Portability GHC only experimental ekmett@gmail.com

Description

Mixed-Mode Automatic Differentiation.

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

Synopsis

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

The `grad` function calculates the gradient of a non-scalar-to-scalar function with `Reverse` AD in a single pass.

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

The `grad'` function calculates the result and gradient of a non-scalar-to-scalar function with `Reverse` AD in a single pass.

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

`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)
```

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> FU f 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` AD in a single pass the gradient is combined element-wise with the argument using the function `g`.

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

# 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 `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 `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

`jacobianWith g f` 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.

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 `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

`jacobianWith' g f` 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.

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 `gradWithM'` from Numeric.AD.Reverse.

gradM :: (Traversable f, Monad m, Num a) => FF f m a -> f a -> m (f a)Source

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

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

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

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.

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

The `gradF'` 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 `gradF`

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

`hessianProduct f wv` computes the product of the hessian `H` of a non-scalar-to-scalar function `f` at `w = fst \$ wv` with a vector `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

`hessianProduct' f wv` computes both the gradient of a non-scalar-to-scalar `f` at `w = fst \$ wv` and the product of the hessian `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 -> aSource

The `diff` function calculates the first derivative of a scalar-to-scalar function by forward-mode `AD`

``` diff sin == cos
```

diffF :: (Functor f, Num a) => UF f a -> a -> f aSource

The `diffF` function calculates the first derivative of scalar-to-nonscalar function by F`orward` `AD`

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
```

diffF' :: (Functor f, Num a) => UF f a -> a -> f (a, a)Source

The `diffF'` function calculates the result and first derivative of a scalar-to-non-scalar function by F`orward` `AD`

# Derivatives (Tower)

diffs :: Num a => UU a -> a -> [a]Source

diffsF :: (Functor f, Num a) => UF f a -> a -> f [a]Source

diffs0 :: Num a => UU a -> a -> [a]Source

diffs0F :: (Functor f, Num a) => UF f a -> a -> f [a]Source

# Directional Derivatives (Forward Mode)

du :: (Functor f, Num a) => FU f a -> f (a, a) -> aSource

du' :: (Functor f, Num a) => FU f a -> f (a, a) -> (a, a)Source

duF :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g aSource

duF' :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g (a, a)Source

# Directional Derivatives (Tower)

dus :: (Functor f, Num a) => FU f a -> f [a] -> [a]Source

dus0 :: (Functor f, Num a) => FU f a -> f [a] -> [a]Source

dusF :: (Functor f, Functor g, Num a) => FF f g a -> f [a] -> g [a]Source

dus0F :: (Functor f, Functor g, Num a) => FF f g a -> f [a] -> g [a]Source

# 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

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

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

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

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

`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.

Constructors

Instances

class Lifted t => Mode t whereSource

Methods

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 :: Num a => t aSource

``` 'zero' = 'lift' 0
```

Instances

 Mode Id Lifted Forward => Mode Forward Lifted Reverse => Mode Reverse Lifted Tower => Mode Tower Mode f => Mode (AD f) Mode f => Mode (:> f) (Mode f, Mode g) => Mode (ComposeMode f g)