Numeric.AD.Rank1.Kahn.Double

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.

Documentation

data KahnDouble

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

Enum KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

succ :: KahnDouble -> KahnDouble

pred :: KahnDouble -> KahnDouble

toEnum :: Int -> KahnDouble

fromEnum :: KahnDouble -> Int

enumFrom :: KahnDouble -> [KahnDouble]

enumFromThen :: KahnDouble -> KahnDouble -> [KahnDouble]

enumFromTo :: KahnDouble -> KahnDouble -> [KahnDouble]

enumFromThenTo :: KahnDouble -> KahnDouble -> KahnDouble -> [KahnDouble]

Eq KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

(==) :: KahnDouble -> KahnDouble -> Bool

(/=) :: KahnDouble -> KahnDouble -> Bool

Floating KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

pi :: KahnDouble

exp :: KahnDouble -> KahnDouble

log :: KahnDouble -> KahnDouble

sqrt :: KahnDouble -> KahnDouble

(**) :: KahnDouble -> KahnDouble -> KahnDouble

logBase :: KahnDouble -> KahnDouble -> KahnDouble

sin :: KahnDouble -> KahnDouble

cos :: KahnDouble -> KahnDouble

tan :: KahnDouble -> KahnDouble

asin :: KahnDouble -> KahnDouble

acos :: KahnDouble -> KahnDouble

atan :: KahnDouble -> KahnDouble

sinh :: KahnDouble -> KahnDouble

cosh :: KahnDouble -> KahnDouble

tanh :: KahnDouble -> KahnDouble

asinh :: KahnDouble -> KahnDouble

acosh :: KahnDouble -> KahnDouble

atanh :: KahnDouble -> KahnDouble

log1p :: KahnDouble -> KahnDouble

expm1 :: KahnDouble -> KahnDouble

log1pexp :: KahnDouble -> KahnDouble

log1mexp :: KahnDouble -> KahnDouble

Fractional KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

(/) :: KahnDouble -> KahnDouble -> KahnDouble

recip :: KahnDouble -> KahnDouble

fromRational :: Rational -> KahnDouble

Num KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

(+) :: KahnDouble -> KahnDouble -> KahnDouble

(-) :: KahnDouble -> KahnDouble -> KahnDouble

(*) :: KahnDouble -> KahnDouble -> KahnDouble

negate :: KahnDouble -> KahnDouble

abs :: KahnDouble -> KahnDouble

signum :: KahnDouble -> KahnDouble

fromInteger :: Integer -> KahnDouble

Ord KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

compare :: KahnDouble -> KahnDouble -> Ordering

(<) :: KahnDouble -> KahnDouble -> Bool

(<=) :: KahnDouble -> KahnDouble -> Bool

(>) :: KahnDouble -> KahnDouble -> Bool

(>=) :: KahnDouble -> KahnDouble -> Bool

max :: KahnDouble -> KahnDouble -> KahnDouble

min :: KahnDouble -> KahnDouble -> KahnDouble

Real KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

toRational :: KahnDouble -> Rational

RealFloat KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

floatRadix :: KahnDouble -> Integer

floatDigits :: KahnDouble -> Int

floatRange :: KahnDouble -> (Int, Int)

decodeFloat :: KahnDouble -> (Integer, Int)

encodeFloat :: Integer -> Int -> KahnDouble

exponent :: KahnDouble -> Int

significand :: KahnDouble -> KahnDouble

scaleFloat :: Int -> KahnDouble -> KahnDouble

isNaN :: KahnDouble -> Bool

isInfinite :: KahnDouble -> Bool

isDenormalized :: KahnDouble -> Bool

isNegativeZero :: KahnDouble -> Bool

isIEEE :: KahnDouble -> Bool

atan2 :: KahnDouble -> KahnDouble -> KahnDouble

RealFrac KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

properFraction :: Integral b => KahnDouble -> (b, KahnDouble)

truncate :: Integral b => KahnDouble -> b

round :: Integral b => KahnDouble -> b

ceiling :: Integral b => KahnDouble -> b

floor :: Integral b => KahnDouble -> b

Show KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

showsPrec :: Int -> KahnDouble -> ShowS

show :: KahnDouble -> String

showList :: [KahnDouble] -> ShowS

MuRef KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Associated Types

type DeRef KahnDouble :: Type -> Type

Methods

mapDeRef :: Applicative f => (forall b. (MuRef b, DeRef KahnDouble ~ DeRef b) => b -> f u) -> KahnDouble -> f (DeRef KahnDouble u)

Erf KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

erf :: KahnDouble -> KahnDouble

erfc :: KahnDouble -> KahnDouble

erfcx :: KahnDouble -> KahnDouble

normcdf :: KahnDouble -> KahnDouble

InvErf KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Methods

inverf :: KahnDouble -> KahnDouble

inverfc :: KahnDouble -> KahnDouble

invnormcdf :: KahnDouble -> KahnDouble

Mode KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Associated Types

type Scalar KahnDouble

Methods

isKnownConstant :: KahnDouble -> Bool

asKnownConstant :: KahnDouble -> Maybe (Scalar KahnDouble)

isKnownZero :: KahnDouble -> Bool

auto :: Scalar KahnDouble -> KahnDouble

(*^) :: Scalar KahnDouble -> KahnDouble -> KahnDouble

(^*) :: KahnDouble -> Scalar KahnDouble -> KahnDouble

(^/) :: KahnDouble -> Scalar KahnDouble -> KahnDouble

zero :: KahnDouble

Jacobian KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

Associated Types

type D KahnDouble

Methods

unary :: (Scalar KahnDouble -> Scalar KahnDouble) -> D KahnDouble -> KahnDouble -> KahnDouble

lift1 :: (Scalar KahnDouble -> Scalar KahnDouble) -> (D KahnDouble -> D KahnDouble) -> KahnDouble -> KahnDouble

lift1_ :: (Scalar KahnDouble -> Scalar KahnDouble) -> (D KahnDouble -> D KahnDouble -> D KahnDouble) -> KahnDouble -> KahnDouble

binary :: (Scalar KahnDouble -> Scalar KahnDouble -> Scalar KahnDouble) -> D KahnDouble -> D KahnDouble -> KahnDouble -> KahnDouble -> KahnDouble

lift2 :: (Scalar KahnDouble -> Scalar KahnDouble -> Scalar KahnDouble) -> (D KahnDouble -> D KahnDouble -> (D KahnDouble, D KahnDouble)) -> KahnDouble -> KahnDouble -> KahnDouble

lift2_ :: (Scalar KahnDouble -> Scalar KahnDouble -> Scalar KahnDouble) -> (D KahnDouble -> D KahnDouble -> D KahnDouble -> (D KahnDouble, D KahnDouble)) -> KahnDouble -> KahnDouble -> KahnDouble

Grad i o o' => Grad (KahnDouble -> i) (Double -> o) (Double -> o')

Defined in Numeric.AD.Internal.Kahn.Double

Methods

pack :: (KahnDouble -> i) -> [KahnDouble] -> KahnDouble

unpack :: (List -> List) -> Double -> o

unpack' :: (List -> (Double, List)) -> Double -> o'

type DeRef KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

type DeRef KahnDouble = Tape

type Scalar KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

type Scalar KahnDouble = Double

type D KahnDouble

Defined in Numeric.AD.Internal.Kahn.Double

type D KahnDouble = Id Double

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

Embed a constant

Gradient

grad :: Traversable f => (f KahnDouble -> KahnDouble) -> f Double -> f Double

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) [1.0,2.0,3.0]
[2.0,1.0,1.0]

grad' :: Traversable f => (f KahnDouble -> KahnDouble) -> f Double -> (Double, f Double)

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] -> x*y+z) [1.0,2.0,3.0]
(5.0,[2.0,1.0,1.0])

gradWith :: Traversable f => (Double -> Double -> b) -> (f KahnDouble -> KahnDouble) -> f Double -> f b

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 => (Double -> Double -> b) -> (f KahnDouble -> KahnDouble) -> f Double -> (Double, f b)

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) => (f KahnDouble -> g KahnDouble) -> f Double -> g (f Double)

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]) [2.0,1.0]
[[0.0,1.0],[1.0,0.0],[1.0,2.0]]

jacobian' :: (Traversable f, Functor g) => (f KahnDouble -> g KahnDouble) -> f Double -> g (Double, f Double)

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]) [2.0,1.0] [(1.0,[0.0,1.0]),(2.0,[1.0,0.0]),(2.0,[1.0,2.0])]

jacobianWith :: (Traversable f, Functor g) => (Double -> Double -> b) -> (f KahnDouble -> g KahnDouble) -> f Double -> g (f b)

'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) => (Double -> Double -> b) -> (f KahnDouble -> g KahnDouble) -> f Double -> g (Double, f b)

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 => (f (On (Kahn KahnDouble)) -> On (Kahn KahnDouble)) -> f Double -> f (f Double)

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) [1.0,2.0]
[[0.0,1.0],[1.0,0.0]]

hessianF :: (Traversable f, Functor g) => (f (On (Kahn KahnDouble)) -> g (On (Kahn KahnDouble))) -> f Double -> g (f (f Double))

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.

Derivatives

diff :: (KahnDouble -> KahnDouble) -> Double -> Double

Compute the derivative of a function.

>>> diff sin 0
1.0

>>> cos 0
1.0

diff' :: (KahnDouble -> KahnDouble) -> Double -> (Double, Double)

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 => (KahnDouble -> f KahnDouble) -> Double -> f Double

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 => (KahnDouble -> f KahnDouble) -> Double -> f (Double, Double)

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)]

Unsafe Variadic Gradient

Variadic combinators for variadic mixed-mode automatic differentiation.

Unfortunately, variadicity comes at the expense of being able to use quantification to avoid sensitivity confusion, so be careful when counting the number of auto calls you use when taking the gradient of a function that takes gradients!

vgrad :: Grad i o o' => i -> o

vgrad' :: Grad i o o' => i -> o'

class Grad i o o' | i -> o o', o -> i o', o' -> i o

Minimal complete definition

pack, unpack, unpack'

Instances

Grad i o o' => Grad (KahnDouble -> i) (Double -> o) (Double -> o')

Defined in Numeric.AD.Internal.Kahn.Double

Methods

pack :: (KahnDouble -> i) -> [KahnDouble] -> KahnDouble

unpack :: (List -> List) -> Double -> o

unpack' :: (List -> (Double, List)) -> Double -> o'