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(Sub)gradient of a function with vector dot products
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I am trying to understand the subgradient of a function for a time series, where for each time instance t:
$f_t: p in Delta_K mapsto ell_t(mathbfpcdot mathbfx_t )in mathbbR_+ $ the function evaluates the losses incurred by the weight vectors. Now from my understanding, if the loss function $ell_t$ is convex and differentiable, the function $f_t$ is convex and differentiable too. If I have a squared loss function of the form: $ell_t(haty_t)=(haty_t-y_t)^2$ where $haty_t=langle mathbfp, mathbfx_t rangle$ and $y_t$ is the revealed observation, how would the derivation of the loss function look like?
My basic math knowledge leads me to:
$ell_t(haty_t)=(haty_t-y_t)^2=haty_t^2-2haty_ty_t+y_t^2$
$ell_t(haty_t)'=2haty_t-2y_t=2(haty_t-y_t) $
Now i found in a paper, that the correct solution for $ell_t(haty_t)'$ might be: $2(haty_t-y_t) haty_t $
I don't understand where the additional $haty_t $ is coming from.
derivatives subgradient differential-algebra
asked Jul 24 at 8:10
UDE_Student
1618
add a comment |Â
up vote
0
down vote
favorite
I am trying to understand the subgradient of a function for a time series, where for each time instance t:
$f_t: p in Delta_K mapsto ell_t(mathbfpcdot mathbfx_t )in mathbbR_+ $ the function evaluates the losses incurred by the weight vectors. Now from my understanding, if the loss function $ell_t$ is convex and differentiable, the function $f_t$ is convex and differentiable too. If I have a squared loss function of the form: $ell_t(haty_t)=(haty_t-y_t)^2$ where $haty_t=langle mathbfp, mathbfx_t rangle$ and $y_t$ is the revealed observation, how would the derivation of the loss function look like?
My basic math knowledge leads me to:
$ell_t(haty_t)=(haty_t-y_t)^2=haty_t^2-2haty_ty_t+y_t^2$
$ell_t(haty_t)'=2haty_t-2y_t=2(haty_t-y_t) $
Now i found in a paper, that the correct solution for $ell_t(haty_t)'$ might be: $2(haty_t-y_t) haty_t $
I don't understand where the additional $haty_t $ is coming from.
derivatives subgradient differential-algebra
asked Jul 24 at 8:10
UDE_Student
1618
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to understand the subgradient of a function for a time series, where for each time instance t:
$f_t: p in Delta_K mapsto ell_t(mathbfpcdot mathbfx_t )in mathbbR_+ $ the function evaluates the losses incurred by the weight vectors. Now from my understanding, if the loss function $ell_t$ is convex and differentiable, the function $f_t$ is convex and differentiable too. If I have a squared loss function of the form: $ell_t(haty_t)=(haty_t-y_t)^2$ where $haty_t=langle mathbfp, mathbfx_t rangle$ and $y_t$ is the revealed observation, how would the derivation of the loss function look like?
My basic math knowledge leads me to:
$ell_t(haty_t)=(haty_t-y_t)^2=haty_t^2-2haty_ty_t+y_t^2$
$ell_t(haty_t)'=2haty_t-2y_t=2(haty_t-y_t) $
Now i found in a paper, that the correct solution for $ell_t(haty_t)'$ might be: $2(haty_t-y_t) haty_t $
I don't understand where the additional $haty_t $ is coming from.
derivatives subgradient differential-algebra
asked Jul 24 at 8:10
UDE_Student
1618
I am trying to understand the subgradient of a function for a time series, where for each time instance t:
$f_t: p in Delta_K mapsto ell_t(mathbfpcdot mathbfx_t )in mathbbR_+ $ the function evaluates the losses incurred by the weight vectors. Now from my understanding, if the loss function $ell_t$ is convex and differentiable, the function $f_t$ is convex and differentiable too. If I have a squared loss function of the form: $ell_t(haty_t)=(haty_t-y_t)^2$ where $haty_t=langle mathbfp, mathbfx_t rangle$ and $y_t$ is the revealed observation, how would the derivation of the loss function look like?
My basic math knowledge leads me to:
$ell_t(haty_t)=(haty_t-y_t)^2=haty_t^2-2haty_ty_t+y_t^2$
$ell_t(haty_t)'=2haty_t-2y_t=2(haty_t-y_t) $
Now i found in a paper, that the correct solution for $ell_t(haty_t)'$ might be: $2(haty_t-y_t) haty_t $
I don't understand where the additional $haty_t $ is coming from.
derivatives subgradient differential-algebra
asked Jul 24 at 8:10
UDE_Student
1618
edited Jul 24 at 8:18
asked Jul 24 at 8:10
UDE_Student
1618
asked Jul 24 at 8:10
UDE_Student
1618
asked Jul 24 at 8:10
UDE_Student
1618
1618
add a comment |Â
add a comment |Â
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