1) X are your inputs, if you have 99 timesteps, then you have 99 vectors of size 13 each. Hence your input to each timestep is a vector that is of size 13. You will need a starting hidden state, unless you have a reason to do otherwise your beginning hidden state can be all 0's. The size of that vector is a hyperparameter you choose.
2) Keep in mind that there are not 99 LSTM cells, there is only 1 LSTM cell that is re-used 99 times for each timestep. The LSTM cell maintains a hidden state and a cell state within it that it passes forward to the next time step. But there is only 1 set of parameters being learned. Those parameters need to be able to handle all timesteps, conditional on the current input, hidden state, and cell state.
3) The cell state is not an output, however it is passed forward as an input to the next timestep. The hidden state h_t will be passed to the output as well as to the next timestep.
4) I'm not quite sure, I need a reference to the term output_dim.
This is an excellent tutorial on LSTMs: http://colah.github.io/posts/2015-08-Understanding-LSTMs/
I'll assume we're talking about recurrent neural nets (RNNs) that produce an output at every time step (if output is only available at the end of the sequence, it only makes sense to run backprop at the end). RNNs in this setting are often trained using truncated backpropagation through time (BPTT), operating sequentially on 'chunks' of a sequence. The procedure looks like this:
- Forward pass: Step through the next $k_1$ time steps, computing the input, hidden, and output states.
- Compute the loss, summed over the previous time steps (see below).
- Backward pass: Compute the gradient of the loss w.r.t. all parameters, accumulating over the previous $k_2$ time steps (this requires having stored all activations for these time steps). Clip gradients to avoid the exploding gradient problem (happens rarely).
- Update parameters (this occurs once per chunk, not incrementally at each time step).
- If processing multiple chunks of a longer sequence, store the hidden state at the last time step (will be used to initialize hidden state for beginning of next chunk). If we've reached the end of the sequence, reset the memory/hidden state and move to the beginning of the next sequence (or beginning of the same sequence, if there's only one).
- Repeat from step 1.
How the loss is summed depends on $k_1$ and $k_2$. For example, when $k_1 = k_2$, the loss is summed over the past $k_1 = k_2$ time steps, but the procedure is different when $k_2 > k_1$ (see Williams and Peng 1990).
Gradient computation and updates are performed every $k_1$ time steps because it's computationally cheaper than updating at every time step. Updating multiple times per sequence (i.e. setting $k_1$ less than the sequence length) can accelerate training because weight updates are more frequent.
Backpropagation is performed for only $k_2$ time steps because it's computationally cheaper than propagating back to the beginning of the sequence (which would require storing and repeatedly processing all time steps). Gradients computed in this manner are an approximation to the 'true' gradient computed over all time steps. But, because of the vanishing gradient problem, gradients will tend to approach zero after some number of time steps; propagating beyond this limit wouldn't give any benefit. Setting $k_2$ too short can limit the temporal scale over which the network can learn. However, the network's memory isn't limited to $k_2$ time steps because the hidden units can store information beyond this period (e.g. see Mikolov 2012 and this post).
Besides computational considerations, the proper settings for $k_1$ and $k_2$ depend on the statistics of the data (e.g. the temporal scale of the structures that are relevant for producing good outputs). They probably also depend on the details of the network. For example, there are a number of architectures, initialization tricks, etc. designed to mitigate the decaying gradient problem.
Your option 1 ('frame-wise backprop') corresponds to setting $k_1$ to $1$ and $k_2$ to the number of time steps from the beginning of the sentence to the current point. Option 2 ('sentence-wise backprop') corresponds to setting both $k_1$ and $k_2$ to the sentence length. Both are valid approaches (with computational/performance considerations as above; #1 would be quite computationally intensive for longer sequences). Neither of these approaches would be called 'truncated' because backpropagation occurs over the entire sequence. Other settings of $k_1$ and $k_2$ are possible; I'll list some examples below.
References describing truncated BPTT (procedure, motivation, practical issues):
- Sutskever (2013). Training recurrent neural networks.
- Mikolov (2012). Statistical Language Models Based on Neural Networks.
- Using vanilla RNNs to process text data as a sequence of words, he recommends setting $k_1$ to 10-20 words and $k_2$ to 5 words
- Performing multiple updates per sequence (i.e. $k_1$ less than the sequence length) works better than updating at the end of the sequence
- Performing updates once per chunk is better than incrementally (which can be unstable)
- Williams and Peng (1990). An efficient gradient-based algorithm for on-line training of recurrent network trajectories.
- Original (?) proposal of the algorithm
- They discuss the choice of $k_1$ and $k_2$ (which they call $h'$ and $h$). They only consider $k_2 \ge k_1$.
- Note: They use the phrase "BPTT(h; h')" or 'the improved algorithm' to refer to what the other references call 'truncated BPTT'. They use the phrase 'truncated BPTT' to mean the special case where $k_1 = 1$.
Other examples using truncated BPTT:
- (Karpathy 2015). char-rnn.
- Description and code
- Vanilla RNN processing text documents one character at a time. Trained to predict the next character. $k_1 = k_2 = 25$ characters. Network used to generate new text in the style of the training document, with amusing results.
- Graves (2014). Generating sequences with recurrent neural networks.
- See section about generating simulated Wikipedia articles. LSTM network processing text data as sequence of bytes. Trained to predict the next byte. $k_1 = k_2 = 100$ bytes. LSTM memory reset every $10,000$ bytes.
- Sak et al. (2014). Long short term memory based recurrent neural network architectures for large vocabulary speech recognition.
- Modified LSTM networks, processing sequences of acoustic features. $k_1 = k_2 = 20$.
- Ollivier et al. (2015). Training recurrent networks online without backtracking.
- Point of this paper was to propose a different learning algorithm, but they did compare it to truncated BPTT. Used vanilla RNNs to predict sequences of symbols. Only mentioning it here to to say that they used $k_1 = k_2 = 15$.
- Hochreiter and Schmidhuber (1997). Long short-term memory.
- They describe a modified procedure for LSTMs
Best Answer
I think you've confused the dimensionality of an LSTM (the number of "units" it has and the sequence length.
An LSTM "cell" is just what library implementors use to describe an object which computes the LSTM update for a single time-step. As such, a regular dense layer has no such concept of "cell".
Usually something like (batch size, sequence length, dims) if you want a sequence, and you can manually index the sequence length dimension to extract the last output.
No, not quite. There is just one cell, which is invoked $T$ times to process a sequence of length $T$.
An LSTM almost always has $N > 1$ units. An LSTM with one unit would have a scalar output. It's usually more helpful to think of an LSTM as having a state of a certain dimension $d$, rather than to think of it as having $d$ "units" inside.