batch normalization, yes or no?

I use Tensorflow 1.14.0 and Keras 2.2.4. The following code implements a simple neural network:

import numpy as np
np.random.seed(1)
import random
random.seed(2)
import tensorflow as tf
tf.set_random_seed(3)

from tensorflow.keras.models import Model, Sequential
from tensorflow.keras.layers import Input, Dense, Activation


x_train=np.random.normal(0,1,(100,12))

model = Sequential()
model.add(Dense(8, input_shape=(12,)))
# model.add(tf.keras.layers.BatchNormalization())
model.add(Activation('linear'))
model.add(Dense(12))
model.add(Activation('linear'))
model.compile(optimizer='adam', loss='mean_squared_error')
model.fit(x_train, x_train,epochs=20, validation_split=0.1, shuffle=False,verbose=2)

The final val_loss after 20 epochs is 0.7751. When I uncomment the only comment line to add the batch normalization layer, the val_loss changes to 1.1230.

My main problem is way more complicated, but the same thing occurs. Since my activation is linear, it does not matter if I put the batch normalization after or before the activation.

Questions: Why batch normalization cannot help? Is there anything I can change so that the batch normalization improves the result without changing the activation functions?

Update after getting a comment:

An NN with one hidden layer and linear activations is kind of like PCA. There are tons of papers on this. For me, this setting gives minimal MSE among all combinations of activation functions for the hidden layer and output.

Some resources that state linear activations mean PCA:

https://arxiv.org/pdf/1702.07800.pdf

https://link.springer.com/article/10.1007/BF00275687

https://www.quora.com/How-can-I-make-a-neural-network-to-work-as-a-PCA

Yes.

The behavior you're observing is a bug - and you don't need BN to see it; plot to the left is for #V1, to the right is for #V2:

#V1
model = Sequential()
model.add(Dense(8, input_shape=(12,)))
#model.add(Activation('linear')) <-- uncomment == #V2
model.add(Dense(12))
model.compile(optimizer='adam', loss='mean_squared_error')

Clearly nonsensical, as Activation('linear') after a layer with activation=None (=='linear') is an identity: model.layers[1].output.name == 'activation/activation/Identity:0'. This can be confirmed further by fetching and plotting intermediate layer outputs, which are identical for 'dense' and 'activation' - will omit here.

So, the activation does literally nothing, except it doesn't - somewhere along the commit chain between 1.14.0 and 2.0.0, this was fixed, though I don't know where. Results w/ BN using TF 2.0.0 w/ Keras 2.3.1 below:

val_loss = 0.840 # without BN
val_loss = 0.819 # with BN

Solution: update to TensorFlow 2.0.0, Keras 2.3.1.

Tip: use Anaconda w/ virtual environment. If you don't have any virtual envs yet, run:

conda create --name tf2_env --clone base
conda activate tf2_env
conda uninstall tensorflow-gpu
conda uninstall keras
conda install -c anaconda tensorflow-gpu==2.0.0
conda install -c conda-forge keras==2.3.1

May be a bit more involved than this, but that's subject of another question.

UPDATE: importing from keras instead of tf.keras also solves the problem.

Disclaimer: BN remains a 'controversial' layer in Keras, yet to be fully fixed - see Relevant Git; I plan on investigating it myself eventually, but for your purposes, this answer's fix should suffice.

I also recommend familiarizing yourself with BN's underlying theory, in particular regarding its train vs. inference operation; in a nutshell, batch sizes under 32 is a pretty bad idea, and dataset should be sufficiently large to allow BN to accurately approximate test-set gamma and beta.

Code used:

x_train=np.random.normal(0, 1, (100, 12))

model = Sequential()
model.add(Dense(8, input_shape=(12,)))
#model.add(Activation('linear'))
#model.add(tf.keras.layers.BatchNormalization())
model.add(Dense(12))
model.compile(optimizer='adam', loss='mean_squared_error')

W_sum_all = []  # fit rewritten to allow runtime weight collection
for _ in range(20):
    for i in range(9):
        x = x_train[i*10:(i+1)*10]
        model.train_on_batch(x, x)

        W_sum_all.append([])
        for layer in model.layers:
            if layer.trainable_weights != []:
                W_sum_all[-1] += [np.sum(layer.get_weights()[0])]
model.evaluate(x[-10:], x[-10:])

plt.plot(W_sum_all)
plt.title("Sum of weights (#V1)", weight='bold', fontsize=14)
plt.legend(labels=["dense", "dense_1"], fontsize=14)
plt.gcf().set_size_inches(7, 4)

Imports/pre-executions:

import numpy as np
np.random.seed(1)
import random
random.seed(2)
import tensorflow as tf
if tf.__version__[0] == '2':
    tf.random.set_seed(3)
else:
    tf.set_random_seed(3)

import matplotlib.pyplot as plt
from tensorflow.keras.models import Model, Sequential
from tensorflow.keras.layers import Input, Dense, Activation