Unbalanced data and weighted cross entropy

Note that weighted_cross_entropy_with_logits is the weighted variant of sigmoid_cross_entropy_with_logits. Sigmoid cross entropy is typically used for binary classification. Yes, it can handle multiple labels, but sigmoid cross entropy basically makes a (binary) decision on each of them -- for example, for a face recognition net, those (not mutually exclusive) labels could be "Does the subject wear glasses?", "Is the subject female?", etc.

In binary classification(s), each output channel corresponds to a binary (soft) decision. Therefore, the weighting needs to happen within the computation of the loss. This is what weighted_cross_entropy_with_logits does, by weighting one term of the cross-entropy over the other.

In mutually exclusive multilabel classification, we use softmax_cross_entropy_with_logits, which behaves differently: each output channel corresponds to the score of a class candidate. The decision comes after, by comparing the respective outputs of each channel.

Weighting in before the final decision is therefore a simple matter of modifying the scores before comparing them, typically by multiplication with weights. For example, for a ternary classification task,

# your class weights
class_weights = tf.constant([[1.0, 2.0, 3.0]])
# deduce weights for batch samples based on their true label
weights = tf.reduce_sum(class_weights * onehot_labels, axis=1)
# compute your (unweighted) softmax cross entropy loss
unweighted_losses = tf.nn.softmax_cross_entropy_with_logits(onehot_labels, logits)
# apply the weights, relying on broadcasting of the multiplication
weighted_losses = unweighted_losses * weights
# reduce the result to get your final loss
loss = tf.reduce_mean(weighted_losses)

You could also rely on tf.losses.softmax_cross_entropy to handle the last three steps.

In your case, where you need to tackle data imbalance, the class weights could indeed be inversely proportional to their frequency in your train data. Normalizing them so that they sum up to one or to the number of classes also makes sense.

Note that in the above, we penalized the loss based on the true label of the samples. We could also have penalized the loss based on the estimated labels by simply defining

weights = class_weights

and the rest of the code need not change thanks to broadcasting magic.

In the general case, you would want weights that depend on the kind of error you make. In other words, for each pair of labels X and Y, you could choose how to penalize choosing label X when the true label is Y. You end up with a whole prior weight matrix, which results in weights above being a full (num_samples, num_classes) tensor. This goes a bit beyond what you want, but it might be useful to know nonetheless that only your definition of the weight tensor need to change in the code above.

See this answer for an alternate solution which works with sparse_softmax_cross_entropy:

import  tensorflow as tf
import numpy as np

np.random.seed(123)
sess = tf.InteractiveSession()

# let's say we have the logits and labels of a batch of size 6 with 5 classes
logits = tf.constant(np.random.randint(0, 10, 30).reshape(6, 5), dtype=tf.float32)
labels = tf.constant(np.random.randint(0, 5, 6), dtype=tf.int32)

# specify some class weightings
class_weights = tf.constant([0.3, 0.1, 0.2, 0.3, 0.1])

# specify the weights for each sample in the batch (without having to compute the onehot label matrix)
weights = tf.gather(class_weights, labels)

# compute the loss
tf.losses.sparse_softmax_cross_entropy(labels, logits, weights).eval()