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

## Question:

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.

## Answers:

The softmax+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). Internally, it first applies softmax to the unscaled output, and then and then computes the cross entropy of those values vs. what they "should" be as defined by the labels.

`tf.nn.softmax`

produces the result of applying the softmax function to an input tensor. The softmax "squishes" the inputs so that `sum(input) = 1`

, and it does the mapping by interpreting the inputs as log-probabilities (logits) and then converting them back into raw probabilities between 0 and 1. The shape of output of a softmax is the same as the input:

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

See this answer for more about why softmax is used extensively in DNNs.

`tf.nn.softmax_cross_entropy_with_logits`

combines the softmax step with the calculation of the cross-entropy loss 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.

`tf.nn.softmax`

computes the forward propagation through a softmax layer. You use it during **evaluation** of the model when you compute the probabilities that the model outputs.

`tf.nn.softmax_cross_entropy_with_logits`

computes the cost for a softmax layer. It is only used during **training**.

The logits are the *unnormalized log probabilities* output the model (the values output before the softmax normalization is applied to them).

**Short version:**

Suppose you have two tensors, where `y_hat`

contains computed scores for each class (for example, from y = W*x +b) and `y_true`

contains one-hot encoded true labels.

```
y_hat = ... # Predicted label, e.g. y = tf.matmul(X, W) + b
y_true = ... # True label, one-hot encoded
```

If you interpret the scores in `y_hat`

as unnormalized log probabilities, then they are **logits**.

Additionally, the total cross-entropy loss computed in this manner:

```
y_hat_softmax = tf.nn.softmax(y_hat)
total_loss = tf.reduce_mean(-tf.reduce_sum(y_true * tf.log(y_hat_softmax), [1]))
```

is essentially equivalent to the total cross-entropy loss computed with the function `softmax_cross_entropy_with_logits()`

:

```
total_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(y_hat, y_true))
```

**Long version:**

In the output layer of your neural network, you will probably compute an array that contains the class scores for each of your training instances, such as from a computation `y_hat = W*x + b`

. To serve as an example, below I’ve created a `y_hat`

as a 2 x 3 array, where the rows correspond to the training instances and the columns correspond to classes. So here there are 2 training instances and 3 classes.

```
import tensorflow as tf
import numpy as np
sess = tf.Session()
# Create example y_hat.
y_hat = tf.convert_to_tensor(np.array([[0.5, 1.5, 0.1],[2.2, 1.3, 1.7]]))
sess.run(y_hat)
# array([[ 0.5, 1.5, 0.1],
# [ 2.2, 1.3, 1.7]])
```

Note that the values are not normalized (i.e. the rows don’t add up to 1). In order to normalize them, we can apply the softmax function, which interprets the input as unnormalized log probabilities (aka **logits**) and outputs normalized linear probabilities.

```
y_hat_softmax = tf.nn.softmax(y_hat)
sess.run(y_hat_softmax)
# array([[ 0.227863 , 0.61939586, 0.15274114],
# [ 0.49674623, 0.20196195, 0.30129182]])
```

It’s important to fully understand what the softmax output is saying. Below I’ve shown a table that more clearly represents the output above. It can be seen that, for example, the probability of training instance 1 being “Class 2” is 0.619. The class probabilities for each training instance are normalized, so the sum of each row is 1.0.

```
Pr(Class 1) Pr(Class 2) Pr(Class 3)
,--------------------------------------
Training instance 1 | 0.227863 | 0.61939586 | 0.15274114
Training instance 2 | 0.49674623 | 0.20196195 | 0.30129182
```

So now we have class probabilities for each training instance, where we can take the argmax() of each row to generate a final classification. From above, we may generate that training instance 1 belongs to “Class 2” and training instance 2 belongs to “Class 1”.

Are these classifications correct? We need to measure against the true labels from the training set. You will need a one-hot encoded `y_true`

array, where again the rows are training instances and columns are classes. Below I’ve created an example `y_true`

one-hot array where the true label for training instance 1 is “Class 2” and the true label for training instance 2 is “Class 3”.

```
y_true = tf.convert_to_tensor(np.array([[0.0, 1.0, 0.0],[0.0, 0.0, 1.0]]))
sess.run(y_true)
# array([[ 0., 1., 0.],
# [ 0., 0., 1.]])
```

Is the probability distribution in `y_hat_softmax`

close to the probability distribution in `y_true`

? We can use cross-entropy loss to measure the error.

We can compute the cross-entropy loss on a row-wise basis and see the results. Below we can see that training instance 1 has a loss of 0.479, while training instance 2 has a higher loss of 1.200. This result makes sense because in our example above, `y_hat_softmax`

showed that training instance 1’s highest probability was for “Class 2”, which matches training instance 1 in `y_true`

; however, the prediction for training instance 2 showed a highest probability for “Class 1”, which does not match the true class “Class 3”.

```
loss_per_instance_1 = -tf.reduce_sum(y_true * tf.log(y_hat_softmax), reduction_indices=[1])
sess.run(loss_per_instance_1)
# array([ 0.4790107 , 1.19967598])
```

What we really want is the total loss over all the training instances. So we can compute:

```
total_loss_1 = tf.reduce_mean(-tf.reduce_sum(y_true * tf.log(y_hat_softmax), reduction_indices=[1]))
sess.run(total_loss_1)
# 0.83934333897877944
```

**Using softmax_cross_entropy_with_logits()**

We can instead compute the total cross entropy loss using the `tf.nn.softmax_cross_entropy_with_logits()`

function, as shown below.

```
loss_per_instance_2 = tf.nn.softmax_cross_entropy_with_logits(y_hat, y_true)
sess.run(loss_per_instance_2)
# array([ 0.4790107 , 1.19967598])
total_loss_2 = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(y_hat, y_true))
sess.run(total_loss_2)
# 0.83934333897877922
```

Note that `total_loss_1`

and `total_loss_2`

produce essentially equivalent results with some small differences in the very final digits. However, you might as well use the second approach: it takes one less line of code and accumulates less numerical error because the softmax is done for you inside of `softmax_cross_entropy_with_logits()`

.

Above answers have enough description for the asked question.

Adding to that, Tensorflow has optimised the operation of applying the activation function then calculating cost using its own activation followed by cost functions. Hence it is a good practice to use: `tf.nn.softmax_cross_entropy()`

over `tf.nn.softmax(); tf.nn.cross_entropy()`

You can find prominent difference between them in a resource intensive model.

**Tensorflow 2.0 Compatible Answer**: The explanations of `dga`

and `stackoverflowuser2010`

are very detailed about Logits and the related Functions.

All those functions, when used in ** Tensorflow 1.x** will work fine, but if you migrate your code from

**to**

`1.x (1.14, 1.15, etc)`

**, using those functions result in error.**

`2.x (2.0, 2.1, etc..)`

Hence, specifying the 2.0 Compatible Calls for all the functions, we discussed above, if we migrate from ** 1.x to 2.x**, for the benefit of the community.

**Functions in 1.x**:

`tf.nn.softmax`

`tf.nn.softmax_cross_entropy_with_logits`

`tf.nn.sparse_softmax_cross_entropy_with_logits`

**Respective Functions when Migrated from 1.x to 2.x**:

`tf.compat.v2.nn.softmax`

`tf.compat.v2.nn.softmax_cross_entropy_with_logits`

`tf.compat.v2.nn.sparse_softmax_cross_entropy_with_logits`

For more information about migration from 1.x to 2.x, please refer this Migration Guide.

One more thing that I would definitely like to highlight as logit is just a raw output, generally the output of last layer. This can be a negative value as well. If we use it as it’s for “cross entropy” evaluation as mentioned below:

```
-tf.reduce_sum(y_true * tf.log(logits))
```

then it wont work. As log of -ve is not defined.

So using o softmax activation, will overcome this problem.

This is my understanding, please correct me if Im wrong.

## Mathematical motivation for term

When we wish to constrain an output between 0 and 1, but our model architecture outputs unconstrained values, we can add a normalisation layer to enforce this.

A common choice is a sigmoid function.^{1} In binary classification this is typically the logistic function, and in multi-class tasks the multinomial logistic function (a.k.a **softmax**).^{2}

If we want to interpret the outputs of our new final layer as ‘probabilities’, then (by implication) the unconstrained inputs to our sigmoid must be `inverse-sigmoid`

(probabilities). In the logistic case this is equivalent to the *log-odds* of our probability (i.e. the log of the odds) a.k.a. **logit**:

That is why the arguments to `softmax`

is called `logits`

in Tensorflow – because under the assumption that `softmax`

is the final layer in the model, and the output *p* is interpreted as a probability, the input *x* to this layer is interpretable as a logit:

## Generalised term

In Machine Learning there is a propensity to generalise terminology borrowed from maths/stats/computer science, hence in Tensorflow `logit`

(by analogy) is used as a synonym for the input to many normalisation functions.

^{}

- While it has nice properties such as being easily diferentiable, and the aforementioned probabilistic interpretation, it is somewhat arbitrary.
`softmax`

might be more accurately called soft*arg*max, as it is a smooth approximation of the argmax function.

Logits are the unnormalized outputs of a neural network. Softmax is a normalization function that squashes the outputs of a neural network so that they are all between 0 and 1 and sum to 1. Softmax_cross_entropy_with_logits is a loss function that takes in the outputs of a neural network (after they have been squashed by softmax) and the true labels for those outputs, and returns a loss value.