Numeric.AD

Mixed-Mode Automatic Differentiation.

Each combinator exported from this module chooses an appropriate AD mode. The following basic operations are supported, modified as appropriate by the suffixes below:

• grad computes the gradient (partial derivatives) of a function at a point
• jacobian computes the Jacobian matrix of a function at a point
• diff computes the derivative of a function at a point
• du computes a directional derivative of a function at a point
• hessian compute the Hessian matrix (matrix of second partial derivatives) of a function at a point

The suffixes have the following meanings:

• ' -- also return the answer
• With lets the user supply a function to blend the input with the output
• F is a version of the base function lifted to return a Traversable (or Functor) result
• s means the function returns all higher derivatives in a list or f-branching Stream
• T means the result is transposed with respect to the traditional formulation.
• 0 means that the resulting derivative list is padded with 0s at the end.

Documentation

data AD s a

Instances

Bounded a => Bounded (AD s a)

Methods

minBound :: AD s a

maxBound :: AD s a

Enum a => Enum (AD s a)

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)

Methods

(==) :: AD s a -> AD s a -> Bool

(/=) :: AD s a -> AD s a -> Bool

Floating a => Floating (AD s a)

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)

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)

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)

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)

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)

Methods

toRational :: AD s a -> Rational

RealFloat a => RealFloat (AD s a)

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)

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)

Methods

showsPrec :: Int -> AD s a -> ShowS

show :: AD s a -> String

showList :: [AD s a] -> ShowS

Erf a => Erf (AD s a)

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)

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)

Associated Types

type Scalar (AD s a) :: *

Methods

isKnownConstant :: AD s a -> Bool

isKnownZero :: AD s a -> Bool

auto :: Scalar (AD s a) -> AD s a

(*^) :: Scalar (AD s a) -> AD s a -> AD s a

(^*) :: AD s a -> Scalar (AD s a) -> AD s a

(^/) :: AD s a -> Scalar (AD s a) -> AD s a

zero :: AD s a

type Scalar (AD s a)

type Scalar (AD s a) = Scalar a

AD modes

class (Num t, Num (Scalar t)) => Mode t where

Minimal complete definition

auto

Associated Types

type Scalar t

Methods

auto :: Scalar t -> t

Embed a constant

Instances

Mode Double

Associated Types

type Scalar Double :: *

Methods

isKnownConstant :: Double -> Bool

isKnownZero :: Double -> Bool

auto :: Scalar Double -> Double

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

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

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

zero :: Double

Mode Float

Associated Types

type Scalar Float :: *

Methods

isKnownConstant :: Float -> Bool

isKnownZero :: Float -> Bool

auto :: Scalar Float -> Float

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

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

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

zero :: Float

Mode Int

Associated Types

type Scalar Int :: *

Methods

isKnownConstant :: Int -> Bool

isKnownZero :: Int -> Bool

auto :: Scalar Int -> Int

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

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

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

zero :: Int

Mode Int8

Associated Types

type Scalar Int8 :: *

Methods

isKnownConstant :: Int8 -> Bool

isKnownZero :: Int8 -> Bool

auto :: Scalar Int8 -> Int8

(*^) :: Scalar Int8 -> Int8 -> Int8

(^*) :: Int8 -> Scalar Int8 -> Int8

(^/) :: Int8 -> Scalar Int8 -> Int8

zero :: Int8

Mode Int16

Associated Types

type Scalar Int16 :: *

Methods

isKnownConstant :: Int16 -> Bool

isKnownZero :: Int16 -> Bool

auto :: Scalar Int16 -> Int16

(*^) :: Scalar Int16 -> Int16 -> Int16

(^*) :: Int16 -> Scalar Int16 -> Int16

(^/) :: Int16 -> Scalar Int16 -> Int16

zero :: Int16

Mode Int32

Associated Types

type Scalar Int32 :: *

Methods

isKnownConstant :: Int32 -> Bool

isKnownZero :: Int32 -> Bool

auto :: Scalar Int32 -> Int32

(*^) :: Scalar Int32 -> Int32 -> Int32

(^*) :: Int32 -> Scalar Int32 -> Int32

(^/) :: Int32 -> Scalar Int32 -> Int32

zero :: Int32

Mode Int64

Associated Types

type Scalar Int64 :: *

Methods

isKnownConstant :: Int64 -> Bool

isKnownZero :: Int64 -> Bool

auto :: Scalar Int64 -> Int64

(*^) :: Scalar Int64 -> Int64 -> Int64

(^*) :: Int64 -> Scalar Int64 -> Int64

(^/) :: Int64 -> Scalar Int64 -> Int64

zero :: Int64

Mode Integer

Associated Types

type Scalar Integer :: *

Methods

isKnownConstant :: Integer -> Bool

isKnownZero :: Integer -> Bool

auto :: Scalar Integer -> Integer

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

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

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

zero :: Integer

Mode Natural

Associated Types

type Scalar Natural :: *

Methods

isKnownConstant :: Natural -> Bool

isKnownZero :: Natural -> Bool

auto :: Scalar Natural -> Natural

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

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

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

zero :: Natural

Mode Word

Associated Types

type Scalar Word :: *

Methods

isKnownConstant :: Word -> Bool

isKnownZero :: Word -> Bool

auto :: Scalar Word -> Word

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

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

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

zero :: Word

Mode Word8

Associated Types

type Scalar Word8 :: *

Methods

isKnownConstant :: Word8 -> Bool

isKnownZero :: Word8 -> Bool

auto :: Scalar Word8 -> Word8

(*^) :: Scalar Word8 -> Word8 -> Word8

(^*) :: Word8 -> Scalar Word8 -> Word8

(^/) :: Word8 -> Scalar Word8 -> Word8

zero :: Word8

Mode Word16

Associated Types

type Scalar Word16 :: *

Methods

isKnownConstant :: Word16 -> Bool

isKnownZero :: Word16 -> Bool

auto :: Scalar Word16 -> Word16

(*^) :: Scalar Word16 -> Word16 -> Word16

(^*) :: Word16 -> Scalar Word16 -> Word16

(^/) :: Word16 -> Scalar Word16 -> Word16

zero :: Word16

Mode Word32

Associated Types

type Scalar Word32 :: *

Methods

isKnownConstant :: Word32 -> Bool

isKnownZero :: Word32 -> Bool

auto :: Scalar Word32 -> Word32

(*^) :: Scalar Word32 -> Word32 -> Word32

(^*) :: Word32 -> Scalar Word32 -> Word32

(^/) :: Word32 -> Scalar Word32 -> Word32

zero :: Word32

Mode Word64

Associated Types

type Scalar Word64 :: *

Methods

isKnownConstant :: Word64 -> Bool

isKnownZero :: Word64 -> Bool

auto :: Scalar Word64 -> Word64

(*^) :: Scalar Word64 -> Word64 -> Word64

(^*) :: Word64 -> Scalar Word64 -> Word64

(^/) :: Word64 -> Scalar Word64 -> Word64

zero :: Word64

Mode ForwardDouble

Associated Types

type Scalar ForwardDouble :: *

Methods

isKnownConstant :: ForwardDouble -> Bool

isKnownZero :: ForwardDouble -> Bool

auto :: Scalar ForwardDouble -> ForwardDouble

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

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

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

zero :: ForwardDouble

Integral a => Mode (Ratio a)

Associated Types

type Scalar (Ratio a) :: *

Methods

isKnownConstant :: Ratio a -> Bool

isKnownZero :: Ratio a -> Bool

auto :: Scalar (Ratio a) -> Ratio a

(*^) :: Scalar (Ratio a) -> Ratio a -> Ratio a

(^*) :: Ratio a -> Scalar (Ratio a) -> Ratio a

(^/) :: Ratio a -> Scalar (Ratio a) -> Ratio a

zero :: Ratio a

RealFloat a => Mode (Complex a)

Associated Types

type Scalar (Complex a) :: *

Methods

isKnownConstant :: Complex a -> Bool

isKnownZero :: Complex a -> Bool

auto :: Scalar (Complex a) -> Complex a

(*^) :: Scalar (Complex a) -> Complex a -> Complex a

(^*) :: Complex a -> Scalar (Complex a) -> Complex a

(^/) :: Complex a -> Scalar (Complex a) -> Complex a

zero :: Complex a

(Mode t, Mode (Scalar t)) => Mode (On t)

Associated Types

type Scalar (On t) :: *

Methods

isKnownConstant :: On t -> Bool

isKnownZero :: On t -> Bool

auto :: Scalar (On t) -> On t

(*^) :: Scalar (On t) -> On t -> On t

(^*) :: On t -> Scalar (On t) -> On t

(^/) :: On t -> Scalar (On t) -> On t

zero :: On t

Num a => Mode (Id a)

Associated Types

type Scalar (Id a) :: *

Methods

isKnownConstant :: Id a -> Bool

isKnownZero :: Id a -> Bool

auto :: Scalar (Id a) -> Id a

(*^) :: Scalar (Id a) -> Id a -> Id a

(^*) :: Id a -> Scalar (Id a) -> Id a

(^/) :: Id a -> Scalar (Id a) -> Id a

zero :: Id a

Num a => Mode (Tower a)

Associated Types

type Scalar (Tower a) :: *

Methods

isKnownConstant :: Tower a -> Bool

isKnownZero :: Tower a -> Bool

auto :: Scalar (Tower a) -> Tower a

(*^) :: Scalar (Tower a) -> Tower a -> Tower a

(^*) :: Tower a -> Scalar (Tower a) -> Tower a

(^/) :: Tower a -> Scalar (Tower a) -> Tower a

zero :: Tower a

Num a => Mode (Sparse a)

Associated Types

type Scalar (Sparse a) :: *

Methods

isKnownConstant :: Sparse a -> Bool

isKnownZero :: Sparse a -> Bool

auto :: Scalar (Sparse a) -> Sparse a

(*^) :: Scalar (Sparse a) -> Sparse a -> Sparse a

(^*) :: Sparse a -> Scalar (Sparse a) -> Sparse a

(^/) :: Sparse a -> Scalar (Sparse a) -> Sparse a

zero :: Sparse a

Num a => Mode (Kahn a)

Associated Types

type Scalar (Kahn a) :: *

Methods

isKnownConstant :: Kahn a -> Bool

isKnownZero :: Kahn a -> Bool

auto :: Scalar (Kahn a) -> Kahn a

(*^) :: Scalar (Kahn a) -> Kahn a -> Kahn a

(^*) :: Kahn a -> Scalar (Kahn a) -> Kahn a

(^/) :: Kahn a -> Scalar (Kahn a) -> Kahn a

zero :: Kahn a

Num a => Mode (Forward a)

Associated Types

type Scalar (Forward a) :: *

Methods

isKnownConstant :: Forward a -> Bool

isKnownZero :: Forward a -> Bool

auto :: Scalar (Forward a) -> Forward a

(*^) :: Scalar (Forward a) -> Forward a -> Forward a

(^*) :: Forward a -> Scalar (Forward a) -> Forward a

(^/) :: Forward a -> Scalar (Forward a) -> Forward a

zero :: Forward a

Mode a => Mode (AD s a)

Associated Types

type Scalar (AD s a) :: *

Methods

isKnownConstant :: AD s a -> Bool

isKnownZero :: AD s a -> Bool

auto :: Scalar (AD s a) -> AD s a

(*^) :: Scalar (AD s a) -> AD s a -> AD s a

(^*) :: AD s a -> Scalar (AD s a) -> AD s a

(^/) :: AD s a -> Scalar (AD s a) -> AD s a

zero :: AD s a

(Reifies * s Tape, Num a) => Mode (Reverse s a)

Associated Types

type Scalar (Reverse s a) :: *

Methods

isKnownConstant :: Reverse s a -> Bool

isKnownZero :: Reverse s a -> Bool

auto :: Scalar (Reverse s a) -> Reverse s a

(*^) :: Scalar (Reverse s a) -> Reverse s a -> Reverse s a

(^*) :: Reverse s a -> Scalar (Reverse s a) -> Reverse s a

(^/) :: Reverse s a -> Scalar (Reverse s a) -> Reverse s a

zero :: Reverse s a

(Num a, Traversable f) => Mode (Dense f a)

Associated Types

type Scalar (Dense f a) :: *

Methods

isKnownConstant :: Dense f a -> Bool

isKnownZero :: Dense f a -> Bool

auto :: Scalar (Dense f a) -> Dense f a

(*^) :: Scalar (Dense f a) -> Dense f a -> Dense f a

(^*) :: Dense f a -> Scalar (Dense f a) -> Dense f a

(^/) :: Dense f a -> Scalar (Dense f a) -> Dense f a

zero :: Dense f a

(Mode a, Mode b, Chosen s, (~) * (Scalar a) (Scalar b)) => Mode (Or s a b)

Associated Types

type Scalar (Or s a b) :: *

Methods

isKnownConstant :: Or s a b -> Bool

isKnownZero :: Or s a b -> Bool

auto :: Scalar (Or s a b) -> Or s a b

(*^) :: Scalar (Or s a b) -> Or s a b -> Or s a b

(^*) :: Or s a b -> Scalar (Or s a b) -> Or s a b

(^/) :: Or s a b -> Scalar (Or s a b) -> Or s a b

zero :: Or s a b

Gradients (Reverse Mode)

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

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

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

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

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)

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)

Higher Order Gradients (Sparse-on-Reverse)

grads :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> Cofree f a

Variadic Gradients (Sparse or Kahn)

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!

class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o

Minimal complete definition

pack, unpack, unpack'

Instances

Num a => Grad (Kahn a) [a] (a, [a]) a

Methods

pack :: Kahn a -> [Kahn a] -> Kahn a

unpack :: ([a] -> [a]) -> [a]

unpack' :: ([a] -> (a, [a])) -> (a, [a])

Grad i o o' a => Grad (Kahn a -> i) (a -> o) (a -> o') a

Methods

pack :: (Kahn a -> i) -> [Kahn a] -> Kahn a

unpack :: ([a] -> [a]) -> a -> o

unpack' :: ([a] -> (a, [a])) -> a -> o'

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

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

class Num a => Grads i o a | i -> a o, o -> a i

Minimal complete definition

packs, unpacks

Instances

Num a => Grads (Sparse a) (Cofree [] a) a

Methods

packs :: Sparse a -> [Sparse a] -> Sparse a

unpacks :: ([a] -> Cofree [] a) -> Cofree [] a

Grads i o a => Grads (Sparse a -> i) (a -> o) a

Methods

packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a

unpacks :: ([a] -> Cofree [] a) -> a -> o

vgrads :: Grads i o a => i -> o

Jacobians (Sparse or Reverse)

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)

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)

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)

'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)

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)

Higher Order Jacobian (Sparse-on-Reverse)

jacobians :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (Cofree f a)

Transposed Jacobians (Forward Mode)

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

A fast, simple, transposed Jacobian computed with Forward mode AD that combines the output with the input.

Hessian (Sparse-On-Reverse)

hessian :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (On (Reverse s (Sparse a))) -> On (Reverse s (Sparse 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 sparse (forward) mode.

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

hessian' :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> (a, f (a, f a))

Hessian Tensors (Sparse or Sparse-On-Reverse)

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

Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function using 'Sparse'-on-'Reverse'

>>> 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]]]

Hessian Tensors (Sparse)

hessianF' :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (a, f (a, f a))

Hessian Vector Products (Forward-On-Reverse)

hessianProduct :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (On (Reverse s (Forward a))) -> On (Reverse s (Forward 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. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.

hessianProduct' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (On (Reverse s (Forward a))) -> On (Reverse s (Forward 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 return the gradient and the directional derivative of the gradient. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.

Derivatives (Forward Mode)

diff :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> a

The diff function calculates the first derivative of a scalar-to-scalar function by forward-mode AD

>>> diff sin 0
1.0

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

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

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

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

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

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

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

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

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

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

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

Derivatives (Tower)

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

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

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

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

Directional Derivatives (Forward Mode)

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

Compute the directional derivative of a function given a zipped up Functor of the input values and their derivatives

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

Compute the answer and directional derivative of a function given a zipped up Functor of the input values and their derivatives

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

Compute a vector of directional derivatives for a function given a zipped up Functor of the input values and their derivatives.

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

Compute a vector of answers and directional derivatives for a function given a zipped up Functor of the input values and their derivatives.

Directional Derivatives (Tower)

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

dus0 :: (Functor f, Num a) => (forall s. f (AD s (Tower a)) -> AD s (Tower a)) -> f [a] -> [a]

dusF :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Tower a)) -> g (AD s (Tower a))) -> f [a] -> g [a]

dus0F :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Tower a)) -> g (AD s (Tower a))) -> f [a] -> g [a]

Taylor Series (Tower)

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

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

Maclaurin Series (Tower)

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

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

Gradient Descent

gradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> [f a]

The gradientDescent function performs a multivariate optimization, based on the naive-gradient-descent in the file stalingrad/examples/flow-tests/pre-saddle-1a.vlad from the VLAD compiler Stalingrad sources. Its output is a stream of increasingly accurate results. (Modulo the usual caveats.)

It uses reverse mode automatic differentiation to compute the gradient.

gradientAscent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> [f a]

Perform a gradient descent using reverse mode automatic differentiation to compute the gradient.

conjugateGradientDescent :: (Traversable f, Ord a, Fractional a) => (forall s. Chosen s => f (Or s (On (Forward (Forward a))) (Kahn a)) -> Or s (On (Forward (Forward a))) (Kahn a)) -> f a -> [f a]

Perform a conjugate gradient descent using reverse mode automatic differentiation to compute the gradient, and using forward-on-forward mode for computing extrema.

>>> let sq x = x * x
>>> let rosenbrock [x,y] = sq (1 - x) + 100 * sq (y - sq x)
>>> rosenbrock [0,0]
1
>>> rosenbrock (conjugateGradientDescent rosenbrock [0, 0] !! 5) < 0.1
True

conjugateGradientAscent :: (Traversable f, Ord a, Fractional a) => (forall s. Chosen s => f (Or s (On (Forward (Forward a))) (Kahn a)) -> Or s (On (Forward (Forward a))) (Kahn a)) -> f a -> [f a]

Perform a conjugate gradient ascent using reverse mode automatic differentiation to compute the gradient.

stochasticGradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Scalar a) -> f (Reverse s a) -> Reverse s a) -> [f (Scalar a)] -> f a -> [f a]

The stochasticGradientDescent function approximates the true gradient of the constFunction by a gradient at a single example. As the algorithm sweeps through the training set, it performs the update for each training example.

It uses reverse mode automatic differentiation to compute the gradient The learning rate is constant through out, and is set to 0.001