From my experience with Keras and assuming you are familiar enough with neural networks:

batch_size denotes the subset size of your training sample (e.g. 100 out of 1000) which is going to be used in order to train the network during its learning process. Each batch trains network in a successive order, taking into account the updated weights coming from the appliance of the previous batch.

return_sequence indicates if a recurrent layer of the network should return its entire output sequence (i.e. a sequence of vectors of specific dimension) to the next layer of the network, or just its last only output which is a single vector of the same dimension. This value can be useful for networks conforming with an RNN architecture.

batch_input_shape defines that the sequential classification of the neural network can accept input data of the defined only batch size, restricting in that way the creation of any variable dimension vector. It is widely used in stacked LSTM networks.

Concerning your last inquiry, I suppose that the smaller batch size eased the overall computational cost -while at the same time- the classification accuracy of your dataset was improved due to the increased number of weights' updates. However, the performance and accuracy of your tested network also depends on various other parameters (like its dataset size, learning rate, momentum, etc), so don't take for granted that a smaller batch size will always wield better results compared to a higher one (a rather small size may 'generate' noise and lead to bad training).

From my experience with Keras and assuming you are familiar enough with neural networks:

batch_size denotes the subset size of your training sample (e.g. 100 out of 1000) which is going to be used in order to train the network during its learning process. Each batch trains network in a successive order, taking into account the updated weights coming from the appliance of the previous batch.

return_sequence indicates if a recurrent layer of the network should return its entire output sequence (i.e. a sequence of vectors of specific dimension) to the next layer of the network, or just its last only output which is a single vector of the same dimension. This value can be useful for networks conforming with an RNN architecture.

batch_input_shape defines that the sequential classification of the neural network can accept input data of the defined only batch size, restricting in that way the creation of any variable dimension vector. It is widely used in stacked LSTM networks.

Concerning your last inquiry, I suppose that the smaller batch size eased the overall computational cost -while at the same time- the classification accuracy of your dataset was improved due to the increased number of weights' updates. However, the performance and accuracy of your tested network also depends on various other parameters (like its dataset size, learning rate, momentum, etc), so don't take for granted that a smaller batch size will always wield better results compared to a higher one (a rather small size may 'generate' noise and lead to bad training).

I really appreciate your help  Mr. Konstantinos Kontakis. It helps.. :)

Thanks

I have the same question, I a priori agreed with Konstatinos but I'm a bit confused seeing this piece of code from Keras Documentation http://keras.io/getting-started/sequential-model-guide/

data_dim = 16

timesteps = 8

nb_classes = 10

batch_size = 32

# expected input batch shape: (batch_size, timesteps, data_dim)

# note that we have to provide the full batch_input_shape since the network is stateful.

# the sample of index i in batch k is the follow-up for the sample i in batch k-1.

model = Sequential()

model.add(LSTM(32, return_sequences=True, stateful=True,

batch_input_shape=(batch_size, timesteps, data_dim)))

model.add(LSTM(32, return_sequences=True, stateful=True))

model.add(LSTM(32, stateful=True))

model.add(Dense(10, activation='softmax'))

If you see the comment " the sample of index i in batch k is the follow-up for the sample i in batch k-1.", it appear that a batch is the unit of information for each time step. But on the other hand the batch_input_shape is a 3D shape, 

Someone can explain me wich is the meaning of this parameters; batch_size, timesteps, data_dim?