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My original post explains why having a state vector necessitates backpropagation through time. If you still don't get it there's nothing else I can do for you, other than recommending that you seriously revise your understanding of RNNs and basic multivariable calculus
It doesn't solve any problems, it creates them. If you don't divide your training corpus into short sequences, you will end having to back propagate through the entire corpus on every time step to find the hidden state gradient. Your error metric should be based on the complete output target, not a partial output. This isn't even possible for models that aren't mapping K to K. This isn't possible for mapping sequences of size K to 1, for example sentiment analysis.
It DOESN'T MATTER how many time steps you want to step before you back propagate, because the hidden state vector is ALWAYS a function of previous inputs (as described in my original post). Therefore if you want to calculate the gradient you will ALWAYS need to account for the previous inputs. which means you need to backpropagate through time.
What do you think "operating on a sequence" means? Because what you described is exactly what operating on a sequence means, and is standard use of an RNN for time sequence modelling.
I've seen the original post. You contradict yourself multiple times. Either you are iterating over time sequence and updating a state vector on each time step (or every nth time step), or you are iterating over a time sequence and not updating a state vector. You can't be updating a state vector without iterating over a time sequence, that is meaningless.
The motivation and benefit of an RNN is that you maintain a sense of state over your time steps - the output of later time steps is influenced by what was seen at earlier time steps. If you are just use a single time step for each training example, then you don't keep any past state and it's not an RNN, just a regular feed forward net. This will work for some things that don't have strong correlation over time steps, but will probably fail for things that do.
I think some clarification here is required. When you say "inputs", are you referring to multiple training examples, or are you referring to multiple time steps within a single training example?
The only sense in which you can feed in multiple inputs at once into an RNN is feeding in multiple training examples, as part of a batch. The batch size is arbitrary, and convergence is guaranteed for any size, but higher batch sizes may lead to more accurate gradient estimations and faster convergence.
