What is logits, softmax and softmax_cross_entropy_with_logits?

Question

I was going through the tensorflow API docs here. In the tensorflow documentation, they used a keyword called logits. What is it? In a lot of methods in the API docs it is written like

tf.nn.softmax(logits, name=None)

If what is written is those logits are only Tensors, why keeping a different name like logits?

Another thing is that there are two methods I could not differentiate. They were

tf.nn.softmax(logits, name=None)
tf.nn.softmax_cross_entropy_with_logits(logits, labels, name=None)

What are the differences between them? The docs are not clear to me. I know what tf.nn.softmax does. But not the other. An example will be really helpful.

Answer 1

Logits simply means that the function operates on the unscaled output of earlier layers and that the relative scale to understand the units is linear. It means, in particular, the sum of the inputs may not equal 1, that the values are not probabilities (you might have an input of 5).

tf.nn.softmax produces just the result of applying the softmax function to an input tensor. The softmax "squishes" the inputs so that sum(input) = 1: it's a way of normalizing. The shape of output of a softmax is the same as the input: it just normalizes the values. The outputs of softmax can be interpreted as probabilities.

a = tf.constant(np.array([[.1, .3, .5, .9]]))
print s.run(tf.nn.softmax(a))
[[ 0.16838508  0.205666    0.25120102  0.37474789]]

In contrast, tf.nn.softmax_cross_entropy_with_logits computes the cross entropy of the result after applying the softmax function (but it does it all together in a more mathematically careful way). It's similar to the result of:

sm = tf.nn.softmax(x)
ce = cross_entropy(sm)

The cross entropy is a summary metric: it sums across the elements. The output of tf.nn.softmax_cross_entropy_with_logits on a shape [2,5] tensor is of shape [2,1] (the first dimension is treated as the batch).

If you want to do optimization to minimize the cross entropy AND you're softmaxing after your last layer, you should use tf.nn.softmax_cross_entropy_with_logits instead of doing it yourself, because it covers numerically unstable corner cases in the mathematically right way. Otherwise, you'll end up hacking it by adding little epsilons here and there.

Edited 2016-02-07: If you have single-class labels, where an object can only belong to one class, you might now consider using tf.nn.sparse_softmax_cross_entropy_with_logits so that you don't have to convert your labels to a dense one-hot array. This function was added after release 0.6.0.