# 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.

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