Numeric.AD

Mixed-Mode Automatic Differentiation.

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

Gradients (Reverse Mode)

grad :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s 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) => (forall s. Mode s => f (AD s a) -> AD s 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) -> (forall s. Mode s => f (AD s a) -> AD s 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) -> (forall s. Mode s => f (AD s a) -> AD s 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)

Jacobians (Mixed Mode)

jacobian :: (Traversable f, Traversable g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f a)

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) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f a)

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) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f b)

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) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f b)

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.

Jacobians (Reverse Mode)

gradF :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s 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) => (forall s. Mode s => f (AD s a) -> g (AD s 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) -> (forall s. Mode s => f (AD s a) -> g (AD s 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) -> (forall s. Mode s => f (AD s a) -> g (AD s 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)

Jacobians (Forward Mode)

jacobianT :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> f (g a)

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

jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> f (g b)

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

Hessian (Forward-On-Reverse)

hessian :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s 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 forward mode.

Hessian Tensors (Forward-On-Mixed)

hessianTensor :: (Traversable f, Traversable g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s 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.

Hessian Vector Products (Forward-On-Reverse)

hessianProduct :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f (a, a) -> f a

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

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 => (forall s. Mode s => AD s a -> AD s a) -> a -> a

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

The diffF function calculates the first derivative of scalar-to-nonscalar function by Forward AD

diff' :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a)

The d'UU function calculates the result and first derivative of scalar-to-scalar function by Forward AD

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

diffF' :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f (a, a)

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

Derivatives (Tower)

diffs :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

diffsF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f [a]

diffs0 :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

diffs0F :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f [a]

Directional Derivatives (Forward Mode)

du :: (Functor f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f (a, a) -> a

du' :: (Functor f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f (a, a) -> (a, a)

duF :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f (a, a) -> g a

duF' :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f (a, a) -> g (a, a)

Taylor Series (Tower)

taylor :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> a -> [a]

taylor0 :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> a -> [a]

Maclaurin Series (Tower)

maclaurin :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

maclaurin0 :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

Monadic Combinators (Forward Mode)

diffM :: (Monad m, Num a) => (forall s. Mode s => AD s a -> m (AD s a)) -> a -> m a

The dUM function calculates the first derivative of scalar-to-scalar monadic function by Forward AD

diffM' :: (Monad m, Num a) => (forall s. Mode s => AD s a -> m (AD s a)) -> a -> m (a, a)

The d'UM function calculates the result and first derivative of a scalar-to-scalar monadic function by Forward AD

Monadic Combinators (Reverse Mode)

gradM :: (Traversable f, Monad m, Num a) => (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (f a)

gradM' :: (Traversable f, Monad m, Num a) => (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (a, f a)

gradWithM :: (Traversable f, Monad m, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (f b)

gradWithM' :: (Traversable f, Monad m, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (a, f b)

Exposed Types

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 g) => Mode (ComposeMode f g)