Numeric.AD - 类型变量逃脱其范围

Question

我正在尝试在 Haskell 中使用 automatic differentiation 来解决非线性控制问题，但在让它工作时遇到了一些问题。我基本上有一个cost 函数，应该在给定初始状态的情况下对其进行优化。类型有：

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)'

我什至不确定我是否正确理解了错误。是不是ref和plant的类型需要访问s，这在gradientDescent的第一个参数范围内？

是否有可能做到这一点？在寻找解决方案时，我尝试将问题简化为最小示例，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采用（对于 any s）[Reverse s a] 并返回 Reverse s a，但 cost ref plant 只能采用 [a]返回一个a。

由于Reference 和Plant 都是Functors，您可以将ref 和plant 从Reference a 和Plant a 转换为Reference (Reverse s a) 和Plant (Reverse s a) @98765444 987654322@.

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