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tf.nn.sigmoid_cross_entropy_with_logits

Computes sigmoid cross entropy given logits.

Used in the notebooks

Used in the tutorials

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 for general multi-class classification and 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)

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).

A Tensor of the same shape as logits with the componentwise logistic losses.

ValueError If logits and labels do not have the same shape.