Copyright | (c) Edward Kmett 2010-2015 |
---|---|

License | BSD3 |

Maintainer | ekmett@gmail.com |

Stability | experimental |

Portability | GHC only |

Safe Haskell | None |

Language | Haskell2010 |

Reverse-mode automatic differentiation using Wengert lists and Data.Reflection

- data Reverse s a
- auto :: Mode t => Scalar t -> t
- grad :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f a
- grad' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f a)
- gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f b
- gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f b)
- jacobian :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f a)
- jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f a)
- jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f b)
- jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f b)
- hessian :: (Traversable f, Num a) => (forall s s'. (Reifies s Tape, Reifies s' Tape) => f (On (Reverse s (Reverse s' a))) -> On (Reverse s (Reverse s' a))) -> f a -> f (f a)
- hessianF :: (Traversable f, Functor g, Num a) => (forall s s'. (Reifies s Tape, Reifies s' Tape) => f (On (Reverse s (Reverse s' a))) -> g (On (Reverse s (Reverse s' a)))) -> f a -> g (f (f a))
- diff :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a) -> a -> a
- diff' :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a) -> a -> (a, a)
- diffF :: (Functor f, Num a) => (forall s. Reifies s Tape => Reverse s a -> f (Reverse s a)) -> a -> f a
- diffF' :: (Functor f, Num a) => (forall s. Reifies s Tape => Reverse s a -> f (Reverse s a)) -> a -> f (a, a)

# Documentation

(Reifies * s Tape, Num a, Bounded a) => Bounded (Reverse s a) | |

(Reifies * s Tape, Num a, Enum a) => Enum (Reverse s a) | |

(Reifies * s Tape, Num a, Eq a) => Eq (Reverse s a) | |

(Reifies * s Tape, Floating a) => Floating (Reverse s a) | |

(Reifies * s Tape, Fractional a) => Fractional (Reverse s a) | |

(Reifies * s Tape, Num a) => Num (Reverse s a) | |

(Reifies * s Tape, Num a, Ord a) => Ord (Reverse s a) | |

(Reifies * s Tape, Real a) => Real (Reverse s a) | |

(Reifies * s Tape, RealFloat a) => RealFloat (Reverse s a) | |

(Reifies * s Tape, RealFrac a) => RealFrac (Reverse s a) | |

Show a => Show (Reverse s a) | |

(Reifies * s Tape, Erf a) => Erf (Reverse s a) | |

(Reifies * s Tape, InvErf a) => InvErf (Reverse s a) | |

(Reifies * s Tape, Num a) => Mode (Reverse s a) | |

(Reifies * s Tape, Num a) => Jacobian (Reverse s a) | |

Typeable (* -> * -> *) Reverse | |

type Scalar (Reverse s a) = a | |

type D (Reverse s a) = Id a |

# Gradient

grad :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f a Source

The `grad`

function calculates the gradient of a non-scalar-to-scalar function with reverse-mode AD in a single pass.

`>>>`

[2,1,1]`grad (\[x,y,z] -> x*y+z) [1,2,3]`

grad' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f a) Source

The `grad'`

function calculates the result and gradient of a non-scalar-to-scalar function with reverse-mode AD in a single pass.

`>>>`

(5,[2,1,1])`grad' (\[x,y,z] -> x*y+z) [1,2,3]`

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f b Source

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f b) Source

# Jacobian

jacobian :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f a) Source

The `jacobian`

function calculates the jacobian of a non-scalar-to-non-scalar function with reverse AD lazily in `m`

passes for `m`

outputs.

`>>>`

[[0,1],[1,0],[1,2]]`jacobian (\[x,y] -> [y,x,x*y]) [2,1]`

jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f a) Source

The `jacobian'`

function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using `m`

invocations of reverse AD,
where `m`

is the output dimensionality. Applying `fmap snd`

to the result will recover the result of `jacobian`

| An alias for `gradF'`

`>>>`

[(1,[0,1]),(2,[1,0]),(2,[1,2])]`jacobian' (\[x,y] -> [y,x,x*y]) [2,1]`

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f b) Source

'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function `f`

with reverse AD lazily in `m`

passes for `m`

outputs.

Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the `g`

.

`jacobian`

==`jacobianWith`

(_ dx -> dx)`jacobianWith`

`const`

== (f x ->`const`

x`<$>`

f x)

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f b) Source

`jacobianWith`

g f' calculates both the result and the Jacobian of a nonscalar-to-nonscalar function `f`

, using `m`

invocations of reverse AD,
where `m`

is the output dimensionality. Applying `fmap snd`

to the result will recover the result of `jacobianWith`

Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the `g`

.

`jacobian'`

==`jacobianWith'`

(_ dx -> dx)

# Hessian

hessian :: (Traversable f, Num a) => (forall s s'. (Reifies s Tape, Reifies s' Tape) => f (On (Reverse s (Reverse s' a))) -> On (Reverse s (Reverse s' a))) -> f a -> f (f a) Source

Compute the hessian via the jacobian of the gradient. gradient is computed in reverse mode and then the jacobian is computed in reverse mode.

However, since the

is square this is not as fast as using the forward-mode Jacobian of a reverse mode gradient provided by `grad`

f :: f a -> f a`hessian`

.

`>>>`

[[0,1],[1,0]]`hessian (\[x,y] -> x*y) [1,2]`

hessianF :: (Traversable f, Functor g, Num a) => (forall s s'. (Reifies s Tape, Reifies s' Tape) => f (On (Reverse s (Reverse s' a))) -> g (On (Reverse s (Reverse s' a)))) -> f a -> g (f (f a)) Source

Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the reverse-mode Jacobian of the reverse-mode Jacobian of the function.

Less efficient than `hessianF`

.

`>>>`

[[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]`hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2]`

# Derivatives

diff :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a) -> a -> a Source

Compute the derivative of a function.

`>>>`

1.0`diff sin 0`

diff' :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a) -> a -> (a, a) Source

The `diff'`

function calculates the result and derivative, as a pair, of a scalar-to-scalar function.

`>>>`

(0.0,1.0)`diff' sin 0`

`>>>`

(1.0,1.0)`diff' exp 0`