], label='train')
pyplot.plot(history.history['val_loss'], label='test')
pyplot.legend()
# plot accuracy during training
pyplot.subplot(212)
pyplot.title('Accuracy')
pyplot.plot(history.history['accuracy'], label='train')
pyplot.plot(history.history['val_accuracy'], label='test')
pyplot.legend()
pyplot.show()
Output:
The
above code first output the binary cross entropy for the model on the train and test
datasets as
Train: 0.840, Test: 0.853
Then
it will plot training and testing loss as shown below:
After
reading this article, finally you came to know the importance of Binary Cross-entropy / log loss.
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