Copyright	(c) Edward Kmett 2010-2015
License	BSD3
Maintainer	ekmett@gmail.com
Stability	experimental
Portability	GHC only
Safe Haskell	None
Language	Haskell2010

Numeric.AD.Mode.Reverse

Contents

Gradient
Jacobian
Hessian
Derivatives

Description

Reverse-mode automatic differentiation using Wengert lists and Data.Reflection

Synopsis

Documentation

data Reverse s a Source #

Instances

(Reifies * s Tape, Num a, Bounded a) => Bounded (Reverse s a) #
Methods minBound :: Reverse s a # maxBound :: Reverse s a #
(Reifies * s Tape, Num a, Enum a) => Enum (Reverse s a) #
Methods succ :: Reverse s a -> Reverse s a # pred :: Reverse s a -> Reverse s a # toEnum :: Int -> Reverse s a # fromEnum :: Reverse s a -> Int # enumFrom :: Reverse s a -> [Reverse s a] # enumFromThen :: Reverse s a -> Reverse s a -> [Reverse s a] # enumFromTo :: Reverse s a -> Reverse s a -> [Reverse s a] # enumFromThenTo :: Reverse s a -> Reverse s a -> Reverse s a -> [Reverse s a] #
(Reifies * s Tape, Num a, Eq a) => Eq (Reverse s a) #
Methods (==) :: Reverse s a -> Reverse s a -> Bool # (/=) :: Reverse s a -> Reverse s a -> Bool #
(Reifies * s Tape, Floating a) => Floating (Reverse s a) #
Methods pi :: Reverse s a # exp :: Reverse s a -> Reverse s a # log :: Reverse s a -> Reverse s a # sqrt :: Reverse s a -> Reverse s a # (**) :: Reverse s a -> Reverse s a -> Reverse s a # logBase :: Reverse s a -> Reverse s a -> Reverse s a # sin :: Reverse s a -> Reverse s a # cos :: Reverse s a -> Reverse s a # tan :: Reverse s a -> Reverse s a # asin :: Reverse s a -> Reverse s a # acos :: Reverse s a -> Reverse s a # atan :: Reverse s a -> Reverse s a # sinh :: Reverse s a -> Reverse s a # cosh :: Reverse s a -> Reverse s a # tanh :: Reverse s a -> Reverse s a # asinh :: Reverse s a -> Reverse s a # acosh :: Reverse s a -> Reverse s a # atanh :: Reverse s a -> Reverse s a # log1p :: Reverse s a -> Reverse s a # expm1 :: Reverse s a -> Reverse s a # log1pexp :: Reverse s a -> Reverse s a # log1mexp :: Reverse s a -> Reverse s a #
(Reifies * s Tape, Fractional a) => Fractional (Reverse s a) #
Methods (/) :: Reverse s a -> Reverse s a -> Reverse s a # recip :: Reverse s a -> Reverse s a # fromRational :: Rational -> Reverse s a #
(Reifies * s Tape, Num a) => Num (Reverse s a) #
Methods (+) :: Reverse s a -> Reverse s a -> Reverse s a # (-) :: Reverse s a -> Reverse s a -> Reverse s a # (*) :: Reverse s a -> Reverse s a -> Reverse s a # negate :: Reverse s a -> Reverse s a # abs :: Reverse s a -> Reverse s a # signum :: Reverse s a -> Reverse s a # fromInteger :: Integer -> Reverse s a #
(Reifies * s Tape, Num a, Ord a) => Ord (Reverse s a) #
Methods compare :: Reverse s a -> Reverse s a -> Ordering # (<) :: Reverse s a -> Reverse s a -> Bool # (<=) :: Reverse s a -> Reverse s a -> Bool # (>) :: Reverse s a -> Reverse s a -> Bool # (>=) :: Reverse s a -> Reverse s a -> Bool # max :: Reverse s a -> Reverse s a -> Reverse s a # min :: Reverse s a -> Reverse s a -> Reverse s a #
(Reifies * s Tape, Real a) => Real (Reverse s a) #
Methods toRational :: Reverse s a -> Rational #
(Reifies * s Tape, RealFloat a) => RealFloat (Reverse s a) #
Methods floatRadix :: Reverse s a -> Integer # floatDigits :: Reverse s a -> Int # floatRange :: Reverse s a -> (Int, Int) # decodeFloat :: Reverse s a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Reverse s a # exponent :: Reverse s a -> Int # significand :: Reverse s a -> Reverse s a # scaleFloat :: Int -> Reverse s a -> Reverse s a # isNaN :: Reverse s a -> Bool # isInfinite :: Reverse s a -> Bool # isDenormalized :: Reverse s a -> Bool # isNegativeZero :: Reverse s a -> Bool # isIEEE :: Reverse s a -> Bool # atan2 :: Reverse s a -> Reverse s a -> Reverse s a #
(Reifies * s Tape, RealFrac a) => RealFrac (Reverse s a) #
Methods properFraction :: Integral b => Reverse s a -> (b, Reverse s a) # truncate :: Integral b => Reverse s a -> b # round :: Integral b => Reverse s a -> b # ceiling :: Integral b => Reverse s a -> b # floor :: Integral b => Reverse s a -> b #
Show a => Show (Reverse s a) Source #
Methods showsPrec :: Int -> Reverse s a -> ShowS # show :: Reverse s a -> String # showList :: [Reverse s a] -> ShowS #
(Reifies * s Tape, Erf a) => Erf (Reverse s a) #
Methods erf :: Reverse s a -> Reverse s a # erfc :: Reverse s a -> Reverse s a # erfcx :: Reverse s a -> Reverse s a # normcdf :: Reverse s a -> Reverse s a #
(Reifies * s Tape, InvErf a) => InvErf (Reverse s a) #
Methods inverf :: Reverse s a -> Reverse s a # inverfc :: Reverse s a -> Reverse s a # invnormcdf :: Reverse s a -> Reverse s a #
(Reifies * s Tape, Num a) => Mode (Reverse s a) Source #
Associated Types type Scalar (Reverse s a) :: * Source # Methods isKnownConstant :: Reverse s a -> Bool Source # isKnownZero :: Reverse s a -> Bool Source # auto :: Scalar (Reverse s a) -> Reverse s a Source # (^) :: Scalar (Reverse s a) -> Reverse s a -> Reverse s a Source # (^) :: Reverse s a -> Scalar (Reverse s a) -> Reverse s a Source # (^/) :: Reverse s a -> Scalar (Reverse s a) -> Reverse s a Source # zero :: Reverse s a Source #
(Reifies * s Tape, Num a) => Jacobian (Reverse s a) Source #
Associated Types type D (Reverse s a) :: * Source # Methods unary :: (Scalar (Reverse s a) -> Scalar (Reverse s a)) -> D (Reverse s a) -> Reverse s a -> Reverse s a Source # lift1 :: (Scalar (Reverse s a) -> Scalar (Reverse s a)) -> (D (Reverse s a) -> D (Reverse s a)) -> Reverse s a -> Reverse s a Source # lift1_ :: (Scalar (Reverse s a) -> Scalar (Reverse s a)) -> (D (Reverse s a) -> D (Reverse s a) -> D (Reverse s a)) -> Reverse s a -> Reverse s a Source # binary :: (Scalar (Reverse s a) -> Scalar (Reverse s a) -> Scalar (Reverse s a)) -> D (Reverse s a) -> D (Reverse s a) -> Reverse s a -> Reverse s a -> Reverse s a Source # lift2 :: (Scalar (Reverse s a) -> Scalar (Reverse s a) -> Scalar (Reverse s a)) -> (D (Reverse s a) -> D (Reverse s a) -> (D (Reverse s a), D (Reverse s a))) -> Reverse s a -> Reverse s a -> Reverse s a Source # lift2_ :: (Scalar (Reverse s a) -> Scalar (Reverse s a) -> Scalar (Reverse s a)) -> (D (Reverse s a) -> D (Reverse s a) -> D (Reverse s a) -> (D (Reverse s a), D (Reverse s a))) -> Reverse s a -> Reverse s a -> Reverse s a Source #
type Scalar (Reverse s a) Source #
type Scalar (Reverse s a) = a
type D (Reverse s a) Source #
type D (Reverse s a) = Id a

auto :: Mode t => Scalar t -> t Source #

Embed a constant

Gradient

grad :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f a Source #

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

>>> grad (\[x,y,z] -> x*y+z) [1,2,3]
[2,1,1]

grad' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f a) Source #

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

>>> grad' (\[x,y,z] -> x*y+z) [1,2,3]
(5,[2,1,1])

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f b Source #

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. Reifies s Tape => f (Reverse s a) -> Reverse s 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-mode 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) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f a) Source #

The jacobian function calculates the jacobian of a non-scalar-to-non-scalar function with reverse AD lazily in m passes for m outputs.

>>> jacobian (\[x,y] -> [y,x,x*y]) [2,1]
[[0,1],[1,0],[1,2]]

jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f a) Source #

The jacobian' 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 jacobian | An alias for gradF'

>>> jacobian' (\[x,y] -> [y,x,x*y]) [2,1]
[(1,[0,1]),(2,[1,0]),(2,[1,2])]

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f b) Source #

'jacobianWith 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.

jacobian == jacobianWith (_ dx -> dx)
jacobianWith const == (f x -> const x <$> f x)

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f b) Source #

jacobianWith 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 jacobianWith

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

jacobian' == jacobianWith' (_ dx -> dx)

Hessian

hessian :: (Traversable f, Num a) => (forall s s'. (Reifies s Tape, Reifies s' Tape) => f (On (Reverse s (Reverse s' a))) -> On (Reverse s (Reverse s' 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 hessian.

>>> hessian (\[x,y] -> x*y) [1,2]
[[0,1],[1,0]]

hessianF :: (Traversable f, Functor g, Num a) => (forall s s'. (Reifies s Tape, Reifies s' Tape) => f (On (Reverse s (Reverse s' a))) -> g (On (Reverse s (Reverse s' a)))) -> f a -> g (f (f a)) Source #

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

Less efficient than hessianF.

>>> hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2]
[[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]

Derivatives

diff :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a) -> a -> a Source #

Compute the derivative of a function.

>>> diff sin 0
1.0

diff' :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a) -> a -> (a, a) Source #

The diff' function calculates the result and derivative, as a pair, of a scalar-to-scalar function.

>>> diff' sin 0
(0.0,1.0)

>>> diff' exp 0
(1.0,1.0)

diffF :: (Functor f, Num a) => (forall s. Reifies s Tape => Reverse s a -> f (Reverse s a)) -> a -> f a Source #

Compute the derivatives of each result of a scalar-to-vector function with regards to its input.

>>> diffF (\a -> [sin a, cos a]) 0
[1.0,0.0]

diffF' :: (Functor f, Num a) => (forall s. Reifies s Tape => Reverse s a -> f (Reverse s a)) -> a -> f (a, a) Source #

Compute the derivatives of each result of a scalar-to-vector function with regards to its input along with the answer.

>>> diffF' (\a -> [sin a, cos a]) 0
[(0.0,1.0),(1.0,0.0)]