Copyright (c) Edward Kmett 2010-2015 BSD3 ekmett@gmail.com experimental GHC only None Haskell2010

Description

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

Synopsis

# Documentation

data Reverse s a Source

Instances

 (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

auto :: Mode t => Scalar t -> t Source

Embed a constant

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.

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

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.

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

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

`grad g f` function calculates the gradient of a non-scalar-to-scalar function `f` with reverse-mode AD in a single pass. The gradient is combined element-wise with the argument using the function `g`.

````grad` == `gradWith` (_ dx -> dx)
`id` == `gradWith` `const`
```

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

`grad' g f` calculates the result and gradient of a non-scalar-to-scalar function `f` with reverse-mode AD in a single pass the gradient is combined element-wise with the argument using the function `g`.

````grad'` == `gradWith'` (_ dx -> dx)
```

# 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.

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

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'`

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

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

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

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`.

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

# Derivatives

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

Compute the derivative of a function.

````>>> ````diff sin 0
```1.0
```

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.

````>>> ````diff' sin 0
```(0.0,1.0)
```
````>>> ````diff' exp 0
```(1.0,1.0)
```

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

Compute the derivatives of each result of a scalar-to-vector function with regards to its input.

````>>> ````diffF (\a -> [sin a, cos a]) 0
```[1.0,0.0]
```

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

Compute the derivatives of each result of a scalar-to-vector function with regards to its input along with the answer.

````>>> ````diffF' (\a -> [sin a, cos a]) 0
```[(0.0,1.0),(1.0,0.0)]
```