What is the difference between a sigmoid followed by the cross entropy and sigmoid_cross_entropy_with_logits in TensorFlow?

Question

When trying to get cross-entropy with sigmoid activation function, there is a difference between

1. loss1 = -tf.reduce_sum(p*tf.log(q), 1)
2. loss2 = tf.reduce_sum(tf.nn.sigmoid_cross_entropy_with_logits(labels=p, logits=logit_q),1)

But they are the same when with softmax activation function.

Following is the sample code:

import tensorflow as tf

sess2 = tf.InteractiveSession()
p = tf.placeholder(tf.float32, shape=[None, 5])
logit_q = tf.placeholder(tf.float32, shape=[None, 5])
q = tf.nn.sigmoid(logit_q)
sess.run(tf.global_variables_initializer())

feed_dict = {p: [[0, 0, 0, 1, 0], [1,0,0,0,0]], logit_q: [[0.2, 0.2, 0.2, 0.2, 0.2], [0.3, 0.3, 0.2, 0.1, 0.1]]}
loss1 = -tf.reduce_sum(p*tf.log(q),1).eval(feed_dict)
loss2 = tf.reduce_sum(tf.nn.sigmoid_cross_entropy_with_logits(labels=p, logits=logit_q),1).eval(feed_dict)

print(p.eval(feed_dict), "\n", q.eval(feed_dict))
print("\n",loss1, "\n", loss2)

Answer 1

You're confusing the cross-entropy for binary and multi-class problems.

Multi-class cross-entropy

The formula that you use is correct and it directly corresponds to tf.nn.softmax_cross_entropy_with_logits:

-tf.reduce_sum(p * tf.log(q), axis=1)

p and q are expected to be probability distributions over N classes. In particular, N can be 2, as in the following example:

p = tf.placeholder(tf.float32, shape=[None, 2])
logit_q = tf.placeholder(tf.float32, shape=[None, 2])
q = tf.nn.softmax(logit_q)

feed_dict = {
  p: [[0, 1],
      [1, 0],
      [1, 0]],
  logit_q: [[0.2, 0.8],
            [0.7, 0.3],
            [0.5, 0.5]]
}

prob1 = -tf.reduce_sum(p * tf.log(q), axis=1)
prob2 = tf.nn.softmax_cross_entropy_with_logits(labels=p, logits=logit_q)
print(prob1.eval(feed_dict))  # [ 0.43748799  0.51301527  0.69314718]
print(prob2.eval(feed_dict))  # [ 0.43748799  0.51301527  0.69314718]

Note that q is computing tf.nn.softmax, i.e. outputs a probability distribution. So it's still multi-class cross-entropy formula, only for N = 2.

Binary cross-entropy

This time the correct formula is

p * -tf.log(q) + (1 - p) * -tf.log(1 - q)

Though mathematically it's a partial case of the multi-class case, the meaning of p and q is different. In the simplest case, each p and q is a number, corresponding to a probability of the class A.

Important: Don't get confused by the common p * -tf.log(q) part and the sum. Previous p was a one-hot vector, now it's a number, zero or one. Same for q - it was a probability distribution, now's it's a number (probability).

If p is a vector, each individual component is considered an independent binary classification. See this answer that outlines the difference between softmax and sigmoid functions in tensorflow. So the definition p = [0, 0, 0, 1, 0] doesn't mean a one-hot vector, but 5 different features, 4 of which are off and 1 is on. The definition q = [0.2, 0.2, 0.2, 0.2, 0.2] means that each of 5 features is on with 20% probability.

This explains the use of sigmoid function before the cross-entropy: its goal is to squash the logit to [0, 1] interval.

The formula above still holds for multiple independent features, and that's exactly what tf.nn.sigmoid_cross_entropy_with_logits computes:

p = tf.placeholder(tf.float32, shape=[None, 5])
logit_q = tf.placeholder(tf.float32, shape=[None, 5])
q = tf.nn.sigmoid(logit_q)

feed_dict = {
  p: [[0, 0, 0, 1, 0],
      [1, 0, 0, 0, 0]],
  logit_q: [[0.2, 0.2, 0.2, 0.2, 0.2],
            [0.3, 0.3, 0.2, 0.1, 0.1]]
}

prob1 = -p * tf.log(q)
prob2 = p * -tf.log(q) + (1 - p) * -tf.log(1 - q)
prob3 = p * -tf.log(tf.sigmoid(logit_q)) + (1-p) * -tf.log(1-tf.sigmoid(logit_q))
prob4 = tf.nn.sigmoid_cross_entropy_with_logits(labels=p, logits=logit_q)
print(prob1.eval(feed_dict))
print(prob2.eval(feed_dict))
print(prob3.eval(feed_dict))
print(prob4.eval(feed_dict))

You should see that the last three tensors are equal, while the prob1 is only a part of cross-entropy, so it contains correct value only when p is 1:

[[ 0.          0.          0.          0.59813893  0.        ]
 [ 0.55435514  0.          0.          0.          0.        ]]
[[ 0.79813886  0.79813886  0.79813886  0.59813887  0.79813886]
 [ 0.5543552   0.85435522  0.79813886  0.74439669  0.74439669]]
[[ 0.7981388   0.7981388   0.7981388   0.59813893  0.7981388 ]
 [ 0.55435514  0.85435534  0.7981388   0.74439663  0.74439663]]
[[ 0.7981388   0.7981388   0.7981388   0.59813893  0.7981388 ]
 [ 0.55435514  0.85435534  0.7981388   0.74439663  0.74439663]]

Now it should be clear that taking a sum of -p * tf.log(q) along axis=1 doesn't make sense in this setting, though it'd be a valid formula in multi-class case.