Keras categorical_crossentropy loss (and accuracy) When training a neural network with keras for the categorical_crossentropy loss, how exactly is the loss defined? I expect it to be the average over all samples of
$$\textstyle\text{loss}(p^\text{true}, p^\text{predict}) = -\sum_i p_i^\text{true} \log p_i^\text{predict}$$
but couldn't find a definitive answer in the docs nor in the code. An authoritative reference is desirable.

Looking at the code I'm not sure if the computation is delegated to tensorflow/theano.

(There is an analogous question concerning the accuracy; the code is clearer, but I don't see a call to mean()?)

PS.
From this code, it appears that loss and accuracy are computed as in loss_and_acc(...) but before the last training epoch (keras version 2.0.4, same results for tensorflow and theano backend).

#!/usr/bin/python3

import numpy as np
from numpy.random import randint, seed

from keras import __version__ as keras_version
from keras.models import Sequential
from keras.layers import Dense

N = 4 # Classes
S = 10 # Samples

nn = Sequential()
nn.add(Dense(input_dim=1, units=N, kernel_initializer='normal', activation='softmax'))
nn.compile(optimizer='adam', loss='categorical_crossentropy', metrics=['accuracy'])

seed(7)
X = np.random.random((S, 1))
Y = np.vstack([np.eye(1, N, k=randint(0, N)) for _ in range(S)])
#for (x, y) in zip(X, Y) : print(x, y)

def loss_and_acc(NN, X, Y) :
    loss = []
    acc  = []
    for (p, q) in zip(Y, NN.predict(X)) :
        loss += [ -sum(a*np.log(b) for (a, b) in zip(p, q) if (b != 0)) ]
        acc  += [ np.argmax(p) == np.argmax(q) ]
    return (np.mean(loss), np.mean(acc))

print("Keras version: ", keras_version)

for _ in range(10) :
    print("Before:  loss = {}, acc = {}".format(*loss_and_acc(nn, X, Y)))
    H = nn.fit(X, Y, epochs=1, verbose=0).history
    print("History: loss = {}, acc = {}".format(H['loss'][-1], H['acc'][-1]))


Output:

Using Theano backend.
Keras version:  2.0.4
Before:  loss = 1.3843669414520263, acc = 0.2
History: loss = 1.3843669891357422, acc = 0.20000000298023224
Before:  loss = 1.3834303855895995, acc = 0.2
History: loss = 1.3834303617477417, acc = 0.20000000298023224
Before:  loss = 1.3824962615966796, acc = 0.3
History: loss = 1.3824962377548218, acc = 0.30000001192092896
Before:  loss = 1.381564486026764, acc = 0.3
History: loss = 1.3815644979476929, acc = 0.30000001192092896
Before:  loss = 1.380635154247284, acc = 0.3
History: loss = 1.380635142326355, acc = 0.30000001192092896
Before:  loss = 1.3797082901000977, acc = 0.3
History: loss = 1.3797082901000977, acc = 0.30000001192092896
Before:  loss = 1.378783941268921, acc = 0.2
History: loss = 1.378783941268921, acc = 0.20000000298023224
Before:  loss = 1.3778621554374695, acc = 0.2
History: loss = 1.3778622150421143, acc = 0.20000000298023224
Before:  loss = 1.3769428968429565, acc = 0.2
History: loss = 1.3769428730010986, acc = 0.20000000298023224
Before:  loss = 1.3760262489318849, acc = 0.3
History: loss = 1.3760262727737427, acc = 0.30000001192092896
I am using keras with tensorflow backend. I checked and the categorical_crossentropy loss in keras is defined as you have defined. This is the part of code (not the whole function definition)-

def categorical_crossentropy(target, output, from_logits=False, axis=-1):
    if not from_logits:
        # scale preds so that the class probas of each sample sum to 1
        output /= tf.reduce_sum(output, axis, 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), axis) 


As you can see in last line, it is returning the sum of product of true values and log of output values for each observation. You can find complete function definition here in line 3176.

For theano backend, it should be same. You can check here in line 1622.