mygrad.nnet.losses.softmax_crossentropy#

mygrad.nnet.losses.softmax_crossentropy(x: ArrayLike, y_true: ArrayLike, *, constant: Optional[bool] = None) → Tensor[source]#

Given the classification scores of C classes for N pieces of data,

computes the NxC softmax classification probabilities. The cross entropy is then computed by using the true classification labels.

log-softmax is used for improved numerical stability.

Parameters

xArrayLike, shape=(N, C): The C class scores for each of the N pieces of data.
y_trueArrayLike, shape=(N,): The correct class-indices, in [0, C), for each datum.
constantbool, optional(default=False): If True, the returned tensor is a constant (it does not back-propagate a gradient)

Returns

The average softmax loss

Raises

ValueError: Bad dimensionalities for x or y_true

Notes

\(N\) is the number of samples in the batch.
\(C\) is the number of possible classes for which scores are provided.

Given the shape-\((N, C)\) tensor of scores, x, the softmax classification probabilities are computed. That is, the score for class-\(k\) of a given datum (\(s_{k}\)) is normalized using the ‘softmax’ transformation:

\[p_{k} = \frac{e^{s_k}}{\sum_{i=1}^{C}{e^{s_i}}}\]

This produces the “prediction probability distribution”, \(p\), for each datum. The cross-entropy loss for that datum is then computed according to the true class-index for that datum, as reported in y_true. That is the “true probability distribution”, \(t\), for the datum is \(1\) for the correct class-index and \(0\) elsewhere.

The cross-entropy loss for that datum is thus:

\[l = - \sum_{k=1}^{C}{t_{k} \log{p_{k}}}\]

Having computed each per-datum cross entropy loss, this function then returns the loss averaged over all \(N\) pieces of data:

\[L = \frac{1}{N}\sum_{i=1}^{N}{l_{i}}\]

Examples

>>> import mygrad as mg
>>> from mygrad.nnet import softmax_crossentropy

Let’s take a simple case where N=1, and C=3. We’ll thus make up classification scores for a single datum. Suppose the scores are identical for the three classes and that the true class is class-0:

>>> x = mg.Tensor([[2., 2., 2.]])  # a shape-(1, 3) tensor of scores
>>> y_true = mg.Tensor([0])  # the correct class for this datum is class-0

Because the scores are identical for all three classes, the softmax normalization will simply produce \(p = [\frac{1}{3}, \frac{1}{3}, \frac{1}{3}]\). Because class-0 is the “true” class, \(t = [1., 0., 0.]\). Thus our softmax cross-entropy loss should be:

\[-(1 \times \log{\frac{1}{3}} + 0 \times \log{\frac{1}{3}} + 0 \times \log{\frac{1}{3}}) = \log(3) \approx 1.099\]

Let’s see that this is what softmax_crossentropy returns:

>>> softmax_crossentropy(x, y_true)
Tensor(1.09861229)

Similarly, suppose a datum’s scores are \([0, 0, 10^6]\), then the softmax normalization will return \(p \approx [0., 0., 1.]\). If the true class for this datum is class-2, then the loss should be nearly 0, since \(p\) and \(t\) are essentially identical:

\[-(0 \times \log{0} + 0 \times \log{0} + 1 \times \log{1}) = -\log(1) = 0\]

Now, let’s construct x and y_true so that they incorporate the scores/labels for both of the data that we have considered:

>>> x = mg.Tensor([[2., 2.,  2.],  # a shape-(2, 3) tensor of scores
...                [0., 0., 1E6]])
>>> y_true = mg.Tensor([0, 2])     # the class IDs for the two data

softmax_crossentropy(x, y_true) will return the average loss of these two data, \(\frac{1}{2}(1.099 + 0) \approx 0.55\):

>>> softmax_crossentropy(x, y_true)
Tensor(0.54930614)