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March 19th, 2017, 03:12 AM   #1
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Derivative Mind-bending!

Hi all,

I've not touched calculus post my maths degree many years ago.

I need to calculate a gradient, to use in back-propagation in a neural network, but it's making my head hurt!

I have the formula

Code:
δ(t)  = tanh[ <w, f(t) > + b + uδ(t−1) ]
where t, t-1 represent time steps

and need to calculate

Code:
dδ(t)/dθ = ∂δ(t)/∂θ + ∂δ(t)/∂δ(t−1) * dδ(t-1)/dθ
where θ comprises the parameter set [w, b, u]. b and u are scalars, but w is a vector of multiple elements, e.g. w1, w2 etc..


The derivative of tanh(x) is 1 - tanh^2(x), so I assume the partial derivative,

Code:
∂δ(t)/dδ(t−1) = 1 - tanh^2[ <w, f(t) > + b + uδ(t−1) ]
The final element (dδ(t-1)/dθ) comes from recursively calculating the gradient, so I'm happy with that.


I'm less sure about the first element, given the θ decomposition into its constituent elements.

Any ideas?!

Last edited by skipjack; March 19th, 2017 at 04:00 PM.
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