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tf.nn.weighted_cross_entropy_with_logits

Computes a weighted cross entropy.

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.

labels = tf.constant([1., 0.5, 0.])
logits = tf.constant([1.5, -0.1, -10.])
tf.nn.weighted_cross_entropy_with_logits(
    labels=labels, logits=logits, pos_weight=tf.constant(1.5)).numpy()
array([3.0211994e-01, 8.8049585e-01, 4.5776367e-05], dtype=float32)
tf.nn.weighted_cross_entropy_with_logits(
    labels=labels, logits=logits, pos_weight=tf.constant(0.5)).numpy()
array([1.00706644e-01, 5.08297503e-01, 4.57763672e-05], dtype=float32)

labels A Tensor of the same type and shape as logits, with values between 0 and 1 inclusive.
logits A Tensor of type float32 or float64, any real numbers.
pos_weight A coefficient to use on the positive examples, typically a scalar but otherwise broadcastable to the shape of logits. Its value should be non-negative.
name A name for the operation (optional).

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

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