tf.compat.v1.nn.weighted_cross_entropy_with_logits

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Computes a weighted cross entropy. (deprecated arguments)

tf.compat.v1.nn.weighted_cross_entropy_with_logits(
    labels=None, logits=None, pos_weight=None, name=None, targets=None
)

This is like sigmoid_cross_entropy_with_logits() except that pos_weight, allows one to trade off recall and precision by up- or down-weighting the cost of a positive error relative to a negative error.

The usual cross-entropy cost is defined as:

labels * -log(sigmoid(logits)) +
    (1 - labels) * -log(1 - sigmoid(logits))

A value pos_weight > 1 decreases the false negative count, hence increasing the recall. Conversely setting pos_weight < 1 decreases the false positive count and increases the precision. This can be seen from the fact that pos_weight is introduced as a multiplicative coefficient for the positive labels term in the loss expression:

labels * -log(sigmoid(logits)) * pos_weight +
    (1 - labels) * -log(1 - sigmoid(logits))

For brevity, let x = logits, z = labels, q = pos_weight. The loss is:

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

Setting l = (1 + (q - 1) * z), to ensure stability and avoid overflow, the implementation uses

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

logits and labels must have the same type and shape.