Portability | GHC only |
---|---|

Stability | experimental |

Maintainer | ekmett@gmail.com |

- Gradients (Reverse Mode)
- Jacobians (Mixed Mode)
- Monadic Gradient/Jacobian (Reverse Mode)
- Functorial Gradient/Jacobian (Reverse Mode)
- Transposed Jacobians (Forward Mode)
- Hessian (Forward-On-Reverse)
- Hessian Tensors (Forward-On-Mixed)
- Hessian Vector Products (Forward-On-Reverse)
- Derivatives (Forward Mode)
- Derivatives (Tower)
- Directional Derivatives (Forward Mode)
- Directional Derivatives (Tower)
- Taylor Series (Tower)
- Maclaurin Series (Tower)
- Monadic Combinators (Forward Mode)
- Exposed Types

Mixed-Mode Automatic Differentiation.

Each combinator exported from this module chooses an appropriate AD mode.

- grad :: (Traversable f, Num a) => FU f a -> f a -> f a
- grad' :: (Traversable f, Num a) => FU f a -> f a -> (a, f a)
- gradWith :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> f b
- gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> (a, f b)
- jacobian :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (f a)
- jacobian' :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (a, f a)
- jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (f b)
- jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (a, f b)
- gradM :: (Traversable f, Monad m, Num a) => FF f m a -> f a -> m (f a)
- gradM' :: (Traversable f, Monad m, Num a) => FF f m a -> f a -> m (a, f a)
- gradWithM :: (Traversable f, Monad m, Num a) => (a -> a -> b) -> FF f m a -> f a -> m (f b)
- gradWithM' :: (Traversable f, Monad m, Num a) => (a -> a -> b) -> FF f m a -> f a -> m (a, f b)
- gradF :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (f a)
- gradF' :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (a, f a)
- gradWithF :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (f b)
- gradWithF' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (a, f b)
- jacobianT :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> f (g a)
- jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> f (g b)
- hessian :: (Traversable f, Num a) => FU f a -> f a -> f (f a)
- hessianTensor :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (f (f a))
- hessianProduct :: (Traversable f, Num a) => FU f a -> f (a, a) -> f a
- hessianProduct' :: (Traversable f, Num a) => FU f a -> f (a, a) -> f (a, a)
- diff :: Num a => UU a -> a -> a
- diffF :: (Functor f, Num a) => UF f a -> a -> f a
- diff' :: Num a => UU a -> a -> (a, a)
- diffF' :: (Functor f, Num a) => UF f a -> a -> f (a, a)
- diffs :: Num a => UU a -> a -> [a]
- diffsF :: (Functor f, Num a) => UF f a -> a -> f [a]
- diffs0 :: Num a => UU a -> a -> [a]
- diffs0F :: (Functor f, Num a) => UF f a -> a -> f [a]
- du :: (Functor f, Num a) => FU f a -> f (a, a) -> a
- du' :: (Functor f, Num a) => FU f a -> f (a, a) -> (a, a)
- duF :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g a
- duF' :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g (a, a)
- dus :: (Functor f, Num a) => FU f a -> f [a] -> [a]
- dus0 :: (Functor f, Num a) => FU f a -> f [a] -> [a]
- dusF :: (Functor f, Functor g, Num a) => FF f g a -> f [a] -> g [a]
- dus0F :: (Functor f, Functor g, Num a) => FF f g a -> f [a] -> g [a]
- taylor :: Fractional a => UU a -> a -> a -> [a]
- taylor0 :: Fractional a => UU a -> a -> a -> [a]
- maclaurin :: Fractional a => UU a -> a -> [a]
- maclaurin0 :: Fractional a => UU a -> a -> [a]
- diffM :: (Monad m, Num a) => UF m a -> a -> m a
- diffM' :: (Monad m, Num a) => UF m a -> a -> m (a, a)
- type UU a = forall s. Mode s => AD s a -> AD s a
- type UF f a = forall s. Mode s => AD s a -> f (AD s a)
- type FU f a = forall s. Mode s => f (AD s a) -> AD s a
- type FF f g a = forall s. Mode s => f (AD s a) -> g (AD s a)
- newtype AD f a = AD {
- runAD :: f a

- class Lifted t => Mode t where

# Gradients (Reverse Mode)

grad :: (Traversable f, Num a) => FU f a -> f a -> f aSource

grad' :: (Traversable f, Num a) => FU f a -> f a -> (a, f a)Source

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> f bSource

function calculates the gradient of a non-scalar-to-scalar function `grad`

g f`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) -> FU f a -> f a -> (a, f b)Source

# Jacobians (Mixed Mode)

jacobian :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (f a)Source

Calculate the Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs.

If you need to support functions where the output is only a `Functor`

or `Monad`

, consider `Numeric.AD.Reverse.jacobian`

or `Numeric.AD.Reverse.gradM`

from Numeric.AD.Reverse.

jacobian' :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (a, f a)Source

Calculate both the answer and Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward- and reverse- mode AD based on the relative, number of inputs and outputs.

If you need to support functions where the output is only a `Functor`

or `Monad`

, consider `Numeric.AD.Reverse.jacobian'`

or `Numeric.AD.Reverse.gradM'`

from Numeric.AD.Reverse.

jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (f b)Source

calculates the Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs.
`jacobianWith`

g f

The resulting Jacobian matrix is then recombined element-wise with the input using `g`

.

If you need to support functions where the output is only a `Functor`

or `Monad`

, consider `Numeric.AD.Reverse.jacobianWith`

or `Numeric.AD.Reverse.gradWithM`

from Numeric.AD.Reverse.

jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (a, f b)Source

calculates the answer and Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs.
`jacobianWith'`

g f

The resulting Jacobian matrix is then recombined element-wise with the input using `g`

.

If you need to support functions where the output is only a `Functor`

or `Monad`

, consider `Numeric.AD.Reverse.jacobianWith'`

or `Numeric.AD.Reverse.gradWithM'`

from Numeric.AD.Reverse.

# Monadic Gradient/Jacobian (Reverse Mode)

gradWithM' :: (Traversable f, Monad m, Num a) => (a -> a -> b) -> FF f m a -> f a -> m (a, f b)Source

# Functorial Gradient/Jacobian (Reverse Mode)

gradF :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> g (f a)Source

The `gradF`

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

passes for `m`

outputs.

gradWithF :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (f b)Source

'gradWithF 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`

.

gradF == gradWithF (\_ dx -> dx) gradWithF const == (\f x -> const x <$> f x)

gradWithF' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (a, f b)Source

`gradWithF`

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 `gradWithF`

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

.

jacobian' == gradWithF' (\_ dx -> dx)

# Transposed Jacobians (Forward Mode)

jacobianT :: (Traversable f, Functor g, Num a) => FF f g a -> f a -> f (g a)Source

A fast, simple transposed Jacobian computed with forward-mode AD.

jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> f (g b)Source

A fast, simple transposed Jacobian computed with forward-mode AD.

# Hessian (Forward-On-Reverse)

hessian :: (Traversable f, Num a) => FU f 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 forward mode.

# Hessian Tensors (Forward-On-Mixed)

hessianTensor :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (f (f a))Source

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

# Hessian Vector Products (Forward-On-Reverse)

hessianProduct :: (Traversable f, Num a) => FU f a -> f (a, a) -> f aSource

computes the product of the hessian `hessianProduct`

f wv`H`

of a non-scalar-to-scalar function `f`

at `w = `

with a vector `fst`

$ wv`v = snd $ wv`

using "Pearlmutter's method" from http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.6143, which states:

H v = (d/dr) grad_w (w + r v) | r = 0

Or in other words, we take the directional derivative of the gradient.

hessianProduct' :: (Traversable f, Num a) => FU f a -> f (a, a) -> f (a, a)Source

computes both the gradient of a non-scalar-to-scalar `hessianProduct'`

f wv`f`

at `w = `

and the product of the hessian `fst`

$ wv`H`

at `w`

with a vector `v = snd $ wv`

using "Pearlmutter's method". The outputs are returned wrapped in the same functor.

H v = (d/dr) grad_w (w + r v) | r = 0

Or in other words, we take the directional derivative of the gradient.

# Derivatives (Forward Mode)

diff' :: Num a => UU a -> a -> (a, a)Source

The `d'UU`

function calculates the result and first derivative of scalar-to-scalar function by F`orward`

`AD`

d' sin == sin &&& cos d' f = f &&& d f

# Derivatives (Tower)

# Directional Derivatives (Forward Mode)

# Directional Derivatives (Tower)

# Taylor Series (Tower)

taylor :: Fractional a => UU a -> a -> a -> [a]Source

taylor0 :: Fractional a => UU a -> a -> a -> [a]Source

# Maclaurin Series (Tower)

maclaurin :: Fractional a => UU a -> a -> [a]Source

maclaurin0 :: Fractional a => UU a -> a -> [a]Source

# Monadic Combinators (Forward Mode)

diffM :: (Monad m, Num a) => UF m a -> a -> m aSource

The `dUM`

function calculates the first derivative of scalar-to-scalar monadic function by F`orward`

`AD`

diffM' :: (Monad m, Num a) => UF m a -> a -> m (a, a)Source

The `d'UM`

function calculates the result and first derivative of a scalar-to-scalar monadic function by F`orward`

`AD`

# Exposed Types

type UU a = forall s. Mode s => AD s a -> AD s aSource

A scalar-to-scalar automatically-differentiable function.

type UF f a = forall s. Mode s => AD s a -> f (AD s a)Source

A scalar-to-non-scalar automatically-differentiable function.

type FU f a = forall s. Mode s => f (AD s a) -> AD s aSource

A non-scalar-to-scalar automatically-differentiable function.

type FF f g a = forall s. Mode s => f (AD s a) -> g (AD s a)Source

A non-scalar-to-non-scalar automatically-differentiable function.

`AD`

serves as a common wrapper for different `Mode`

instances, exposing a traditional
numerical tower. Universal quantification is used to limit the actions in user code to
machinery that will return the same answers under all AD modes, allowing us to use modes
interchangeably as both the type level "brand" and dictionary, providing a common API.

Typeable1 f => Typeable1 (AD f) | |

Primal f => Primal (AD f) | |

Mode f => Mode (AD f) | |

Lifted f => Lifted (AD f) | |

Var (AD Reverse) | |

Iso (f a) (AD f a) | |

(Num a, Lifted f, Bounded a) => Bounded (AD f a) | |

(Num a, Lifted f, Enum a) => Enum (AD f a) | |

(Num a, Lifted f, Eq a) => Eq (AD f a) | |

(Lifted f, Floating a) => Floating (AD f a) | |

(Lifted f, Fractional a) => Fractional (AD f a) | |

(Typeable1 f, Typeable a, Data (f a), Data a) => Data (AD f a) | |

(Lifted f, Num a) => Num (AD f a) | |

(Num a, Lifted f, Ord a) => Ord (AD f a) | |

(Lifted f, Real a) => Real (AD f a) | |

(Lifted f, RealFloat a) => RealFloat (AD f a) | |

(Lifted f, RealFrac a) => RealFrac (AD f a) | |

(Lifted f, Show a) => Show (AD f a) | |

(Typeable1 f, Typeable a) => Typeable (AD f a) |

class Lifted t => Mode t whereSource

lift :: Num a => a -> t aSource

Embed a constant

(<+>) :: Num a => t a -> t a -> t aSource

Vector sum

(*^) :: Num a => a -> t a -> t aSource

Scalar-vector multiplication

(^*) :: Num a => t a -> a -> t aSource

Vector-scalar multiplication

(^/) :: Fractional a => t a -> a -> t aSource

Scalar division

'zero' = 'lift' 0