I wansn't able to use the class_weight parameter yet, but in the mean time i've found another way to apply class weighting to each output layer.

Current solution

In this keras issue they have supplied an easy method to apply class weights via a custom loss that implements the required class weighing.

def weighted_categorical_crossentropy(y_true, y_pred, weights):
    nb_cl = len(weights)
    final_mask = K.zeros_like(y_pred[:, 0])
    y_pred_max = K.max(y_pred, axis=1)
    y_pred_max = K.reshape(y_pred_max, (K.shape(y_pred)[0], 1))
    y_pred_max_mat = K.cast(K.equal(y_pred, y_pred_max), K.floatx())
    for c_p, c_t in product(range(nb_cl), range(nb_cl)):
        final_mask += (weights[c_t, c_p] * y_pred_max_mat[:, c_p] * y_true[:, c_t])
    return K.categorical_crossentropy(y_pred, y_true) * final_mask

where weights is a CxC matrix (where C is the number of classes) that defines the class weights.
More precisely, weights[i, j] defines the weight for an example of class i which was falsely classified as class j.

So how do we use it?

Keras allows to assign a loss function for each output.
so we could assign each output a loss fucntion with the correct weights matrix.

For example, to satisfy the request i made in the question we could suggest the following code.

# Define the weight matrices
w1 = np.ones((2, 2))
w1[1, 0] = 10
w1[1, 1] = 10

w2 = np.ones((3, 3))
w2[0, 0] = 5
w2[0, 1] = 5
w2[0, 2] = 5
w2[2, 0] = 10
w2[2, 1] = 10
w2[2, 2] = 10    

# Define the weighted loss functions
from functools import partial
loss1 = partial(weighted_categorical_crossentropy, weights=w1)
loss2 = partial(weighted_categorical_crossentropy, weights=w2)

# Finally, apply the loss functions to the outputs
model.compile(loss={'output1': loss1, 'output2': loss2}, optimizer='adam')

And that accomplishes the request :)

Edit

There is a small edition that needs to be made.
The loss functions must have a name, so we can supply this with the following:

loss1.__name__ = 'loss1'
loss2.__name__ = 'loss2'

You have a list of outputs. You can simply pass a list of class_weight for each output as follows:

class_weight = [{0: 1, 1: 10},{0: 5, 1: 1, 2: 10}]

Pass a dictionary in the following format to class_weight parameter in fit_generator:

{ 'output1': {0: ratio_1 , 1: ratio_2} , 'output2': {0: ratio_3 , 1: ratio_4}}

You can use class_weight from sklearn.utils to calculate class weights from your data

EDIT: This approach works in TF 2.1.0 and earlier versions only. Thanks for replies.

References:

https://github.com/keras-team/keras/issues/4735#issuecomment-267473722 https://scikit-learn.org/stable/modules/generated/sklearn.utils.class_weight.compute_class_weight.html

Here's my solution for sparse categorical crossentropy for a Keras model with multiple outputs in TF2. I think it looks fairly clean but it might be horrifically inefficient, idk.

First create a dictionary where the key is the name set in the output Dense layers and the value is a 1D constant tensor. The value in index 0 of the tensor is the loss weight of class 0, a value is required for all classes present in each output even if it is just 1 or 0.

Compile your model with

model.compile(optimizer=optimizer,
              loss={k: class_loss(v) for k, v in class_weights.items()})

where class_loss() is defined in the following manner

def class_loss(class_weight):
  """Returns a loss function for a specific class weight tensor

  Params:
    class_weight: 1-D constant tensor of class weights

  Returns:
    A loss function where each loss is scaled according to the observed class"""
  def loss(y_obs, y_pred):
    y_obs = tf.dtypes.cast(y_obs, tf.int32)
    hothot = tf.one_hot(tf.reshape(y_obs, [-1]), depth=class_weight.shape[0])
    weight = tf.math.multiply(class_weight, hothot)
    weight = tf.reduce_sum(weight, axis=-1)
    losses = tf.compat.v1.losses.sparse_softmax_cross_entropy(labels=y_obs,
                                                              logits=y_pred,
                                                              weights=weight)
    return losses
  return loss

If someone has a better suggestion than using tf.compat.v1 then please let me know. I don't feel confident that it will stick around through future versions of Tensorflow. I also posted this answer here: https://github.com/keras-team/keras/issues/11735#issuecomment-641775516

EDIT: Be aware that this is for an output with a linear output rather than a softmax output! You have to softmax the outputs afterwards if you want softmax values (but if you just want the predictions ranked then logits still work).