Computing partial derivative of certain line integrals
$begingroup$
Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $mathbf{w}$, and (2) input data ${bf x} in {mathbb R}^d$. Fix $i in [d]$, consider the following path integral:
$$C_i({bf x}, {bf w}) = {bf x}_i int_0^1frac{partial F}{partial {bf x}_i}(alpha {bf x})dalpha $$
Basically, one can think of $C_i({bf x}, {bf w})$ as the "contribution" of the $i$-th dimension to the final output -- by integrating along a line from $bf 0$ to $bf x$.
My question is whether there are known methods (references, etc.) that give "backpropagation-like" algorithms for computing:
$$frac{partial C_i({bf x}, {bf w})}{partial {bf w}_i}$$
That is, basically I want to understand the first order information of the contribution of the $i$-th dimension w.r.t. ${bf w}_i$.
Thanks in advance for any insight.
partial-derivative computer-science machine-learning control-theory neural-networks
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
$endgroup$
add a comment |
$begingroup$
Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $mathbf{w}$, and (2) input data ${bf x} in {mathbb R}^d$. Fix $i in [d]$, consider the following path integral:
$$C_i({bf x}, {bf w}) = {bf x}_i int_0^1frac{partial F}{partial {bf x}_i}(alpha {bf x})dalpha $$
Basically, one can think of $C_i({bf x}, {bf w})$ as the "contribution" of the $i$-th dimension to the final output -- by integrating along a line from $bf 0$ to $bf x$.
My question is whether there are known methods (references, etc.) that give "backpropagation-like" algorithms for computing:
$$frac{partial C_i({bf x}, {bf w})}{partial {bf w}_i}$$
That is, basically I want to understand the first order information of the contribution of the $i$-th dimension w.r.t. ${bf w}_i$.
Thanks in advance for any insight.
partial-derivative computer-science machine-learning control-theory neural-networks
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
$endgroup$
add a comment |
$begingroup$
Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $mathbf{w}$, and (2) input data ${bf x} in {mathbb R}^d$. Fix $i in [d]$, consider the following path integral:
$$C_i({bf x}, {bf w}) = {bf x}_i int_0^1frac{partial F}{partial {bf x}_i}(alpha {bf x})dalpha $$
Basically, one can think of $C_i({bf x}, {bf w})$ as the "contribution" of the $i$-th dimension to the final output -- by integrating along a line from $bf 0$ to $bf x$.
My question is whether there are known methods (references, etc.) that give "backpropagation-like" algorithms for computing:
$$frac{partial C_i({bf x}, {bf w})}{partial {bf w}_i}$$
That is, basically I want to understand the first order information of the contribution of the $i$-th dimension w.r.t. ${bf w}_i$.
Thanks in advance for any insight.
partial-derivative computer-science machine-learning control-theory neural-networks
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
$endgroup$
Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $mathbf{w}$, and (2) input data ${bf x} in {mathbb R}^d$. Fix $i in [d]$, consider the following path integral:
$$C_i({bf x}, {bf w}) = {bf x}_i int_0^1frac{partial F}{partial {bf x}_i}(alpha {bf x})dalpha $$
Basically, one can think of $C_i({bf x}, {bf w})$ as the "contribution" of the $i$-th dimension to the final output -- by integrating along a line from $bf 0$ to $bf x$.
My question is whether there are known methods (references, etc.) that give "backpropagation-like" algorithms for computing:
$$frac{partial C_i({bf x}, {bf w})}{partial {bf w}_i}$$
That is, basically I want to understand the first order information of the contribution of the $i$-th dimension w.r.t. ${bf w}_i$.
Thanks in advance for any insight.
partial-derivative computer-science machine-learning control-theory neural-networks
partial-derivative computer-science machine-learning control-theory neural-networks
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
asked Dec 14 '18 at 16:06
Xi WuXi Wu
7117
7117
add a comment |
add a comment |
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