[CppAD] Arbitrary levels of autodifferentiation

[Andrew Spielberg]

Hi Brad - so just to verify, in your example, is there no way to use g to
initialize a new ADFun< AD<double> > and then take gradients of that, while
maintaining the dependency on the original independent variables?

-Andy S.



On Mon, Jun 4, 2018 at 8:05 AM, Brad Bell <bradbell at seanet.com> wrote:

> When a function f(x) is recorded, CppAD allows for the computation of
> arbitrary order derivatives of f(x). The multilevel option; e.g., AD<
> AD<double> > allows one to use arbitrary order derivatives of f(x) to
> express a function g(x) and then take derivatives of g(x).
>
> As you have noticed, in this case the function f(x) would be ADFun<
> AD<double> > and g(x) would be ADFun<double>.
>
> It is possible that you could do what you are asking for using template
> meta programmings, but I have never tried.
>
> I think that the research code stalingrad does automatic differentiation
> with the derivatives having the same type as the original functions; see
>         https://urldefense.proofpoint.com/v2/url?u=https-3A__github.
> com_Functional-2DAutoDiff_STALINGRAD&d=DwIGaQ&c=Ngd-ta5yRYsq
> eUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=DI01O1qukvwbwT9tWYo85y
> ppLHUSzhHiWawx-7crnog&m=_25pbHruWRmD1L2XClVRt07j_oFK0I3mg9CG
> jbh94fw&s=TYrwlh51kdextj7wgvj3pTmut23D6bXYrNHhV8VfGLk&e=
>
> On 06/03/2018 04:35 PM, Andrew Spielberg wrote:
>
>> Hi all,
>>
>> I am trying to come up with a way to get arbitrary levels of
>> AutoDifferentiation using CppAD.  Given a function f : R-->R, I would like
>> to be able to differentiate expressions such as, say:
>>
>> g(x) = (f'(x) + f''(x)) * f(x)
>>
>> in order to get dg/dx.
>>
>> Now, obviously, I can try to rewrite this as a single function just in f
>> whose derivative will be equal to dg/dx, but it that is not always
>> possible.  To this end, I am aware of the multiple levels example:
>>
>> https://urldefense.proofpoint.com/v2/url?u=https-3A__coin-2D
>> or.github.io_CppAD_doc_mul-5Flevel.cpp.htm&d=DwIGaQ&c=Ngd-ta
>> 5yRYsqeUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=DI01O1qukvwbwT9tWYo8
>> 5yppLHUSzhHiWawx-7crnog&m=_25pbHruWRmD1L2XClVRt07j_oFK0I3mg9
>> CGjbh94fw&s=7YUvdyn-IO5rWGWTGCWB1L5EUjF6-MgpfQT-8HnYGK8&e= <
>> https://urldefense.proofpoint.com/v2/url?u=https-3A__coin-
>> 2Dor.github.io_CppAD_doc_mul-5Flevel.cpp.htm&d=DwIGaQ&c=
>> Ngd-ta5yRYsqeUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=DI01O1qukvwbwT
>> 9tWYo85yppLHUSzhHiWawx-7crnog&m=_25pbHruWRmD1L2XClVRt07j_oFK
>> 0I3mg9CGjbh94fw&s=7YUvdyn-IO5rWGWTGCWB1L5EUjF6-MgpfQT-8HnYGK8&e=>
>>
>> However, as far as I can tell, the number of differentiations here must
>> be known a priori, and the correct number of nestings of ADFun must be
>> chosen up front.  In some cases though (say, when dealing with series of
>> unknown length), that number won't be known up front.
>>
>> I am wondering if there is a way to get arbitrary levels of
>> autodifferentiation somehow, so that I can treat gradients not unlike an
>> arithmetic function.  I was thinking, it might be possible with only two
>> levels of ADFun - every time a ADFun<ADFun<T>> is differentiated, it
>> returns an ADFun<T> - and then maybe this can be rewrapped in a new ADFun<
>> > before being reused.  I don't know if this works though.
>>
>> Are arbitrary levels possible?  If not, is there another way to achieve
>> what I want?
>>
>>
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>> ppLHUSzhHiWawx-7crnog&m=_25pbHruWRmD1L2XClVRt07j_oFK0I3mg9CG
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> ppLHUSzhHiWawx-7crnog&m=_25pbHruWRmD1L2XClVRt07j_oFK0I3mg9CG
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