ad-0.28: Automatic Differentiation

Portability GHC only experimental ekmett@gmail.com

Numeric.AD.Reverse

Description

Mixed-Mode Automatic Differentiation.

For reverse mode AD we use `System.Mem.StableName.StableName` to recover sharing information from the tape to avoid combinatorial explosion, and thus run asymptotically faster than it could without such sharing information, but the use of side-effects contained herein is benign.

Synopsis

# Gradient

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)
id == gradWith const
```

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)
```

# Jacobian

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

An alias for `gradF`

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

An alias for `gradF'`

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

An alias for `gradWithF`.

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

An alias for `gradWithF'`

# Hessian

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 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 `Numeric.AD.hessian` in Numeric.AD.

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

Compute the hessian via the reverse-mode jacobian of the reverse-mode gradient of a non-scalar-to-scalar monadic action.

While this is less efficient than `Numeric.AD.hessianTensor` from Numeric.AD or `Numeric.AD.Forward.hessianTensor` from Numeric.AD.Forward, the type signature is more permissive with regards to the output non-scalar, and it may be more efficient if only a few coefficients of the result are consumed.

hessianTensor :: (Traversable f, Functor 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.

While this is less efficient than `Numeric.AD.hessianTensor` from Numeric.AD or `Numeric.AD.Forward.hessianTensor` from Numeric.AD.Forward, the type signature is more permissive with regards to the output non-scalar, and it may be more efficient if only a few coefficients of the result are consumed.

# Derivatives

diff :: Num a => UU a -> a -> aSource

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

The `d'` function calculates the value and derivative, as a pair, of a scalar-to-scalar function.

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

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

# Monadic Combinators

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

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

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

# Synonyms

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)
```

# 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.

newtype AD f 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

 AD FieldsrunAD :: f a

Instances

 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) (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)

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 (Iterated f) (Mode f, Mode g) => Mode (ComposeMode f g)