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

Numeric.AD

Contents

AD modes
Gradients (Reverse Mode)
Higher Order Gradients (Sparse-on-Reverse)
Variadic Gradients (Sparse or Kahn)
Jacobians (Sparse or Reverse)
Higher Order Jacobian (Sparse-on-Reverse)
Transposed Jacobians (Forward Mode)
Hessian (Sparse-On-Reverse)
Hessian Tensors (Sparse or Sparse-On-Reverse)
Hessian Tensors (Sparse)
Hessian Vector Products (Forward-On-Reverse)
Derivatives (Forward Mode)
Derivatives (Tower)
Directional Derivatives (Forward Mode)
Directional Derivatives (Tower)
Taylor Series (Tower)
Maclaurin Series (Tower)
Gradient Descent

Description

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.

Synopsis

data AD s a
class (Num t, Num (Scalar t)) => Mode t where
- auto :: Scalar t -> t
type family Scalar t
grad :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f a
grad' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f a)
gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f b
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)
grads :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> Cofree f a
class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i 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
vgrads :: Grads i o a => i -> o
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)
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)
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' :: (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)
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)
jacobianT :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g a)
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)
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)
hessian' :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> (a, f (a, f a))
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))
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))
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' :: (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)
diff :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> a
diffF :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f a
diff' :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> (a, a)
diffF' :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f (a, a)
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]
du :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> a
du' :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> (a, a)
duF :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g a
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)
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 :: 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 :: 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]
gradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> [f a]
gradientAscent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> [f a]
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]
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]
stochasticGradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => e -> f (Reverse s a) -> Reverse s a) -> [e] -> f a -> [f 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

AD modes

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

Methods

auto :: Scalar t -> t Source #

Embed a constant

Instances

Mode Double Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Double :: Type Source # Methods isKnownConstant :: Double -> Bool Source # isKnownZero :: Double -> Bool Source # auto :: Scalar Double -> Double Source # (^) :: Scalar Double -> Double -> Double Source # (^) :: Double -> Scalar Double -> Double Source # (^/) :: Double -> Scalar Double -> Double Source # zero :: Double Source #
Mode Float Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Float :: Type Source # Methods isKnownConstant :: Float -> Bool Source # isKnownZero :: Float -> Bool Source # auto :: Scalar Float -> Float Source # (^) :: Scalar Float -> Float -> Float Source # (^) :: Float -> Scalar Float -> Float Source # (^/) :: Float -> Scalar Float -> Float Source # zero :: Float Source #
Mode Int Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Int :: Type Source # Methods isKnownConstant :: Int -> Bool Source # isKnownZero :: Int -> Bool Source # auto :: Scalar Int -> Int Source # (^) :: Scalar Int -> Int -> Int Source # (^) :: Int -> Scalar Int -> Int Source # (^/) :: Int -> Scalar Int -> Int Source # zero :: Int Source #
Mode Int8 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Int8 :: Type Source # Methods isKnownConstant :: Int8 -> Bool Source # isKnownZero :: Int8 -> Bool Source # auto :: Scalar Int8 -> Int8 Source # (^) :: Scalar Int8 -> Int8 -> Int8 Source # (^) :: Int8 -> Scalar Int8 -> Int8 Source # (^/) :: Int8 -> Scalar Int8 -> Int8 Source # zero :: Int8 Source #
Mode Int16 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Int16 :: Type Source # Methods isKnownConstant :: Int16 -> Bool Source # isKnownZero :: Int16 -> Bool Source # auto :: Scalar Int16 -> Int16 Source # (^) :: Scalar Int16 -> Int16 -> Int16 Source # (^) :: Int16 -> Scalar Int16 -> Int16 Source # (^/) :: Int16 -> Scalar Int16 -> Int16 Source # zero :: Int16 Source #
Mode Int32 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Int32 :: Type Source # Methods isKnownConstant :: Int32 -> Bool Source # isKnownZero :: Int32 -> Bool Source # auto :: Scalar Int32 -> Int32 Source # (^) :: Scalar Int32 -> Int32 -> Int32 Source # (^) :: Int32 -> Scalar Int32 -> Int32 Source # (^/) :: Int32 -> Scalar Int32 -> Int32 Source # zero :: Int32 Source #
Mode Int64 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Int64 :: Type Source # Methods isKnownConstant :: Int64 -> Bool Source # isKnownZero :: Int64 -> Bool Source # auto :: Scalar Int64 -> Int64 Source # (^) :: Scalar Int64 -> Int64 -> Int64 Source # (^) :: Int64 -> Scalar Int64 -> Int64 Source # (^/) :: Int64 -> Scalar Int64 -> Int64 Source # zero :: Int64 Source #
Mode Integer Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Integer :: Type Source # Methods isKnownConstant :: Integer -> Bool Source # isKnownZero :: Integer -> Bool Source # auto :: Scalar Integer -> Integer Source # (^) :: Scalar Integer -> Integer -> Integer Source # (^) :: Integer -> Scalar Integer -> Integer Source # (^/) :: Integer -> Scalar Integer -> Integer Source # zero :: Integer Source #
Mode Natural Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Natural :: Type Source # Methods isKnownConstant :: Natural -> Bool Source # isKnownZero :: Natural -> Bool Source # auto :: Scalar Natural -> Natural Source # (^) :: Scalar Natural -> Natural -> Natural Source # (^) :: Natural -> Scalar Natural -> Natural Source # (^/) :: Natural -> Scalar Natural -> Natural Source # zero :: Natural Source #
Mode Word Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Word :: Type Source # Methods isKnownConstant :: Word -> Bool Source # isKnownZero :: Word -> Bool Source # auto :: Scalar Word -> Word Source # (^) :: Scalar Word -> Word -> Word Source # (^) :: Word -> Scalar Word -> Word Source # (^/) :: Word -> Scalar Word -> Word Source # zero :: Word Source #
Mode Word8 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Word8 :: Type Source # Methods isKnownConstant :: Word8 -> Bool Source # isKnownZero :: Word8 -> Bool Source # auto :: Scalar Word8 -> Word8 Source # (^) :: Scalar Word8 -> Word8 -> Word8 Source # (^) :: Word8 -> Scalar Word8 -> Word8 Source # (^/) :: Word8 -> Scalar Word8 -> Word8 Source # zero :: Word8 Source #
Mode Word16 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Word16 :: Type Source # Methods isKnownConstant :: Word16 -> Bool Source # isKnownZero :: Word16 -> Bool Source # auto :: Scalar Word16 -> Word16 Source # (^) :: Scalar Word16 -> Word16 -> Word16 Source # (^) :: Word16 -> Scalar Word16 -> Word16 Source # (^/) :: Word16 -> Scalar Word16 -> Word16 Source # zero :: Word16 Source #
Mode Word32 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Word32 :: Type Source # Methods isKnownConstant :: Word32 -> Bool Source # isKnownZero :: Word32 -> Bool Source # auto :: Scalar Word32 -> Word32 Source # (^) :: Scalar Word32 -> Word32 -> Word32 Source # (^) :: Word32 -> Scalar Word32 -> Word32 Source # (^/) :: Word32 -> Scalar Word32 -> Word32 Source # zero :: Word32 Source #
Mode Word64 Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar Word64 :: Type Source # Methods isKnownConstant :: Word64 -> Bool Source # isKnownZero :: Word64 -> Bool Source # auto :: Scalar Word64 -> Word64 Source # (^) :: Scalar Word64 -> Word64 -> Word64 Source # (^) :: Word64 -> Scalar Word64 -> Word64 Source # (^/) :: Word64 -> Scalar Word64 -> Word64 Source # zero :: Word64 Source #
Mode ForwardDouble Source #
Instance details Defined in Numeric.AD.Internal.Forward.Double Associated Types type Scalar ForwardDouble :: Type Source # Methods isKnownConstant :: ForwardDouble -> Bool Source # isKnownZero :: ForwardDouble -> Bool Source # auto :: Scalar ForwardDouble -> ForwardDouble Source # (^) :: Scalar ForwardDouble -> ForwardDouble -> ForwardDouble Source # (^) :: ForwardDouble -> Scalar ForwardDouble -> ForwardDouble Source # (^/) :: ForwardDouble -> Scalar ForwardDouble -> ForwardDouble Source # zero :: ForwardDouble Source #
Integral a => Mode (Ratio a) Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar (Ratio a) :: Type Source # Methods isKnownConstant :: Ratio a -> Bool Source # isKnownZero :: Ratio a -> Bool Source # auto :: Scalar (Ratio a) -> Ratio a Source # (^) :: Scalar (Ratio a) -> Ratio a -> Ratio a Source # (^) :: Ratio a -> Scalar (Ratio a) -> Ratio a Source # (^/) :: Ratio a -> Scalar (Ratio a) -> Ratio a Source # zero :: Ratio a Source #
RealFloat a => Mode (Complex a) Source #
Instance details Defined in Numeric.AD.Mode Associated Types type Scalar (Complex a) :: Type Source # Methods isKnownConstant :: Complex a -> Bool Source # isKnownZero :: Complex a -> Bool Source # auto :: Scalar (Complex a) -> Complex a Source # (^) :: Scalar (Complex a) -> Complex a -> Complex a Source # (^) :: Complex a -> Scalar (Complex a) -> Complex a Source # (^/) :: Complex a -> Scalar (Complex a) -> Complex a Source # zero :: Complex a Source #
(Mode t, Mode (Scalar t)) => Mode (On t) Source #
Instance details Defined in Numeric.AD.Internal.On Associated Types type Scalar (On t) :: Type Source # Methods isKnownConstant :: On t -> Bool Source # isKnownZero :: On t -> Bool Source # auto :: Scalar (On t) -> On t Source # (^) :: Scalar (On t) -> On t -> On t Source # (^) :: On t -> Scalar (On t) -> On t Source # (^/) :: On t -> Scalar (On t) -> On t Source # zero :: On t Source #
Num a => Mode (Id a) Source #
Instance details Defined in Numeric.AD.Internal.Identity Associated Types type Scalar (Id a) :: Type Source # Methods isKnownConstant :: Id a -> Bool Source # isKnownZero :: Id a -> Bool Source # auto :: Scalar (Id a) -> Id a Source # (^) :: Scalar (Id a) -> Id a -> Id a Source # (^) :: Id a -> Scalar (Id a) -> Id a Source # (^/) :: Id a -> Scalar (Id a) -> Id a Source # zero :: Id a Source #
Num a => Mode (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower Associated Types type Scalar (Tower a) :: Type Source # Methods isKnownConstant :: Tower a -> Bool Source # isKnownZero :: Tower a -> Bool Source # auto :: Scalar (Tower a) -> Tower a Source # (^) :: Scalar (Tower a) -> Tower a -> Tower a Source # (^) :: Tower a -> Scalar (Tower a) -> Tower a Source # (^/) :: Tower a -> Scalar (Tower a) -> Tower a Source # zero :: Tower a Source #
Num a => Mode (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Associated Types type Scalar (Sparse a) :: Type Source # Methods isKnownConstant :: Sparse a -> Bool Source # isKnownZero :: Sparse a -> Bool Source # auto :: Scalar (Sparse a) -> Sparse a Source # (^) :: Scalar (Sparse a) -> Sparse a -> Sparse a Source # (^) :: Sparse a -> Scalar (Sparse a) -> Sparse a Source # (^/) :: Sparse a -> Scalar (Sparse a) -> Sparse a Source # zero :: Sparse a Source #
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 => Mode (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward Associated Types type Scalar (Forward a) :: Type Source # Methods isKnownConstant :: Forward a -> Bool Source # isKnownZero :: Forward a -> Bool Source # auto :: Scalar (Forward a) -> Forward a Source # (^) :: Scalar (Forward a) -> Forward a -> Forward a Source # (^) :: Forward a -> Scalar (Forward a) -> Forward a Source # (^/) :: Forward a -> Scalar (Forward a) -> Forward a Source # zero :: Forward a Source #
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 #
(Reifies s Tape, Num a) => Mode (Reverse s a) Source #
Instance details Defined in Numeric.AD.Internal.Reverse Associated Types type Scalar (Reverse s a) :: Type 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 #
(Num a, Traversable f) => Mode (Dense f a) Source #
Instance details Defined in Numeric.AD.Internal.Dense Associated Types type Scalar (Dense f a) :: Type Source # Methods isKnownConstant :: Dense f a -> Bool Source # isKnownZero :: Dense f a -> Bool Source # auto :: Scalar (Dense f a) -> Dense f a Source # (^) :: Scalar (Dense f a) -> Dense f a -> Dense f a Source # (^) :: Dense f a -> Scalar (Dense f a) -> Dense f a Source # (^/) :: Dense f a -> Scalar (Dense f a) -> Dense f a Source # zero :: Dense f a Source #
(Mode a, Mode b, Chosen s, Scalar a ~ Scalar b) => Mode (Or s a b) Source #
Instance details Defined in Numeric.AD.Internal.Or Associated Types type Scalar (Or s a b) :: Type Source # Methods isKnownConstant :: Or s a b -> Bool Source # isKnownZero :: Or s a b -> Bool Source # auto :: Scalar (Or s a b) -> Or s a b Source # (^) :: Scalar (Or s a b) -> Or s a b -> Or s a b Source # (^) :: Or s a b -> Scalar (Or s a b) -> Or s a b Source # (^/) :: Or s a b -> Scalar (Or s a b) -> Or s a b Source # zero :: Or s a b Source #

type family Scalar t Source #

Instances

type Scalar Double Source #
Instance details Defined in Numeric.AD.Mode type Scalar Double = Double
type Scalar Float Source #
Instance details Defined in Numeric.AD.Mode type Scalar Float = Float
type Scalar Int Source #
Instance details Defined in Numeric.AD.Mode type Scalar Int = Int
type Scalar Int8 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Int8 = Int8
type Scalar Int16 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Int16 = Int16
type Scalar Int32 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Int32 = Int32
type Scalar Int64 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Int64 = Int64
type Scalar Integer Source #
Instance details Defined in Numeric.AD.Mode type Scalar Integer = Integer
type Scalar Natural Source #
Instance details Defined in Numeric.AD.Mode type Scalar Natural = Natural
type Scalar Word Source #
Instance details Defined in Numeric.AD.Mode type Scalar Word = Word
type Scalar Word8 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Word8 = Word8
type Scalar Word16 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Word16 = Word16
type Scalar Word32 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Word32 = Word32
type Scalar Word64 Source #
Instance details Defined in Numeric.AD.Mode type Scalar Word64 = Word64
type Scalar ForwardDouble Source #
Instance details Defined in Numeric.AD.Internal.Forward.Double type Scalar ForwardDouble = Double
type Scalar (Ratio a) Source #
Instance details Defined in Numeric.AD.Mode type Scalar (Ratio a) = Ratio a
type Scalar (Complex a) Source #
Instance details Defined in Numeric.AD.Mode type Scalar (Complex a) = Complex a
type Scalar (On t) Source #
Instance details Defined in Numeric.AD.Internal.On type Scalar (On t) = Scalar (Scalar t)
type Scalar (Id a) Source #
Instance details Defined in Numeric.AD.Internal.Identity type Scalar (Id a) = a
type Scalar (Tower a) Source #
Instance details Defined in Numeric.AD.Internal.Tower type Scalar (Tower a) = a
type Scalar (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse type Scalar (Sparse a) = a
type Scalar (Kahn a) Source #
Instance details Defined in Numeric.AD.Internal.Kahn type Scalar (Kahn a) = a
type Scalar (Forward a) Source #
Instance details Defined in Numeric.AD.Internal.Forward type Scalar (Forward a) = a
type Scalar (AD s a) Source #
Instance details Defined in Numeric.AD.Internal.Type type Scalar (AD s a) = Scalar a
type Scalar (Reverse s a) Source #
Instance details Defined in Numeric.AD.Internal.Reverse type Scalar (Reverse s a) = a
type Scalar (Dense f a) Source #
Instance details Defined in Numeric.AD.Internal.Dense type Scalar (Dense f a) = a
type Scalar (Or s a b) Source #
Instance details Defined in Numeric.AD.Internal.Or type Scalar (Or s a b) = Scalar a

Gradients (Reverse Mode)

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 (\[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) 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)

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 Source #

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 Source #

Minimal complete definition

pack, unpack, unpack'

Instances

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 #

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

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

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

Minimal complete definition

packs, unpacks

Instances

Num a => Grads (Sparse a) (Cofree [] a) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: Sparse a -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> Cofree [] a Source #
Grads i o a => Grads (Sparse a -> i) (a -> o) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> a -> o Source #

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

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) 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)

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) Source #

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) Source #

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) Source #

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) Source #

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)) Source #

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)) Source #

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 :: 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]]]

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)) Source #

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 Source #

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) Source #

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 Source #

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 Source #

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) Source #

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) Source #

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] Source #

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

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

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

Directional Derivatives (Forward Mode)

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

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) Source #

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 Source #

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) Source #

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] Source #

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

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

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

Taylor Series (Tower)

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

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

Maclaurin Series (Tower)

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

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

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] Source #

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] Source #

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] Source #

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] Source #

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 => e -> f (Reverse s a) -> Reverse s a) -> [e] -> f a -> [f a] Source #

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