# tf.nn.sigmoid_cross_entropy_with_logits

|---|
| [![](https://www.tensorflow.org/images/GitHub-Mark-32px.png) View source on GitHub](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/nn_impl.py#L152-L244) |

Computes sigmoid cross entropy given `logits`.

    tf.nn.sigmoid_cross_entropy_with_logits(
        labels=None, logits=None, name=None
    )

Measures the probability error in tasks with two outcomes in which each
outcome is independent and need not have a fully certain label. For instance,
one could perform a regression where the probability of an event happening is
known and used as a label. This loss may also be used for binary
classification, where labels are either zero or one.

For brevity, let `x = logits`, `z = labels`. The logistic loss is

      z * -log(sigmoid(x)) + (1 - z) * -log(1 - sigmoid(x))
    = z * -log(1 / (1 + exp(-x))) + (1 - z) * -log(exp(-x) / (1 + exp(-x)))
    = z * log(1 + exp(-x)) + (1 - z) * (-log(exp(-x)) + log(1 + exp(-x)))
    = z * log(1 + exp(-x)) + (1 - z) * (x + log(1 + exp(-x))
    = (1 - z) * x + log(1 + exp(-x))
    = x - x * z + log(1 + exp(-x))

For x \< 0, to avoid overflow in exp(-x), we reformulate the above

      x - x * z + log(1 + exp(-x))
    = log(exp(x)) - x * z + log(1 + exp(-x))
    = - x * z + log(1 + exp(x))

Hence, to ensure stability and avoid overflow, the implementation uses this
equivalent formulation

    max(x, 0) - x * z + log(1 + exp(-abs(x)))

`logits` and `labels` must have the same type and shape.

    logits = tf.constant([1., -1., 0., 1., -1., 0., 0.])
    labels = tf.constant([0., 0., 0., 1., 1., 1., 0.5])
    tf.nn.sigmoid_cross_entropy_with_logits(
        labels=labels, logits=logits).numpy()
    array([1.3132617, 0.3132617, 0.6931472, 0.3132617, 1.3132617, 0.6931472,
           0.6931472], dtype=float32)

Compared to the losses which handle multiple outcomes,
[`tf.nn.softmax_cross_entropy_with_logits`](https://www.tensorflow.org/versions/r2.14/api_docs/python/tf/nn/softmax_cross_entropy_with_logits) for general multi-class
classification and [`tf.nn.sparse_softmax_cross_entropy_with_logits`](https://www.tensorflow.org/versions/r2.14/api_docs/python/tf/nn/sparse_softmax_cross_entropy_with_logits) for more
efficient multi-class classification with hard labels,
`sigmoid_cross_entropy_with_logits` is a slight simplification for binary
classification:

      sigmoid(x) = softmax([x, 0])[0]

\\\[\\frac{1}{1 + e\^{-x} } = \\frac{e\^x}{e\^x + e\^0}\\\]

While `sigmoid_cross_entropy_with_logits` works for soft binary labels
(probabilities between 0 and 1), it can also be used for binary classification
where the labels are hard. There is an equivalence between all three symbols
in this case, with a probability 0 indicating the second class or 1 indicating
the first class:

    sigmoid_logits = tf.constant([1., -1., 0.])
    softmax_logits = tf.stack([sigmoid_logits, tf.zeros_like(sigmoid_logits)],
                              axis=-1)
    soft_binary_labels = tf.constant([1., 1., 0.])
    soft_multiclass_labels = tf.stack(
        [soft_binary_labels, 1. - soft_binary_labels], axis=-1)
    hard_labels = tf.constant([0, 0, 1])
    tf.nn.sparse_softmax_cross_entropy_with_logits(
        labels=hard_labels, logits=softmax_logits).numpy()
    array([0.31326166, 1.3132616 , 0.6931472 ], dtype=float32)
    tf.nn.softmax_cross_entropy_with_logits(
        labels=soft_multiclass_labels, logits=softmax_logits).numpy()
    array([0.31326166, 1.3132616, 0.6931472], dtype=float32)
    tf.nn.sigmoid_cross_entropy_with_logits(
        labels=soft_binary_labels, logits=sigmoid_logits).numpy()
    array([0.31326166, 1.3132616, 0.6931472], dtype=float32)

<br />

<br />

<br />

| ## Args ||
|---|---|
| `labels` | A `Tensor` of the same type and shape as `logits`. Between 0 and 1, inclusive. |
| `logits` | A `Tensor` of type `float32` or `float64`. Any real number. |
| `name` | A name for the operation (optional). |

<br />

<br />

<br />

<br />

| ## Returns ||
|---|---|
| A `Tensor` of the same shape as `logits` with the componentwise logistic losses. ||

<br />

<br />

<br />

<br />

| ## Raises ||
|---|---|
| `ValueError` | If `logits` and `labels` do not have the same shape. |

<br />