Neural network for multi label classification with large number of classes outputs only zero

Tensorflow has a loss function weighted_cross_entropy_with_logits, which can be used to give more weight to the 1's. So it should be applicable to a sparse multi-label classification setting like yours.

From the documentation:

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 argument pos_weight is used as a multiplier for the positive targets

If you use the tensorflow backend in Keras, you can use the loss function like this (Keras 2.1.1):

import tensorflow as tf
import keras.backend.tensorflow_backend as tfb

POS_WEIGHT = 10  # multiplier for positive targets, needs to be tuned

def weighted_binary_crossentropy(target, output):
    """
    Weighted binary crossentropy between an output tensor 
    and a target tensor. POS_WEIGHT is used as a multiplier 
    for the positive targets.

    Combination of the following functions:
    * keras.losses.binary_crossentropy
    * keras.backend.tensorflow_backend.binary_crossentropy
    * tf.nn.weighted_cross_entropy_with_logits
    """
    # transform back to logits
    _epsilon = tfb._to_tensor(tfb.epsilon(), output.dtype.base_dtype)
    output = tf.clip_by_value(output, _epsilon, 1 - _epsilon)
    output = tf.log(output / (1 - output))
    # compute weighted loss
    loss = tf.nn.weighted_cross_entropy_with_logits(targets=target,logits=output, pos_weight=POS_WEIGHT)
    return tf.reduce_mean(loss, axis=-1)

Then in your model:

model.compile(loss=weighted_binary_crossentropy, ...)

I have not found many resources yet which report well working values for the pos_weight in relation to the number of classes, average active classes, etc.

Update for tensorflow 2.6.0:

I was going to write a comment but there are many things that needs to be changed for @tobigue answer to work. And I am not entirely sure if everything is correct with my answer. To make things work:

1. You need to replace import keras.backend.tensorflow_backend as tfb with import keras.backend as tfb
2. The target parameter in tf.nn.weighted_cross_entropy_with_logits needs to be changed to labels
3. tf.log needs to be called like this: tf.math.log
4. To make this custom loss function to work with keras, you need to import get_custom_objects and define the custom loss function as a loss function. So, from keras.utils.generic_utils import get_custom_objects and then before you compile the model you need to: get_custom_objects().update({"weighted_binary_crossentropy": weighted_binary_crossentropy})
5. I also encountered this error but it may not be the same for everyone. The error is: TypeError: Input 'y' of 'Mul' Op has type float32 that does not match type int32 of argument 'x'. To fix this error, I have converted the target to float32 like this: target = tf.cast(target, tf.float32)

So, the final code that I am using is this:

import tensorflow as tf
import keras.backend as tfb
from keras.utils.generic_utils import get_custom_objects

POS_WEIGHT = 10  # multiplier for positive targets, needs to be tuned
def weighted_binary_crossentropy(target, output):
    """
    Weighted binary crossentropy between an output tensor
    and a target tensor. POS_WEIGHT is used as a multiplier
    for the positive targets.

    Combination of the following functions:
    * keras.losses.binary_crossentropy
    * keras.backend.tensorflow_backend.binary_crossentropy
    * tf.nn.weighted_cross_entropy_with_logits
    """
    # transform back to logits
    _epsilon = tfb._to_tensor(tfb.epsilon(), output.dtype.base_dtype)
    output = tf.clip_by_value(output, _epsilon, 1 - _epsilon)
    output = tf.math.log(output / (1 - output))
    # compute weighted loss
    target = tf.cast(target, tf.float32)
    loss = tf.nn.weighted_cross_entropy_with_logits(labels=target,
                                                    logits=output,
                                                    pos_weight=POS_WEIGHT)
    return tf.reduce_mean(loss, axis=-1)

Then in your model

get_custom_objects().update({"weighted_binary_crossentropy": weighted_binary_crossentropy})
model.compile(loss='weighted_binary_crossentropy', ...)