University of Michigan - Ann Arbor

In a recurrent neural network (RNN) processing a sequence of length $$n$$, updating the $$d$$-dimensional hidden state involves a multiplication with a $$d 	imes d$$ weight matrix, leading to a computational cost of $$\mathcal{O}(d^2)$$ per step. Consequently, the total computational complexity for the entire recurrent layer is $$\mathcal{O}(nd^2)$$. Because the hidden state must be updated iteratively one token at a time, there are $$\mathcal{O}(n)$$ sequential operations that cannot be parallelized, and the maximum path length between any two tokens is similarly $$\mathcal{O}(n)$$.

RNN Sequence Processing Complexity

Match each structural component of an RNN to its corresponding definition or dimension.

For an RNN layer processing an input sequence of length n with a d-dimensional hidden state, the total computational complexity across the entire layer is ___.

Analyze the computational constraints of this architecture by identifying the fundamental limitation that prevents GPU parallelization across time, and state the formal complexity for both the sequential operations and the maximum path length between tokens.

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