数字.转义其作用域的AD类型变量

Question

我试图在Haskell中使用自动微分来解决非线性控制问题，但是有一些问题让它工作。我基本上有一个cost函数，它应该在给定初始状态的情况下被优化。类型为:

data Reference a = Reference a deriving Functor
data Plant a = Plant a deriving Functor
optimize :: (RealFloat a) => Reference a -> Plant a -> [a] -> [[a]]
optimize ref plant initialInputs = gradientDescent (cost ref plant) initialInputs
cost :: (RealFloat a) => Reference a -> Plant a -> [a] -> a
cost = ...

这会导致以下错误消息:

Couldn't match expected type `Reference
                                (Numeric.AD.Internal.Reverse.Reverse s a)'
            with actual type `t'
  because type variable `s' would escape its scope
This (rigid, skolem) type variable is bound by
  a type expected by the context:
    Data.Reflection.Reifies s Numeric.AD.Internal.Reverse.Tape =>
    [Numeric.AD.Internal.Reverse.Reverse s a]
    -> Numeric.AD.Internal.Reverse.Reverse s a
  at test.hs:13:5-50
Relevant bindings include
  initialInputs :: [a] (bound at test.hs:12:20)
  ref :: t (bound at test.hs:12:10)
  optimize :: t -> t1 -> [a] -> [[a]] (bound at test.hs:12:1)
In the first argument of `cost', namely `ref'
In the first argument of `gradientDescent', namely
  `(cost ref plant)'

我甚至不确定我是否正确理解了这个错误。是否，ref和plant的类型需要访问s，这是gradientDescent的第一个参数的范围内?

这个可行吗?在寻找解决方案时，我尝试将问题简化为最小的示例，并发现以下定义会产生类似的错误消息:

optimize f inputs = gradientDescent f inputs 

这看起来很奇怪，因为optimize = gradientDescent不会产生任何错误。

Answer 1

cost ref plant的类型为[a] -> a，其中a与optimize的签名中的a相同

optimize :: (RealFloat a) => Reference a -> Plant a -> [a] -> [[a]]
                                       ^          ^
                                       |          ------------
                                       ------------------v   v
optimize ref plant initialInputs = gradientDescent (cost ref plant) initialInputs
                                                         ^   ^
                                   -----------------------   |
                                   v          v---------------
cost :: (RealFloat a) => Reference a -> Plant a -> [a] -> a
cost = ...

但是gradientDescent的类型是

gradientDescent :: (Traversable f, Fractional a, Ord a) =>
                   (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> 
                   f a -> [f a]

gradientDescent的第一个参数需要能够取[Reverse s a](对于任何 s)并返回Reverse s a，但cost ref plant只能取[a]并返回a。

由于Reference和Plant都是Functor s，可以将ref和plant从Reference a和Plant a转换为Reference (Reverse s a)和Plant (Reverse s a)，通过fmap替换auto。

optimize :: (RealFloat a) => Reference a -> Plant a -> [a] -> [[a]]
optimize ref plant initialInputs = gradientDescent (cost (fmap auto ref) (fmap auto plant)) initialInputs