# What is the time complexity of the forward pass and back-propagation of the sequence-to-sequence model with and without attention?

I keep looking through the literature, but can't seem to find any information regarding the time complexity of the forward pass and back-propagation of the sequence-to-sequence RNN encoder-decoder model, with and without attention.

The paper Attention is All You Need by Vaswani et. al in 2017 states the forward pass cost is $$O(n^3)$$, which makes sense to me (with 1 hidden layer). In terms of $$X$$ the input length, and $$Y$$ the output length, it then looks like $$O(X^3 + Y^3)$$, which I understand.

However, for training, it seems to me like one back-propagation is at worst $$O(X^3 + Y^3)$$, and we do $$Y$$ of them, so $$O(Y(X^3 Y^3))$$.

This is the following diagram, where the green blocks are the hidden states, the red ones are the input text and the blue ones are output text. If I were to add global attention, as introduced by Luong et. al in 2015, the attention adds an extra $$X^2$$ in there due to attention multiplication, to make an overall inference of $$O(X^3 + X^2 Y^3)$$, and training even worse at $$O(XY(X^3 + X^2 Y^3))$$ since it needs to learn attention weights too.

The following diagram shows the sequence-to-sequence model with attention, where $$h$$'s are the hidden states, $$c$$ is the context vector and $$y$$ is the output word, $$Y$$ of such output words and $$X$$ such inputs. This setup is described in the paper Effective Approaches to Attention-based Neural Machine Translation, by Luong et al. in 2015. Is my intuition correct?

• I encourage you to focus on one of the models, because your question isn't very simple, because one needs to investigate both models. Furthermore, I suggest you focus either on the forward pass or back-propagation. But these are just suggestions. You can ask different separate questions. One e.g. for the forward pass, one for the model with attention. This will help users to focus on one problem. – nbro Apr 9 '20 at 22:51