Deriving the gradient hidden to hidden weights for backpropagation through time in a reccurent neural network

Question

I'm currently working on deriving the the gradients of a simple recurrent neural networks weights with respect to the loss to update the weights through backpropagation. It's a super simple network, so the derivative isn't the challenge, it's figuring out what exactly $\frac{\partial h_1}{\partial W_{hh}}$ is. When $h_1$ is the first hidden node and has no $W_{hh}$ coming into it from a previous node what happens here? Is there something that is supposed to be connected to $h_1$ (other than $x_1$ through $W_{xh}$)? Below is the gradient I'm looking at for reference and I'm looking for the very last term when $k=1$.

Answer 1

Derivative of unrelated variables

If $h_1$ is no function of $W_{hh}$ (which is the case for the first time-step), then $$\frac{\partial h_1}{\partial W_{hh}} = 0$$ This will then lead to the whole summand for $k=1$ being $0$ and we could rewrite it as

$$\frac{\partial L}{\partial W_{hh}}=\sum_t^T\sum_{k=2}^{t+1}\ldots$$

(Note the $k=2$ part, here).

The first time-step

So why not just simply start with $k=2$?
Because based on the link you gave, the first timestep is $t=0$:

NOTE: At the first time step, t=0, there are no previous outputs, so they are typically assumed to be all zeros.

This means that $$h_1 = f_h(X_1, h_0)$$ but $$\frac{\partial h_0}{\partial W_{hh}} = 0$$ This is the reason, that the sum in the derivative starts at $k=1$ and not $k=0$.