# What are logits? What is the difference between softmax and softmax_cross_entropy_with_logits?

## Issue

In the tensorflow API docs they use a keyword called `logits`. What is it? A lot of methods are written like:

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

If `logits` is just a generic `Tensor` input, why is it named `logits`?

Secondly, what is the difference between the following two methods?

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

I know what `tf.nn.softmax` does, but not the other. An example would be really helpful.

## Solution

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.