How to train the network only on one output when there are multiple outputs?

The accepted answer won't work in general if the custom loss can't be applied to the outputs you're trying to ignore, e.g. if they have the wrong shapes. In that case you can assign a dummy loss function to those outputs:

labels = [labels_for_relevant_output, dummy_labels_for_ignored_output]

def dummy_loss(y_true, y_pred):
    return 0.0

model.compile(loss = [custom_loss_function, dummy_loss])
model.fit(x, labels)

I only want to train the network on output y2.

Based on Keras functional API guide you can achieve that with

model1 = Model(input=x, output=[y2,y3])   
model1.compile(optimizer='sgd', loss=custom_loss_function,
                  loss_weights=[1., 0.0])

What is the shape/structure of the y_pred and y_true argument in loss function when multiple outputs are used? Can I access them as above? Is it y_pred[0] or y_pred[:,0]

In keras multi-output models loss function is applied for each output separately. In pseudo-code:

loss = sum( [ loss_function( output_true, output_pred ) for ( output_true, output_pred ) in zip( outputs_data, outputs_model ) ] )

The functionality to do loss function on multiple outputs seems unavailable to me. One probably could achieve that by incorporating the loss function as a layer of the network.

Sharapolas' answer is right.

However, there is a better way than using a layer for building custom loss functions with complex interdependence of several outputs of a model.

The method I know is being used in practice is by never calling model.compile but only model._make_predict_function(). From there on, you can go on and build a custom optimizer method by calling model.output in there. This will give you all outputs, [y2,y3] in your case. When doing your magic with it, get a keras.optimizer and use it's get_update method using your model.trainable_weights and your loss. Finally, return a keras.function with a list of the inputs required (in your case only model.input) and the updates you just got from the optimizer.get_update call. This function now replaces model.fit.

The above is often used in PolicyGradient algorithms, like A3C or PPO. Here is an example of what I tried to explain: https://github.com/Hyeokreal/Actor-Critic-Continuous-Keras/blob/master/a2c_continuous.py Look at build_model and critic_optimizer methods and read kreas.backend.function documentation to understand what happens.

I found this way to have frequently problems with the session management and does not appear to work in tf-2.0 keras at all currently. Hence, if anyone knows a method, please let me know. I came here looking for one :)