Loss function for class imbalanced binary classifier in Tensor flow

You can add class weights to the loss function, by multiplying logits. Regular cross entropy loss is this:

loss(x, class) = -log(exp(x[class]) / (\sum_j exp(x[j])))
               = -x[class] + log(\sum_j exp(x[j]))

in weighted case:

loss(x, class) = weights[class] * -x[class] + log(\sum_j exp(weights[class] * x[j]))

So by multiplying logits, you are re-scaling predictions of each class by its class weight.

For example:

ratio = 31.0 / (500.0 + 31.0)
class_weight = tf.constant([ratio, 1.0 - ratio])
logits = ... # shape [batch_size, 2]
weighted_logits = tf.mul(logits, class_weight) # shape [batch_size, 2]
xent = tf.nn.softmax_cross_entropy_with_logits(
  weighted_logits, labels, name="xent_raw")

There is a standard losses function now that supports weights per batch:

tf.losses.sparse_softmax_cross_entropy(labels=label, logits=logits, weights=weights)

Where weights should be transformed from class weights to a weight per example (with shape [batch_size]). See documentation here.

The code you proposed seems wrong to me. The loss should be multiplied by the weight, I agree.

But if you multiply the logit by the class weights, you end with:

weights[class] * -x[class] + log( \sum_j exp(x[j] * weights[class]) )

The second term is not equal to:

weights[class] * log(\sum_j exp(x[j]))

To show this, we can be rewrite the latter as:

log( (\sum_j exp(x[j]) ^ weights[class] )

So here is the code I'm proposing:

ratio = 31.0 / (500.0 + 31.0)
class_weight = tf.constant([[ratio, 1.0 - ratio]])
logits = ... # shape [batch_size, 2]

weight_per_label = tf.transpose( tf.matmul(labels
                           , tf.transpose(class_weight)) ) #shape [1, batch_size]
# this is the weight for each datapoint, depending on its label

xent = tf.mul(weight_per_label
         , tf.nn.softmax_cross_entropy_with_logits(logits, labels, name="xent_raw") #shape [1, batch_size]
loss = tf.reduce_mean(xent) #shape 1

Use tf.nn.weighted_cross_entropy_with_logits() and set pos_weight to 1 / (expected ratio of positives).