Project Plan

First Milestone

• implement required module signature
• implement a vanilla implementation
• represent a formula using functor
• evaluate the formula using ocaml category

Second Milestone

• replace additive type with float
  • in order to deal with generic zero object issue
  • I choose not to have parameterized type and use float only
    • 0.0 is zero object
• implement the forward AAD version

Airport Milestone

• airport_typeclass: compare haskell type class vs ocaml modular
• haskell foundation: type class/type family
• basic Haskell type class trick
• concat implementation on Additive constraints
• hacking solution to introduce additive

X Milestone

• represent linear map as generalized matrices
• represent linear map as continuation of linear map
• RAD implementation

Helper function

• linear map: scale (d x r)
• scalarD

scalarD f d = D (\x -> let r = f x in (r, scale (d x r)))

• scalarR f d = scalarD f (const d)
  • const :: a -> b -> a
  • d :: (s -> s)
  • target :: s -> (s -> s)
  • const :: (s -> s) -> s -> (s -> s)
  • const d :: s -> (s -> s)
  • (const d) x r = d r $$ \frac{d(\frac{1}{x})}{dx} = -(\frac{1}{x})^2 $$
• scalarX f d = scalarD f (const . d)
  • const :: a -> b -> a
  • (.) :: (b -> c) -> (a -> b) -> (a -> c)
  • d :: s -> s
  • (.) :: (s -> (s -> s)) -> (s -> s) -> (s -> (s -> s))
  • const :: s -> (s -> s)
  • (const . d) :: s -> (s -> s)
  • (const . d) x r = ((const . d) x) r = (const (d x)) r = d x

$$ \frac{d(cos(x))}{x} = - sin(x) $$

$$ \begin{aligned} d(a/b) &= \frac{a'b -ab'}{b^2}\\ D_(x * y) (a, b) (da, db) &= a\times db + b \times da \\ D_(x * \frac{1}{y}) (a, \frac{1}{b}) (da, d(\frac{1}{b})) &= a \times d(\frac{1}{b}) + \frac{1}{b} \times da\\ & = - a \times db \frac{1}{b^2} + \frac{da\times b}{b^2} \\ & = \frac{da\times b - db\times a}{b^2} \end{aligned} $$

• chain rule $$ D(f . g) (a) = Df(g(a)). Dg(a) $$