Consider an F function (think of neural networks) with two sets of parameters: (1) model parameters $ mathbf {w} $and (2) input data $ { bf x} in { mathbb R} ^ d $. To repair $ i in [d]$, consider the following path integral:

$$ C_i ({ bf x}, { bf w}) = { bf x} _i int_0 ^ 1 frac { partial F} { partial { bf x} _i} ( alpha { bf x}) d alpha $$

Basically, we can think of $ C_i ({ bf x}, { bf w}) $ as "contribution" of the $ i $dimension to the final result – by integrating along a line from $ bf $ 0 at $ bf x $.

My question is: are there any known methods (references, etc.) giving "backpropagation" algorithms for calculation:

$$ frac { partial C_i ({ bf x}, { bf w})} { partial { bf w} _i} $$

That is to say, basically I want to understand the first-rate information from the contribution of the $ i $-th dimension w.r.t. $ { bf w} _i $.

Thank you in advance for any idea.