Numeric.AD.Reverse

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.

Gradient

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

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)

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 b

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)

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)

An alias for gradF

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

An alias for gradF'

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

An alias for gradWithF.

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

An alias for gradWithF'

Hessian

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

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))

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))

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 -> a

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

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 a

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

Monadic Combinators

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

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

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)

Synonyms

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

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)

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)

'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)

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 a

A scalar-to-scalar automatically-differentiable function.

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

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

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

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)

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

newtype AD f a

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

Fields

runAD :: 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 where

Methods

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

Embed a constant

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

Vector sum

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

Scalar-vector multiplication

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

Vector-scalar multiplication

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

Scalar division

zero :: Num a => t a

 '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)