How is the categorical_crossentropy implemented in keras?

As an answer to "Do you happen to know what the epsilon and tf.clip_by_value is doing?",
it is ensuring that output != 0, because tf.log(0) returns a division by zero error.
(I don't have points to comment but thought I'd contribute)

I see that you used the tensorflow tag, so I guess this is the backend you are using?

def categorical_crossentropy(output, target, from_logits=False):
"""Categorical crossentropy between an output tensor and a target tensor.
# Arguments
    output: A tensor resulting from a softmax
        (unless `from_logits` is True, in which
        case `output` is expected to be the logits).
    target: A tensor of the same shape as `output`.
    from_logits: Boolean, whether `output` is the
        result of a softmax, or is a tensor of logits.
# Returns
    Output tensor.

This code comes from the keras source code. Looking directly at the code should answer all your questions :) If you need more info just ask !

EDIT :

Here is the code that interests you :

 # Note: tf.nn.softmax_cross_entropy_with_logits
# expects logits, Keras expects probabilities.
if not from_logits:
    # scale preds so that the class probas of each sample sum to 1
    output /= tf.reduce_sum(output,
                            reduction_indices=len(output.get_shape()) - 1,
                            keep_dims=True)
    # manual computation of crossentropy
    epsilon = _to_tensor(_EPSILON, output.dtype.base_dtype)
    output = tf.clip_by_value(output, epsilon, 1. - epsilon)
    return - tf.reduce_sum(target * tf.log(output),
                          reduction_indices=len(output.get_shape()) - 1)

If you look at the return, they sum it... :)