Define a sequence $(\mathbf{y})_{i=0}^N$ in $\mathbb{R}^n$ such that:
$$\mathbf{y}_{k+1} = \mathbf{y}_{k} + \lambda \nabla_\mathbf{y} E(\mathbf{y}_k,\mathbf{w}), \quad k=0,1,\ldots,N-1,$$
where $\lambda$ is a constant, $\mathbf{w}\in\mathbb{R}^m$, and $E:\mathbb{R}^{n+m}\to \mathbb{R}$ is some differentiable function.

Let $Q:\mathbb{R}^{n}\to \mathbb{R}$ be a differentiable function and $L=Q(\mathbf{y} _N)$. 

Applying the chain rule we have:
$$\frac{dL}{d\mathbf{w}} = \sum_{k=1}^N\frac{\partial \mathbf{y}_k}{\partial \mathbf{w}} \frac{dQ}{d\mathbf{y}_k}\qquad (1)$$
and
$$\frac{dQ}{d\mathbf{y}_k} = \frac{\partial \mathbf{y}_{k+1}}{\partial \mathbf{y}_{k}} \frac{dQ}{d\mathbf{y}_{k+1}}.\qquad (2)$$

(Source: [this paper][2], equation (12) and the one between (12) and (13).)

**My questions:** how to obtain $(1)$ and $(2)$?

I can show that if $(2)$ holds then $(1)$ holds. But I cannot see why $(2)$ holds. Consider $k=N-1$ for example: 

Applying [the chain rule][1] we have:
\begin{align}\frac{dQ}{d\mathbf{y}_{N-1}} 
&= \frac{dQ(\mathbf{y}_{N}(\mathbf{w},\mathbf{y}_{N-1}))}{d\mathbf{y}_{N-1}} \\
&= \frac{d\mathbf{y}_{N}(\mathbf{w},\mathbf{y}_{N-1})}{d\mathbf{y}_{N-1}}\frac{dQ}{d\mathbf{y}_{N}} \\
&= \begin{bmatrix}\frac{d\mathbf{w}}{d\mathbf{y}_{N-1}} & \mathbf{I}\end{bmatrix}\begin{bmatrix}\frac{\partial \mathbf{y}_{N}}{\partial \mathbf{w}} \\ \frac{\partial \mathbf{y}_{N}}{\partial \mathbf{y}_{N-1}}\end{bmatrix}\frac{dQ}{d\mathbf{y}_{N}} \\
&= \frac{d\mathbf{w}}{d\mathbf{y}_{N-1}}\frac{\partial \mathbf{y}_{N}}{\partial \mathbf{w}}\frac{dQ}{d\mathbf{y}_{N}} + \frac{\partial \mathbf{y}_{N}}{\partial \mathbf{y}_{N-1}}\frac{dQ}{d\mathbf{y}_{N}},
\end{align}
which has an extra term $\frac{d\mathbf{w}}{d\mathbf{y}_{N-1}}\frac{\partial \mathbf{y}_{N}}{\partial \mathbf{w}}\frac{dQ}{d\mathbf{y}_{N}}$ compared to $(2)$.

Thank you in advance for your help.


  [1]: https://math.stackexchange.com/questions/1566376/derivation-of-the-multivariate-chain-rule/1567200#1567200
  [2]: http://proceedings.mlr.press/v22/domke12/domke12.pdf