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.Kahn

Contents

Gradient
Jacobian
Hessian
Derivatives

Description

This module provides reverse-mode Automatic Differentiation using post-hoc linear time topological sorting.

For reverse mode AD we use 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

data AD s a
data Kahn a
auto :: Mode t => Scalar t -> t
grad :: (Traversable f, Num a) => (forall s. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> f a
grad' :: (Traversable f, Num a) => (forall s. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> (a, f a)
gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> f b
gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> (a, f b)
jacobian :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Kahn a)) -> g (AD s (Kahn a))) -> f a -> g (f a)
jacobian' :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Kahn a)) -> g (AD s (Kahn a))) -> f a -> g (a, f a)
jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Kahn a)) -> g (AD s (Kahn a))) -> f a -> g (f b)
jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Kahn a)) -> g (AD s (Kahn a))) -> f a -> g (a, f b)
hessian :: (Traversable f, Num a) => (forall s. f (AD s (On (Kahn (Kahn a)))) -> AD s (On (Kahn (Kahn a)))) -> f a -> f (f a)
hessianF :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (On (Kahn (Kahn a)))) -> g (AD s (On (Kahn (Kahn a))))) -> f a -> g (f (f a))
diff :: Num a => (forall s. AD s (Kahn a) -> AD s (Kahn a)) -> a -> a
diff' :: Num a => (forall s. AD s (Kahn a) -> AD s (Kahn a)) -> a -> (a, a)
diffF :: (Functor f, Num a) => (forall s. AD s (Kahn a) -> f (AD s (Kahn a))) -> a -> f a
diffF' :: (Functor f, Num a) => (forall s. AD s (Kahn a) -> f (AD s (Kahn a))) -> a -> f (a, a)

Documentation

data AD s a Source #

Instances

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

data Kahn a Source #

Kahn is a Mode using reverse-mode automatic differentiation that provides fast diffFU, diff2FU, grad, grad2 and a fast jacobian when you have a significantly smaller number of outputs than inputs.

Instances

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

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

Embed a constant

Gradient

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

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

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

>>> grad (\[x,y] -> x**y) [0,2]
[0.0,NaN]

grad' :: (Traversable f, Num a) => (forall s. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> (a, f a) Source #

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

>>> grad' (\[x,y,z] -> 4*x*exp y+cos z) [1,2,3]
(28.566231899122155,[29.5562243957226,29.5562243957226,-0.1411200080598672])

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> f b Source #

grad g f function calculates the gradient of a non-scalar-to-scalar function f with kahn-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. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> (a, f b) Source #

grad' g f calculates the result and gradient of a non-scalar-to-scalar function f with kahn-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. f (AD s (Kahn a)) -> g (AD s (Kahn a))) -> f a -> g (f a) Source #

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

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

>>> jacobian (\[x,y] -> [exp y,cos x,x+y]) [1,2]
[[0.0,7.38905609893065],[-0.8414709848078965,0.0],[1.0,1.0]]

jacobian' :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Kahn a)) -> g (AD s (Kahn 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 kahn AD, where m is the output dimensionality. Applying fmap snd to the result will recover the result of jacobian | An alias for gradF'

ghci> 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. f (AD s (Kahn a)) -> g (AD s (Kahn a))) -> f a -> g (f b) Source #

'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function f with kahn 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. f (AD s (Kahn a)) -> g (AD s (Kahn 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 kahn 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. f (AD s (On (Kahn (Kahn a)))) -> AD s (On (Kahn (Kahn a)))) -> f a -> f (f a) Source #

Compute the hessian via the jacobian of the gradient. gradient is computed in Kahn mode and then the jacobian is computed in Kahn 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. f (AD s (On (Kahn (Kahn a)))) -> g (AD s (On (Kahn (Kahn a))))) -> f a -> g (f (f a)) Source #

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

Less efficient than hessianF.

>>> hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2 :: RDouble]
[[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.131204383757,-2.471726672005],[-2.471726672005,1.131204383757]]]

Derivatives

diff :: Num a => (forall s. AD s (Kahn a) -> AD s (Kahn a)) -> a -> a Source #

Compute the derivative of a function.

>>> diff sin 0
1.0

>>> cos 0
1.0

diff' :: Num a => (forall s. AD s (Kahn a) -> AD s (Kahn a)) -> a -> (a, a) Source #

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

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

diffF :: (Functor f, Num a) => (forall s. AD s (Kahn a) -> f (AD s (Kahn a))) -> a -> f a Source #

Compute the derivatives of a function that returns a vector with regards to its single input.

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

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

Compute the derivatives of a function that returns a vector with regards to its single input as well as the primal answer.

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