Numeric.Backprop.Explicit

Provides "explicit" versions of all of the functions in Numeric.Backprop. Instead of relying on a Backprop instance, allows you to manually provide zero, add, and one on a per-value basis.

It is recommended you use Numeric.Backprop or Numeric.Backprop.Num instead, unless your type has no Num instance, or you else you want to avoid defining orphan Backprop instances for external types. Can also be useful if mixing and matching styles.

See Numeric.Backprop for fuller documentation on using these functions.

WARNING: API of this module can be considered only "semi-stable"; while the API of Numeric.Backprop and Numeric.Backprop.Num are kept consistent, some argument order changes might happen in this module to reflect changes in underlying implementation.

Since: 0.2.0.0

Types

data BVar s a

A BVar s a is a value of type a that can be "backpropagated".

Functions referring to BVars are tracked by the library and can be automatically differentiated to get their gradients and results.

For simple numeric values, you can use its Num, Fractional, and Floating instances to manipulate them as if they were the numbers they represent.

If a contains items, the items can be accessed and extracted using lenses. A Lens' b a can be used to access an a inside a b, using ^^. (viewVar):

(^.)  ::        a -> Lens' a b ->        b
(^^.) :: BVar s a -> Lens' a b -> BVar s b

There is also ^^? (previewVar), to use a Prism' or Traversal' to extract a target that may or may not be present (which can implement pattern matching), ^^.. (toListOfVar) to use a Traversal' to extract all targets inside a BVar, and .~~ (setVar) to set and update values inside a BVar.

If you have control over your data type definitions, you can also use splitBV and joinBV to manipulate data types by easily extracting fields out of a BVar of data types and creating BVars of data types out of BVars of their fields. See Numeric.Backprop for a tutorial on this use pattern.

For more complex operations, libraries can provide functions on BVars using liftOp and related functions. This is how you can create primitive functions that users can use to manipulate your library's values. See https://backprop.jle.im/08-equipping-your-library.html for a detailed guide.

For example, the hmatrix library has a matrix-vector multiplication function, #> :: L m n -> R n -> L m.

A library could instead provide a function #> :: BVar (L m n) -> BVar (R n) -> BVar (R m), which the user can then use to manipulate their BVars of L m ns and R ns, etc.

See Numeric.Backprop and documentation for liftOp for more information.

Instances

BVGroup s ([] :: [Type]) (K1 i a :: Type -> Type) (K1 i (BVar s a) :: Type -> Type)

Defined in Numeric.Backprop.Explicit

Methods

gsplitBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> BVar s (K1 i a ()) -> K1 i (BVar s a) ()

gjoinBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> K1 i (BVar s a) () -> BVar s (K1 i a ())

IsoHKD (BVar s :: Type -> Type) (a :: Type)

Since: 0.2.6.3

Defined in Numeric.Backprop.Internal

Associated Types

type HKD (BVar s) a :: Type

Methods

unHKD :: HKD (BVar s) a -> BVar s a

toHKD :: BVar s a -> HKD (BVar s) a

Eq a => Eq (BVar s a)

Compares the values inside the BVar.

Since: 0.1.5.0

Defined in Numeric.Backprop.Internal

Methods

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

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

(Floating a, Reifies s W) => Floating (BVar s a)

Defined in Numeric.Backprop.Internal

Methods

pi :: BVar s a

exp :: BVar s a -> BVar s a

log :: BVar s a -> BVar s a

sqrt :: BVar s a -> BVar s a

(**) :: BVar s a -> BVar s a -> BVar s a

logBase :: BVar s a -> BVar s a -> BVar s a

sin :: BVar s a -> BVar s a

cos :: BVar s a -> BVar s a

tan :: BVar s a -> BVar s a

asin :: BVar s a -> BVar s a

acos :: BVar s a -> BVar s a

atan :: BVar s a -> BVar s a

sinh :: BVar s a -> BVar s a

cosh :: BVar s a -> BVar s a

tanh :: BVar s a -> BVar s a

asinh :: BVar s a -> BVar s a

acosh :: BVar s a -> BVar s a

atanh :: BVar s a -> BVar s a

log1p :: BVar s a -> BVar s a

expm1 :: BVar s a -> BVar s a

log1pexp :: BVar s a -> BVar s a

log1mexp :: BVar s a -> BVar s a

(Fractional a, Reifies s W) => Fractional (BVar s a)

Defined in Numeric.Backprop.Internal

Methods

(/) :: BVar s a -> BVar s a -> BVar s a

recip :: BVar s a -> BVar s a

fromRational :: Rational -> BVar s a

(Num a, Reifies s W) => Num (BVar s a)

Defined in Numeric.Backprop.Internal

Methods

(+) :: BVar s a -> BVar s a -> BVar s a

(-) :: BVar s a -> BVar s a -> BVar s a

(*) :: BVar s a -> BVar s a -> BVar s a

negate :: BVar s a -> BVar s a

abs :: BVar s a -> BVar s a

signum :: BVar s a -> BVar s a

fromInteger :: Integer -> BVar s a

Ord a => Ord (BVar s a)

Compares the values inside the BVar.

Since: 0.1.5.0

Defined in Numeric.Backprop.Internal

Methods

compare :: BVar s a -> BVar s a -> Ordering

(<) :: BVar s a -> BVar s a -> Bool

(<=) :: BVar s a -> BVar s a -> Bool

(>) :: BVar s a -> BVar s a -> Bool

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

max :: BVar s a -> BVar s a -> BVar s a

min :: BVar s a -> BVar s a -> BVar s a

NFData a => NFData (BVar s a)

This will force the value inside, as well.

Defined in Numeric.Backprop.Internal

Methods

rnf :: BVar s a -> ()

(Backprop a, Reifies s W) => Backprop (BVar s a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Internal

Methods

zero :: BVar s a -> BVar s a

add :: BVar s a -> BVar s a -> BVar s a

one :: BVar s a -> BVar s a

type HKD (BVar s :: Type -> Type) (a :: Type)

Defined in Numeric.Backprop.Internal

type HKD (BVar s :: Type -> Type) (a :: Type) = BVar s a

data W

An ephemeral Wengert Tape in the environment. Used internally to track of the computational graph of variables.

For the end user, one can just imagine Reifies s W as a required constraint on s that allows backpropagation to work.

class Backprop a where

Class of values that can be backpropagated in general.

For instances of Num, these methods can be given by zeroNum, addNum, and oneNum. There are also generic options given in Numeric.Backprop.Class for functors, IsList instances, and Generic instances.

instance Backprop Double where
    zero = zeroNum
    add = addNum
    one = oneNum

If you leave the body of an instance declaration blank, GHC Generics will be used to derive instances if the type has a single constructor and each field is an instance of Backprop.

To ensure that backpropagation works in a sound way, should obey the laws:

identity

• add x (zero y) = x

• add (zero x) y = y

Also implies preservation of information, making zipWith (+) an illegal implementation for lists and vectors.

This is only expected to be true up to potential "extra zeroes" in x and y in the result.

commutativity

• add x y = add y x

associativity

• add x (add y z) = add (add x y) z

idempotence

• zero . zero = zero

• one . one = one

unital

• one = gradBP id

Note that not all values in the backpropagation process needs all of these methods: Only the "final result" needs one, for example. These are all grouped under one typeclass for convenience in defining instances, and also to talk about sensible laws. For fine-grained control, use the "explicit" versions of library functions (for example, in Numeric.Backprop.Explicit) instead of Backprop based ones.

This typeclass replaces the reliance on Num of the previous API (v0.1). Num is strictly more powerful than Backprop, and is a stronger constraint on types than is necessary for proper backpropagating. In particular, fromInteger is a problem for many types, preventing useful backpropagation for lists, variable-length vectors (like Data.Vector) and variable-size matrices from linear algebra libraries like hmatrix and accelerate.

Since: 0.2.0.0

Minimal complete definition

Nothing

Methods

zero :: a -> a

"Zero out" all components of a value. For scalar values, this should just be const 0. For vectors and matrices, this should set all components to zero, the additive identity.

Should be idempotent:

• zero . zero = zero

Should be as lazy as possible. This behavior is observed for all instances provided by this library.

See zeroNum for a pre-built definition for instances of Num and zeroFunctor for a definition for instances of Functor. If left blank, will automatically be genericZero, a pre-built definition for instances of Generic whose fields are all themselves instances of Backprop.

add :: a -> a -> a

Add together two values of a type. To combine contributions of gradients, so should be information-preserving:

• add x (zero y) = x

• add (zero x) y = y

Should be as strict as possible. This behavior is observed for all instances provided by this library.

See addNum for a pre-built definition for instances of Num and addIsList for a definition for instances of IsList. If left blank, will automatically be genericAdd, a pre-built definition for instances of Generic with one constructor whose fields are all themselves instances of Backprop.

one :: a -> a

One all components of a value. For scalar values, this should just be const 1. For vectors and matrices, this should set all components to one, the multiplicative identity.

As the library uses it, the most important law is:

• one = gradBP id

That is, one x is the gradient of the identity function with respect to its input.

Ideally should be idempotent:

• one . one = one

Should be as lazy as possible. This behavior is observed for all instances provided by this library.

See oneNum for a pre-built definition for instances of Num and oneFunctor for a definition for instances of Functor. If left blank, will automatically be genericOne, a pre-built definition for instances of Generic whose fields are all themselves instances of Backprop.

zero :: (Generic a, GZero (Rep a)) => a -> a

"Zero out" all components of a value. For scalar values, this should just be const 0. For vectors and matrices, this should set all components to zero, the additive identity.

Should be idempotent:

• zero . zero = zero

Should be as lazy as possible. This behavior is observed for all instances provided by this library.

See zeroNum for a pre-built definition for instances of Num and zeroFunctor for a definition for instances of Functor. If left blank, will automatically be genericZero, a pre-built definition for instances of Generic whose fields are all themselves instances of Backprop.

add :: (Generic a, GAdd (Rep a)) => a -> a -> a

Add together two values of a type. To combine contributions of gradients, so should be information-preserving:

• add x (zero y) = x

• add (zero x) y = y

Should be as strict as possible. This behavior is observed for all instances provided by this library.

See addNum for a pre-built definition for instances of Num and addIsList for a definition for instances of IsList. If left blank, will automatically be genericAdd, a pre-built definition for instances of Generic with one constructor whose fields are all themselves instances of Backprop.

one :: (Generic a, GOne (Rep a)) => a -> a

One all components of a value. For scalar values, this should just be const 1. For vectors and matrices, this should set all components to one, the multiplicative identity.

As the library uses it, the most important law is:

• one = gradBP id

That is, one x is the gradient of the identity function with respect to its input.

Ideally should be idempotent:

• one . one = one

Should be as lazy as possible. This behavior is observed for all instances provided by this library.

See oneNum for a pre-built definition for instances of Num and oneFunctor for a definition for instances of Functor. If left blank, will automatically be genericOne, a pre-built definition for instances of Generic whose fields are all themselves instances of Backprop.

Instances

Backprop Double

Defined in Numeric.Backprop.Class

Methods

zero :: Double -> Double

add :: Double -> Double -> Double

one :: Double -> Double

Backprop Float

Defined in Numeric.Backprop.Class

Methods

zero :: Float -> Float

add :: Float -> Float -> Float

one :: Float -> Float

Backprop Int

Defined in Numeric.Backprop.Class

Methods

zero :: Int -> Int

add :: Int -> Int -> Int

one :: Int -> Int

Backprop Integer

Defined in Numeric.Backprop.Class

Methods

zero :: Integer -> Integer

add :: Integer -> Integer -> Integer

one :: Integer -> Integer

Backprop Natural

Since: 0.2.1.0

Defined in Numeric.Backprop.Class

Methods

zero :: Natural -> Natural

add :: Natural -> Natural -> Natural

one :: Natural -> Natural

Backprop Word

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Word -> Word

add :: Word -> Word -> Word

one :: Word -> Word

Backprop Word8

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Word8 -> Word8

add :: Word8 -> Word8 -> Word8

one :: Word8 -> Word8

Backprop Word16

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Word16 -> Word16

add :: Word16 -> Word16 -> Word16

one :: Word16 -> Word16

Backprop Word32

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Word32 -> Word32

add :: Word32 -> Word32 -> Word32

one :: Word32 -> Word32

Backprop Word64

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Word64 -> Word64

add :: Word64 -> Word64 -> Word64

one :: Word64 -> Word64

Backprop ()

add is strict, but zero and one are lazy in their arguments.

Defined in Numeric.Backprop.Class

Methods

zero :: () -> ()

add :: () -> () -> ()

one :: () -> ()

Backprop Void

Defined in Numeric.Backprop.Class

Methods

zero :: Void -> Void

add :: Void -> Void -> Void

one :: Void -> Void

Backprop a => Backprop [a]

add assumes the shorter list has trailing zeroes, and the result has the length of the longest input.

Defined in Numeric.Backprop.Class

Methods

zero :: [a] -> [a]

add :: [a] -> [a] -> [a]

one :: [a] -> [a]

Backprop a => Backprop (Maybe a)

Nothing is treated the same as Just 0. However, zero, add, and one preserve Nothing if all inputs are also Nothing.

Defined in Numeric.Backprop.Class

Methods

zero :: Maybe a -> Maybe a

add :: Maybe a -> Maybe a -> Maybe a

one :: Maybe a -> Maybe a

Integral a => Backprop (Ratio a)

Defined in Numeric.Backprop.Class

Methods

zero :: Ratio a -> Ratio a

add :: Ratio a -> Ratio a -> Ratio a

one :: Ratio a -> Ratio a

RealFloat a => Backprop (Complex a)

Defined in Numeric.Backprop.Class

Methods

zero :: Complex a -> Complex a

add :: Complex a -> Complex a -> Complex a

one :: Complex a -> Complex a

Backprop a => Backprop (First a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: First a -> First a

add :: First a -> First a -> First a

one :: First a -> First a

Backprop a => Backprop (Last a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Last a -> Last a

add :: Last a -> Last a -> Last a

one :: Last a -> Last a

Backprop a => Backprop (Option a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Option a -> Option a

add :: Option a -> Option a -> Option a

one :: Option a -> Option a

Backprop a => Backprop (Identity a)

Defined in Numeric.Backprop.Class

Methods

zero :: Identity a -> Identity a

add :: Identity a -> Identity a -> Identity a

one :: Identity a -> Identity a

Backprop a => Backprop (First a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: First a -> First a

add :: First a -> First a -> First a

one :: First a -> First a

Backprop a => Backprop (Last a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Last a -> Last a

add :: Last a -> Last a -> Last a

one :: Last a -> Last a

Backprop a => Backprop (Dual a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Dual a -> Dual a

add :: Dual a -> Dual a -> Dual a

one :: Dual a -> Dual a

Backprop a => Backprop (Sum a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Sum a -> Sum a

add :: Sum a -> Sum a -> Sum a

one :: Sum a -> Sum a

Backprop a => Backprop (Product a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Product a -> Product a

add :: Product a -> Product a -> Product a

one :: Product a -> Product a

Backprop a => Backprop (NonEmpty a)

add assumes the shorter list has trailing zeroes, and the result has the length of the longest input.

Defined in Numeric.Backprop.Class

Methods

zero :: NonEmpty a -> NonEmpty a

add :: NonEmpty a -> NonEmpty a -> NonEmpty a

one :: NonEmpty a -> NonEmpty a

Backprop a => Backprop (IntMap a)

zero and one replace all current values, and add merges keys from both maps, adding in the case of double-occurrences.

Defined in Numeric.Backprop.Class

Methods

zero :: IntMap a -> IntMap a

add :: IntMap a -> IntMap a -> IntMap a

one :: IntMap a -> IntMap a

Backprop a => Backprop (Seq a)

add assumes the shorter sequence has trailing zeroes, and the result has the length of the longest input.

Defined in Numeric.Backprop.Class

Methods

zero :: Seq a -> Seq a

add :: Seq a -> Seq a -> Seq a

one :: Seq a -> Seq a

(Unbox a, Backprop a) => Backprop (Vector a)

Defined in Numeric.Backprop.Class

Methods

zero :: Vector a -> Vector a

add :: Vector a -> Vector a -> Vector a

one :: Vector a -> Vector a

(Storable a, Backprop a) => Backprop (Vector a)

Defined in Numeric.Backprop.Class

Methods

zero :: Vector a -> Vector a

add :: Vector a -> Vector a -> Vector a

one :: Vector a -> Vector a

(Prim a, Backprop a) => Backprop (Vector a)

Defined in Numeric.Backprop.Class

Methods

zero :: Vector a -> Vector a

add :: Vector a -> Vector a -> Vector a

one :: Vector a -> Vector a

Backprop a => Backprop (Vector a)

Defined in Numeric.Backprop.Class

Methods

zero :: Vector a -> Vector a

add :: Vector a -> Vector a -> Vector a

one :: Vector a -> Vector a

Backprop (Label field)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: Label field -> Label field

add :: Label field -> Label field -> Label field

one :: Label field -> Label field

Backprop a => Backprop (Identity a)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: Identity a -> Identity a

add :: Identity a -> Identity a -> Identity a

one :: Identity a -> Identity a

Backprop a => Backprop (Thunk a)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: Thunk a -> Thunk a

add :: Thunk a -> Thunk a -> Thunk a

one :: Thunk a -> Thunk a

Backprop t => Backprop (ElField ((,) s t))

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: ElField (s, t) -> ElField (s, t)

add :: ElField (s, t) -> ElField (s, t) -> ElField (s, t)

one :: ElField (s, t) -> ElField (s, t)

Num a => Backprop (NumBP a)

Defined in Numeric.Backprop.Class

Methods

zero :: NumBP a -> NumBP a

add :: NumBP a -> NumBP a -> NumBP a

one :: NumBP a -> NumBP a

Backprop a => Backprop (r -> a)

add adds together results; zero and one act on results.

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: (r -> a) -> r -> a

add :: (r -> a) -> (r -> a) -> r -> a

one :: (r -> a) -> r -> a

Backprop (V1 p)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: V1 p -> V1 p

add :: V1 p -> V1 p -> V1 p

one :: V1 p -> V1 p

Backprop (U1 p)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: U1 p -> U1 p

add :: U1 p -> U1 p -> U1 p

one :: U1 p -> U1 p

(Backprop a, Backprop b) => Backprop (a, b)

add is strict

Defined in Numeric.Backprop.Class

Methods

zero :: (a, b) -> (a, b)

add :: (a, b) -> (a, b) -> (a, b)

one :: (a, b) -> (a, b)

(Backprop a, Backprop b) => Backprop (Arg a b)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Arg a b -> Arg a b

add :: Arg a b -> Arg a b -> Arg a b

one :: Arg a b -> Arg a b

Backprop (Proxy a)

Defined in Numeric.Backprop.Class

Methods

zero :: Proxy a -> Proxy a

add :: Proxy a -> Proxy a -> Proxy a

one :: Proxy a -> Proxy a

(Backprop a, Ord k) => Backprop (Map k a)

zero and one replace all current values, and add merges keys from both maps, adding in the case of double-occurrences.

Defined in Numeric.Backprop.Class

Methods

zero :: Map k a -> Map k a

add :: Map k a -> Map k a -> Map k a

one :: Map k a -> Map k a

Backprop (SField field)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: SField field -> SField field

add :: SField field -> SField field -> SField field

one :: SField field -> SField field

(Applicative f, Backprop a) => Backprop (ABP f a)

Defined in Numeric.Backprop.Class

Methods

zero :: ABP f a -> ABP f a

add :: ABP f a -> ABP f a -> ABP f a

one :: ABP f a -> ABP f a

(Vector v a, Num a) => Backprop (NumVec v a)

Defined in Numeric.Backprop.Class

Methods

zero :: NumVec v a -> NumVec v a

add :: NumVec v a -> NumVec v a -> NumVec v a

one :: NumVec v a -> NumVec v a

(Backprop a, Reifies s W) => Backprop (BVar s a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Internal

Methods

zero :: BVar s a -> BVar s a

add :: BVar s a -> BVar s a -> BVar s a

one :: BVar s a -> BVar s a

(Backprop a, Backprop b, Backprop c) => Backprop (a, b, c)

add is strict

Defined in Numeric.Backprop.Class

Methods

zero :: (a, b, c) -> (a, b, c)

add :: (a, b, c) -> (a, b, c) -> (a, b, c)

one :: (a, b, c) -> (a, b, c)

(Backprop a, Applicative m) => Backprop (Kleisli m r a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Kleisli m r a -> Kleisli m r a

add :: Kleisli m r a -> Kleisli m r a -> Kleisli m r a

one :: Kleisli m r a -> Kleisli m r a

Backprop w => Backprop (Const w a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Const w a -> Const w a

add :: Const w a -> Const w a -> Const w a

one :: Const w a -> Const w a

(ReifyConstraint Backprop f rs, RMap rs, RApply rs, IsoXRec f rs) => Backprop (XRec f rs)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: XRec f rs -> XRec f rs

add :: XRec f rs -> XRec f rs -> XRec f rs

one :: XRec f rs -> XRec f rs

Backprop (HKD t a) => Backprop (XData t a)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: XData t a -> XData t a

add :: XData t a -> XData t a -> XData t a

one :: XData t a -> XData t a

(ReifyConstraint Backprop f rs, RMap rs, RApply rs, RecApplicative rs, NatToInt (RLength rs), RPureConstrained (IndexableField rs) rs) => Backprop (ARec f rs)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: ARec f rs -> ARec f rs

add :: ARec f rs -> ARec f rs -> ARec f rs

one :: ARec f rs -> ARec f rs

(ReifyConstraint Backprop f rs, RMap rs, RApply rs, Storable (Rec f rs)) => Backprop (SRec f rs)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: SRec f rs -> SRec f rs

add :: SRec f rs -> SRec f rs -> SRec f rs

one :: SRec f rs -> SRec f rs

(ReifyConstraint Backprop f rs, RMap rs, RApply rs) => Backprop (Rec f rs)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: Rec f rs -> Rec f rs

add :: Rec f rs -> Rec f rs -> Rec f rs

one :: Rec f rs -> Rec f rs

Backprop w => Backprop (Const w a)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: Const w a -> Const w a

add :: Const w a -> Const w a -> Const w a

one :: Const w a -> Const w a

Backprop a => Backprop (K1 i a p)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: K1 i a p -> K1 i a p

add :: K1 i a p -> K1 i a p -> K1 i a p

one :: K1 i a p -> K1 i a p

(Backprop (f p), Backprop (g p)) => Backprop ((f :*: g) p)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: (f :*: g) p -> (f :*: g) p

add :: (f :*: g) p -> (f :*: g) p -> (f :*: g) p

one :: (f :*: g) p -> (f :*: g) p

(Backprop a, Backprop b, Backprop c, Backprop d) => Backprop (a, b, c, d)

add is strict

Defined in Numeric.Backprop.Class

Methods

zero :: (a, b, c, d) -> (a, b, c, d)

add :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d)

one :: (a, b, c, d) -> (a, b, c, d)

(Backprop (f a), Backprop (g a)) => Backprop (Product f g a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Product f g a -> Product f g a

add :: Product f g a -> Product f g a -> Product f g a

one :: Product f g a -> Product f g a

Backprop (f p) => Backprop (M1 i c f p)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: M1 i c f p -> M1 i c f p

add :: M1 i c f p -> M1 i c f p -> M1 i c f p

one :: M1 i c f p -> M1 i c f p

Backprop (f (g a)) => Backprop ((f :.: g) a)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: (f :.: g) a -> (f :.: g) a

add :: (f :.: g) a -> (f :.: g) a -> (f :.: g) a

one :: (f :.: g) a -> (f :.: g) a

(Backprop a, Backprop b, Backprop c, Backprop d, Backprop e) => Backprop (a, b, c, d, e)

add is strict

Defined in Numeric.Backprop.Class

Methods

zero :: (a, b, c, d, e) -> (a, b, c, d, e)

add :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e)

one :: (a, b, c, d, e) -> (a, b, c, d, e)

Backprop (f (g a)) => Backprop (Compose f g a)

Since: 0.2.2.0

Defined in Numeric.Backprop.Class

Methods

zero :: Compose f g a -> Compose f g a

add :: Compose f g a -> Compose f g a -> Compose f g a

one :: Compose f g a -> Compose f g a

Backprop (f (g a)) => Backprop (Compose f g a)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: Compose f g a -> Compose f g a

add :: Compose f g a -> Compose f g a -> Compose f g a

one :: Compose f g a -> Compose f g a

Backprop (op (f a) (g a)) => Backprop (Lift op f g a)

Since: 0.2.6.3

Defined in Numeric.Backprop.Class

Methods

zero :: Lift op f g a -> Lift op f g a

add :: Lift op f g a -> Lift op f g a -> Lift op f g a

one :: Lift op f g a -> Lift op f g a

newtype ABP f a

A newtype wrapper over an f a for Applicative f that gives a free Backprop instance (as well as Num etc. instances).

Useful for performing backpropagation over functions that require some monadic context (like IO) to perform.

Since: 0.2.1.0

Constructors

ABP

Fields

• runABP :: f a

Instances

Monad f => Monad (ABP f)

Defined in Numeric.Backprop.Class

Methods

(>>=) :: ABP f a -> (a -> ABP f b) -> ABP f b

(>>) :: ABP f a -> ABP f b -> ABP f b

return :: a -> ABP f a

fail :: String -> ABP f a

Functor f => Functor (ABP f)

Defined in Numeric.Backprop.Class

Methods

fmap :: (a -> b) -> ABP f a -> ABP f b

(<$) :: a -> ABP f b -> ABP f a

Applicative f => Applicative (ABP f)

Defined in Numeric.Backprop.Class

Methods

pure :: a -> ABP f a

(<*>) :: ABP f (a -> b) -> ABP f a -> ABP f b

liftA2 :: (a -> b -> c) -> ABP f a -> ABP f b -> ABP f c

(*>) :: ABP f a -> ABP f b -> ABP f b

(<*) :: ABP f a -> ABP f b -> ABP f a

Foldable f => Foldable (ABP f)

Defined in Numeric.Backprop.Class

Methods

fold :: Monoid m => ABP f m -> m

foldMap :: Monoid m => (a -> m) -> ABP f a -> m

foldr :: (a -> b -> b) -> b -> ABP f a -> b

foldr' :: (a -> b -> b) -> b -> ABP f a -> b

foldl :: (b -> a -> b) -> b -> ABP f a -> b

foldl' :: (b -> a -> b) -> b -> ABP f a -> b

foldr1 :: (a -> a -> a) -> ABP f a -> a

foldl1 :: (a -> a -> a) -> ABP f a -> a

toList :: ABP f a -> [a]

null :: ABP f a -> Bool

length :: ABP f a -> Int

elem :: Eq a => a -> ABP f a -> Bool

maximum :: Ord a => ABP f a -> a

minimum :: Ord a => ABP f a -> a

sum :: Num a => ABP f a -> a

product :: Num a => ABP f a -> a

Traversable f => Traversable (ABP f)

Defined in Numeric.Backprop.Class

Methods

traverse :: Applicative f0 => (a -> f0 b) -> ABP f a -> f0 (ABP f b)

sequenceA :: Applicative f0 => ABP f (f0 a) -> f0 (ABP f a)

mapM :: Monad m => (a -> m b) -> ABP f a -> m (ABP f b)

sequence :: Monad m => ABP f (m a) -> m (ABP f a)

Alternative f => Alternative (ABP f)

Defined in Numeric.Backprop.Class

Methods

empty :: ABP f a

(<|>) :: ABP f a -> ABP f a -> ABP f a

some :: ABP f a -> ABP f [a]

many :: ABP f a -> ABP f [a]

MonadPlus f => MonadPlus (ABP f)

Defined in Numeric.Backprop.Class

Methods

mzero :: ABP f a

mplus :: ABP f a -> ABP f a -> ABP f a

Eq (f a) => Eq (ABP f a)

Defined in Numeric.Backprop.Class

Methods

(==) :: ABP f a -> ABP f a -> Bool

(/=) :: ABP f a -> ABP f a -> Bool

(Applicative f, Floating a) => Floating (ABP f a)

Defined in Numeric.Backprop.Class

Methods

pi :: ABP f a

exp :: ABP f a -> ABP f a

log :: ABP f a -> ABP f a

sqrt :: ABP f a -> ABP f a

(**) :: ABP f a -> ABP f a -> ABP f a

logBase :: ABP f a -> ABP f a -> ABP f a

sin :: ABP f a -> ABP f a

cos :: ABP f a -> ABP f a

tan :: ABP f a -> ABP f a

asin :: ABP f a -> ABP f a

acos :: ABP f a -> ABP f a

atan :: ABP f a -> ABP f a

sinh :: ABP f a -> ABP f a

cosh :: ABP f a -> ABP f a

tanh :: ABP f a -> ABP f a

asinh :: ABP f a -> ABP f a

acosh :: ABP f a -> ABP f a

atanh :: ABP f a -> ABP f a

log1p :: ABP f a -> ABP f a

expm1 :: ABP f a -> ABP f a

log1pexp :: ABP f a -> ABP f a

log1mexp :: ABP f a -> ABP f a

(Applicative f, Fractional a) => Fractional (ABP f a)

Defined in Numeric.Backprop.Class

Methods

(/) :: ABP f a -> ABP f a -> ABP f a

recip :: ABP f a -> ABP f a

fromRational :: Rational -> ABP f a

(Typeable f, Typeable a, Data (f a)) => Data (ABP f a)

Defined in Numeric.Backprop.Class

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> ABP f a -> c (ABP f a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (ABP f a)

toConstr :: ABP f a -> Constr

dataTypeOf :: ABP f a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (ABP f a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (ABP f a))

gmapT :: (forall b. Data b => b -> b) -> ABP f a -> ABP f a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> ABP f a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> ABP f a -> r

gmapQ :: (forall d. Data d => d -> u) -> ABP f a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> ABP f a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> ABP f a -> m (ABP f a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> ABP f a -> m (ABP f a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> ABP f a -> m (ABP f a)

(Applicative f, Num a) => Num (ABP f a)

Defined in Numeric.Backprop.Class

Methods

(+) :: ABP f a -> ABP f a -> ABP f a

(-) :: ABP f a -> ABP f a -> ABP f a

(*) :: ABP f a -> ABP f a -> ABP f a

negate :: ABP f a -> ABP f a

abs :: ABP f a -> ABP f a

signum :: ABP f a -> ABP f a

fromInteger :: Integer -> ABP f a

Ord (f a) => Ord (ABP f a)

Defined in Numeric.Backprop.Class

Methods

compare :: ABP f a -> ABP f a -> Ordering

(<) :: ABP f a -> ABP f a -> Bool

(<=) :: ABP f a -> ABP f a -> Bool

(>) :: ABP f a -> ABP f a -> Bool

(>=) :: ABP f a -> ABP f a -> Bool

max :: ABP f a -> ABP f a -> ABP f a

min :: ABP f a -> ABP f a -> ABP f a

Read (f a) => Read (ABP f a)

Defined in Numeric.Backprop.Class

Methods

readsPrec :: Int -> ReadS (ABP f a)

readList :: ReadS [ABP f a]

readPrec :: ReadPrec (ABP f a)

readListPrec :: ReadPrec [ABP f a]

Show (f a) => Show (ABP f a)

Defined in Numeric.Backprop.Class

Methods

showsPrec :: Int -> ABP f a -> ShowS

show :: ABP f a -> String

showList :: [ABP f a] -> ShowS

Generic (ABP f a)

Defined in Numeric.Backprop.Class

Associated Types

type Rep (ABP f a) :: Type -> Type

Methods

from :: ABP f a -> Rep (ABP f a) x

to :: Rep (ABP f a) x -> ABP f a

NFData (f a) => NFData (ABP f a)

Defined in Numeric.Backprop.Class

Methods

rnf :: ABP f a -> ()

(Applicative f, Backprop a) => Backprop (ABP f a)

Defined in Numeric.Backprop.Class

Methods

zero :: ABP f a -> ABP f a

add :: ABP f a -> ABP f a -> ABP f a

one :: ABP f a -> ABP f a

type Rep (ABP f a)

Defined in Numeric.Backprop.Class

type Rep (ABP f a) = D1 (MetaData "ABP" "Numeric.Backprop.Class" "backprop-0.2.6.3-AbmhPNL895h5gLgyaH73Nz" True) (C1 (MetaCons "ABP" PrefixI True) (S1 (MetaSel (Just "runABP") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (f a))))

newtype NumBP a

A newtype wrapper over an instance of Num that gives a free Backprop instance.

Useful for things like DerivingVia, or for avoiding orphan instances.

Since: 0.2.1.0

Constructors

NumBP

Fields

• runNumBP :: a

Instances

Monad NumBP

Defined in Numeric.Backprop.Class

Methods

(>>=) :: NumBP a -> (a -> NumBP b) -> NumBP b

(>>) :: NumBP a -> NumBP b -> NumBP b

return :: a -> NumBP a

fail :: String -> NumBP a

Functor NumBP

Defined in Numeric.Backprop.Class

Methods

fmap :: (a -> b) -> NumBP a -> NumBP b

(<$) :: a -> NumBP b -> NumBP a

Applicative NumBP

Defined in Numeric.Backprop.Class

Methods

pure :: a -> NumBP a

(<*>) :: NumBP (a -> b) -> NumBP a -> NumBP b

liftA2 :: (a -> b -> c) -> NumBP a -> NumBP b -> NumBP c

(*>) :: NumBP a -> NumBP b -> NumBP b

(<*) :: NumBP a -> NumBP b -> NumBP a

Foldable NumBP

Defined in Numeric.Backprop.Class

Methods

fold :: Monoid m => NumBP m -> m

foldMap :: Monoid m => (a -> m) -> NumBP a -> m

foldr :: (a -> b -> b) -> b -> NumBP a -> b

foldr' :: (a -> b -> b) -> b -> NumBP a -> b

foldl :: (b -> a -> b) -> b -> NumBP a -> b

foldl' :: (b -> a -> b) -> b -> NumBP a -> b

foldr1 :: (a -> a -> a) -> NumBP a -> a

foldl1 :: (a -> a -> a) -> NumBP a -> a

toList :: NumBP a -> [a]

null :: NumBP a -> Bool

length :: NumBP a -> Int

elem :: Eq a => a -> NumBP a -> Bool

maximum :: Ord a => NumBP a -> a

minimum :: Ord a => NumBP a -> a

sum :: Num a => NumBP a -> a

product :: Num a => NumBP a -> a

Traversable NumBP

Defined in Numeric.Backprop.Class

Methods

traverse :: Applicative f => (a -> f b) -> NumBP a -> f (NumBP b)

sequenceA :: Applicative f => NumBP (f a) -> f (NumBP a)

mapM :: Monad m => (a -> m b) -> NumBP a -> m (NumBP b)

sequence :: Monad m => NumBP (m a) -> m (NumBP a)

Eq a => Eq (NumBP a)

Defined in Numeric.Backprop.Class

Methods

(==) :: NumBP a -> NumBP a -> Bool

(/=) :: NumBP a -> NumBP a -> Bool

Floating a => Floating (NumBP a)

Defined in Numeric.Backprop.Class

Methods

pi :: NumBP a

exp :: NumBP a -> NumBP a

log :: NumBP a -> NumBP a

sqrt :: NumBP a -> NumBP a

(**) :: NumBP a -> NumBP a -> NumBP a

logBase :: NumBP a -> NumBP a -> NumBP a

sin :: NumBP a -> NumBP a

cos :: NumBP a -> NumBP a

tan :: NumBP a -> NumBP a

asin :: NumBP a -> NumBP a

acos :: NumBP a -> NumBP a

atan :: NumBP a -> NumBP a

sinh :: NumBP a -> NumBP a

cosh :: NumBP a -> NumBP a

tanh :: NumBP a -> NumBP a

asinh :: NumBP a -> NumBP a

acosh :: NumBP a -> NumBP a

atanh :: NumBP a -> NumBP a

log1p :: NumBP a -> NumBP a

expm1 :: NumBP a -> NumBP a

log1pexp :: NumBP a -> NumBP a

log1mexp :: NumBP a -> NumBP a

Fractional a => Fractional (NumBP a)

Defined in Numeric.Backprop.Class

Methods

(/) :: NumBP a -> NumBP a -> NumBP a

recip :: NumBP a -> NumBP a

fromRational :: Rational -> NumBP a

Data a => Data (NumBP a)

Defined in Numeric.Backprop.Class

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> NumBP a -> c (NumBP a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (NumBP a)

toConstr :: NumBP a -> Constr

dataTypeOf :: NumBP a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (NumBP a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (NumBP a))

gmapT :: (forall b. Data b => b -> b) -> NumBP a -> NumBP a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> NumBP a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> NumBP a -> r

gmapQ :: (forall d. Data d => d -> u) -> NumBP a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> NumBP a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> NumBP a -> m (NumBP a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> NumBP a -> m (NumBP a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> NumBP a -> m (NumBP a)

Num a => Num (NumBP a)

Defined in Numeric.Backprop.Class

Methods

(+) :: NumBP a -> NumBP a -> NumBP a

(-) :: NumBP a -> NumBP a -> NumBP a

(*) :: NumBP a -> NumBP a -> NumBP a

negate :: NumBP a -> NumBP a

abs :: NumBP a -> NumBP a

signum :: NumBP a -> NumBP a

fromInteger :: Integer -> NumBP a

Ord a => Ord (NumBP a)

Defined in Numeric.Backprop.Class

Methods

compare :: NumBP a -> NumBP a -> Ordering

(<) :: NumBP a -> NumBP a -> Bool

(<=) :: NumBP a -> NumBP a -> Bool

(>) :: NumBP a -> NumBP a -> Bool

(>=) :: NumBP a -> NumBP a -> Bool

max :: NumBP a -> NumBP a -> NumBP a

min :: NumBP a -> NumBP a -> NumBP a

Read a => Read (NumBP a)

Defined in Numeric.Backprop.Class

Methods

readsPrec :: Int -> ReadS (NumBP a)

readList :: ReadS [NumBP a]

readPrec :: ReadPrec (NumBP a)

readListPrec :: ReadPrec [NumBP a]

Show a => Show (NumBP a)

Defined in Numeric.Backprop.Class

Methods

showsPrec :: Int -> NumBP a -> ShowS

show :: NumBP a -> String

showList :: [NumBP a] -> ShowS

Generic (NumBP a)

Defined in Numeric.Backprop.Class

Associated Types

type Rep (NumBP a) :: Type -> Type

Methods

from :: NumBP a -> Rep (NumBP a) x

to :: Rep (NumBP a) x -> NumBP a

NFData a => NFData (NumBP a)

Defined in Numeric.Backprop.Class

Methods

rnf :: NumBP a -> ()

Num a => Backprop (NumBP a)

Defined in Numeric.Backprop.Class

Methods

zero :: NumBP a -> NumBP a

add :: NumBP a -> NumBP a -> NumBP a

one :: NumBP a -> NumBP a

type Rep (NumBP a)

Defined in Numeric.Backprop.Class

type Rep (NumBP a) = D1 (MetaData "NumBP" "Numeric.Backprop.Class" "backprop-0.2.6.3-AbmhPNL895h5gLgyaH73Nz" True) (C1 (MetaCons "NumBP" PrefixI True) (S1 (MetaSel (Just "runNumBP") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))

Explicit zero, add, and one

newtype ZeroFunc a

"Zero out" all components of a value. For scalar values, this should just be const 0. For vectors and matrices, this should set all components to zero, the additive identity.

Should be idempotent: Applying the function twice is the same as applying it just once.

Each type should ideally only have one ZeroFunc. This coherence constraint is given by the typeclass Backprop.

Since: 0.2.0.0

Constructors

ZF

Fields

• runZF :: a -> a

zfNum :: Num a => ZeroFunc a

If a type has a Num instance, this is the canonical ZeroFunc.

Since: 0.2.0.0

zfNums :: RPureConstrained Num as => Rec ZeroFunc as

ZeroFuncs for every item in a type level list based on their Num instances

Since: 0.2.0.0

zeroFunc :: Backprop a => ZeroFunc a

The canonical ZeroFunc for instances of Backprop.

Since: 0.2.0.0

zeroFuncs :: RPureConstrained Backprop as => Rec ZeroFunc as

Generate an ZeroFunc for every type in a type-level list, if every type has an instance of Backprop.

Since: 0.2.0.0

zfFunctor :: (Backprop a, Functor f) => ZeroFunc (f a)

zeroFunc for instances of Functor

Since: 0.2.1.0

newtype AddFunc a

Add together two values of a type. To combine contributions of gradients, so should ideally be information-preserving.

See laws for Backprop for the laws this should be expected to preserve. Namely, it should be commutative and associative, with an identity for a valid ZeroFunc.

Each type should ideally only have one AddFunc. This coherence constraint is given by the typeclass Backprop.

Since: 0.2.0.0

Constructors

AF

Fields

• runAF :: a -> a -> a

afNum :: Num a => AddFunc a

If a type has a Num instance, this is the canonical AddFunc.

Since: 0.2.0.0

afNums :: RPureConstrained Num as => Rec AddFunc as

ZeroFuncs for every item in a type level list based on their Num instances

Since: 0.2.0.0

addFunc :: Backprop a => AddFunc a

The canonical AddFunc for instances of Backprop.

Since: 0.2.0.0

addFuncs :: RPureConstrained Backprop as => Rec AddFunc as

Generate an AddFunc for every type in a type-level list, if every type has an instance of Backprop.

Since: 0.2.0.0

newtype OneFunc a

One all components of a value. For scalar values, this should just be const 1. For vectors and matrices, this should set all components to one, the multiplicative identity.

Should be idempotent: Applying the function twice is the same as applying it just once.

Each type should ideally only have one OneFunc. This coherence constraint is given by the typeclass Backprop.

Since: 0.2.0.0

Constructors

OF

Fields

• runOF :: a -> a

ofNum :: Num a => OneFunc a

If a type has a Num instance, this is the canonical OneFunc.

Since: 0.2.0.0

ofNums :: RPureConstrained Num as => Rec OneFunc as

ZeroFuncs for every item in a type level list based on their Num instances

Since: 0.2.0.0

oneFunc :: Backprop a => OneFunc a

The canonical OneFunc for instances of Backprop.

Since: 0.2.0.0

oneFuncs :: RPureConstrained Backprop as => Rec OneFunc as

Generate an OneFunc for every type in a type-level list, if every type has an instance of Backprop.

Since: 0.2.0.0

ofFunctor :: (Backprop a, Functor f) => OneFunc (f a)

OneFunc for instances of Functor

Since: 0.2.1.0

Running

backprop :: ZeroFunc a -> OneFunc b -> (forall s. Reifies s W => BVar s a -> BVar s b) -> a -> (b, a)

backprop, but with explicit zero and one.

evalBP :: (forall s. Reifies s W => BVar s a -> BVar s b) -> a -> b

Turn a function BVar s a -> BVar s b into the function a -> b that it represents.

Benchmarks show that this should have virtually no overhead over directly writing a a -> b. BVar is, in this situation, a zero-cost abstraction, performance-wise.

See documentation of backprop for more information.

gradBP :: ZeroFunc a -> OneFunc b -> (forall s. Reifies s W => BVar s a -> BVar s b) -> a -> a

gradBP, but with explicit zero and one.

backpropWith :: ZeroFunc a -> (forall s. Reifies s W => BVar s a -> BVar s b) -> a -> (b, b -> a)

backpropWith, but with explicit zero.

Note that argument order changed in v0.2.4.

Multiple inputs

evalBP0 :: (forall s. Reifies s W => BVar s a) -> a

evalBP but with no arguments. Useful when everything is just given through constVar.

backprop2 :: ZeroFunc a -> ZeroFunc b -> OneFunc c -> (forall s. Reifies s W => BVar s a -> BVar s b -> BVar s c) -> a -> b -> (c, (a, b))

backprop2, but with explicit zero and one.

evalBP2 :: (forall s. Reifies s W => BVar s a -> BVar s b -> BVar s c) -> a -> b -> c

evalBP for a two-argument function. See backprop2 for notes.

gradBP2 :: ZeroFunc a -> ZeroFunc b -> OneFunc c -> (forall s. Reifies s W => BVar s a -> BVar s b -> BVar s c) -> a -> b -> (a, b)

gradBP2 with explicit zero and one.

backpropWith2 :: ZeroFunc a -> ZeroFunc b -> (forall s. Reifies s W => BVar s a -> BVar s b -> BVar s c) -> a -> b -> (c, c -> (a, b))

backpropWith2, but with explicit zero.

Note that argument order changed in v0.2.4.

Since: 0.2.0.0

backpropN :: forall as b. Rec ZeroFunc as -> OneFunc b -> (forall s. Reifies s W => Rec (BVar s) as -> BVar s b) -> Rec Identity as -> (b, Rec Identity as)

backpropN, but with explicit zero and one.

evalBPN :: forall as b. (forall s. Reifies s W => Rec (BVar s) as -> BVar s b) -> Rec Identity as -> b

evalBP generalized to multiple inputs of different types. See documentation for backpropN for more details.

gradBPN :: Rec ZeroFunc as -> OneFunc b -> (forall s. Reifies s W => Rec (BVar s) as -> BVar s b) -> Rec Identity as -> Rec Identity as

gradBP, Nbut with explicit zero and one.

backpropWithN :: forall as b. Rec ZeroFunc as -> (forall s. Reifies s W => Rec (BVar s) as -> BVar s b) -> Rec Identity as -> (b, b -> Rec Identity as)

backpropWithN, but with explicit zero and one.

Note that argument order changed in v0.2.4.

Since: 0.2.0.0

class RPureConstrained (c :: u -> Constraint) (ts :: [u])

Build a record whose elements are derived solely from a constraint satisfied by each.

Minimal complete definition

rpureConstrained

Instances

RPureConstrained (c :: u -> Constraint) ([] :: [u])

Defined in Data.Vinyl.Core

Methods

rpureConstrained :: (forall (a :: u0). c a => f a) -> Rec f []

(c x, RPureConstrained c xs) => RPureConstrained (c :: a -> Constraint) (x ': xs :: [a])

Defined in Data.Vinyl.Core

Methods

rpureConstrained :: (forall (a0 :: u). c a0 => f a0) -> Rec f (x ': xs)

Manipulating BVar

constVar :: a -> BVar s a

Lift a value into a BVar representing a constant value.

This value will not be considered an input, and its gradients will not be backpropagated.

auto :: a -> BVar s a

Shorter alias for constVar, inspired by the ad library.

Since: 0.2.0.0

coerceVar :: Coercible a b => BVar s a -> BVar s b

Coerce a BVar contents. Useful for things like newtype wrappers.

Since: 0.1.5.2

viewVar :: forall a b s. Reifies s W => AddFunc a -> ZeroFunc b -> Lens' b a -> BVar s b -> BVar s a

viewVar, but with explicit add and zero.

setVar :: forall a b s. Reifies s W => AddFunc a -> AddFunc b -> ZeroFunc a -> Lens' b a -> BVar s a -> BVar s b -> BVar s b

setVar, but with explicit add and zero.

overVar :: Reifies s W => AddFunc a -> AddFunc b -> ZeroFunc a -> ZeroFunc b -> Lens' b a -> (BVar s a -> BVar s a) -> BVar s b -> BVar s b

overVar with explicit add and zero.

Since: 0.2.4.0

sequenceVar :: forall t a s. (Reifies s W, Traversable t) => AddFunc a -> ZeroFunc a -> BVar s (t a) -> t (BVar s a)

sequenceVar, but with explicit add and zero.

collectVar :: forall t a s. (Reifies s W, Foldable t, Functor t) => AddFunc a -> ZeroFunc a -> t (BVar s a) -> BVar s (t a)

collectVar, but with explicit add and zero.

previewVar :: forall b a s. Reifies s W => AddFunc a -> ZeroFunc b -> Traversal' b a -> BVar s b -> Maybe (BVar s a)

previewVar, but with explicit add and zero.

toListOfVar :: forall b a s. Reifies s W => AddFunc a -> ZeroFunc b -> Traversal' b a -> BVar s b -> [BVar s a]

toListOfVar, but with explicit add and zero.

With Isomorphisms

isoVar :: Reifies s W => AddFunc a -> (a -> b) -> (b -> a) -> BVar s a -> BVar s b

isoVar with explicit add and zero.

isoVar2 :: Reifies s W => AddFunc a -> AddFunc b -> (a -> b -> c) -> (c -> (a, b)) -> BVar s a -> BVar s b -> BVar s c

isoVar2 with explicit add and zero.

isoVar3 :: Reifies s W => AddFunc a -> AddFunc b -> AddFunc c -> (a -> b -> c -> d) -> (d -> (a, b, c)) -> BVar s a -> BVar s b -> BVar s c -> BVar s d

isoVar3 with explicit add and zero.

isoVarN :: Reifies s W => Rec AddFunc as -> (Rec Identity as -> b) -> (b -> Rec Identity as) -> Rec (BVar s) as -> BVar s b

isoVarN with explicit add and zero.

With Ops

liftOp :: forall as b s. Reifies s W => Rec AddFunc as -> Op as b -> Rec (BVar s) as -> BVar s b

liftOp, but with explicit add and zero.

liftOp1 :: forall a b s. Reifies s W => AddFunc a -> Op '[a] b -> BVar s a -> BVar s b

liftOp1, but with explicit add and zero.

liftOp2 :: forall a b c s. Reifies s W => AddFunc a -> AddFunc b -> Op '[a, b] c -> BVar s a -> BVar s b -> BVar s c

liftOp2, but with explicit add and zero.

liftOp3 :: forall a b c d s. Reifies s W => AddFunc a -> AddFunc b -> AddFunc c -> Op '[a, b, c] d -> BVar s a -> BVar s b -> BVar s c -> BVar s d

liftOp3, but with explicit add and zero.

Generics

splitBV

Arguments

:: (Generic (z f), Generic (z (BVar s)), BVGroup s as (Rep (z f)) (Rep (z (BVar s))), Reifies s W)

=> AddFunc (Rep (z f) ())

-> Rec AddFunc as

-> ZeroFunc (z f)

-> Rec ZeroFunc as

-> BVar s (z f)

BVar of value

-> z (BVar s)

BVars of fields

splitBV with explicit add and zero.

Since: 0.2.2.0

joinBV

Arguments

:: (Generic (z f), Generic (z (BVar s)), BVGroup s as (Rep (z f)) (Rep (z (BVar s))), Reifies s W)

=> AddFunc (z f)

-> Rec AddFunc as

-> ZeroFunc (Rep (z f) ())

-> Rec ZeroFunc as

-> z (BVar s)

BVars of fields

-> BVar s (z f)

BVar of combined value

joinBV with explicit add and zero.

Since: 0.2.2.0

class BVGroup s as i o | o -> i, i -> as

Helper class for generically "splitting" and "joining" BVars into constructors. See splitBV and joinBV.

See Numeric.Backprop for a tutorial on how to use this.

Instances should be available for types made with one constructor whose fields are all instances of Backprop, with a Generic instance.

Since: 0.2.2.0

Minimal complete definition

gsplitBV, gjoinBV

Instances

BVGroup s as i o => BVGroup s as (M1 p c i) (M1 p c o)

Defined in Numeric.Backprop.Explicit

Methods

gsplitBV :: Rec AddFunc as -> Rec ZeroFunc as -> BVar s (M1 p c i ()) -> M1 p c o ()

gjoinBV :: Rec AddFunc as -> Rec ZeroFunc as -> M1 p c o () -> BVar s (M1 p c i ())

BVGroup s ([] :: [Type]) (U1 :: Type -> Type) (U1 :: Type -> Type)

Defined in Numeric.Backprop.Explicit

Methods

gsplitBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> BVar s (U1 ()) -> U1 ()

gjoinBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> U1 () -> BVar s (U1 ())

BVGroup s ([] :: [Type]) (V1 :: Type -> Type) (V1 :: Type -> Type)

Defined in Numeric.Backprop.Explicit

Methods

gsplitBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> BVar s (V1 ()) -> V1 ()

gjoinBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> V1 () -> BVar s (V1 ())

BVGroup s ([] :: [Type]) (K1 i a :: Type -> Type) (K1 i (BVar s a) :: Type -> Type)

Defined in Numeric.Backprop.Explicit

Methods

gsplitBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> BVar s (K1 i a ()) -> K1 i (BVar s a) ()

gjoinBV :: Rec AddFunc [] -> Rec ZeroFunc [] -> K1 i (BVar s a) () -> BVar s (K1 i a ())

(Reifies s W, BVGroup s as i1 o1, BVGroup s bs i2 o2, cs ~ (as ++ bs), RecApplicative as) => BVGroup s (i1 () ': (i2 () ': cs)) (i1 :+: i2) (o1 :+: o2)

This instance is possible but it is not clear when it would be useful

Defined in Numeric.Backprop.Explicit

Methods

gsplitBV :: Rec AddFunc (i1 () ': (i2 () ': cs)) -> Rec ZeroFunc (i1 () ': (i2 () ': cs)) -> BVar s ((i1 :+: i2) ()) -> (o1 :+: o2) ()

gjoinBV :: Rec AddFunc (i1 () ': (i2 () ': cs)) -> Rec ZeroFunc (i1 () ': (i2 () ': cs)) -> (o1 :+: o2) () -> BVar s ((i1 :+: i2) ())

(Reifies s W, BVGroup s as i1 o1, BVGroup s bs i2 o2, cs ~ (as ++ bs), RecApplicative as) => BVGroup s (i1 () ': (i2 () ': cs)) (i1 :*: i2) (o1 :*: o2)

Defined in Numeric.Backprop.Explicit

Methods

gsplitBV :: Rec AddFunc (i1 () ': (i2 () ': cs)) -> Rec ZeroFunc (i1 () ': (i2 () ': cs)) -> BVar s ((i1 :*: i2) ()) -> (o1 :*: o2) ()

gjoinBV :: Rec AddFunc (i1 () ': (i2 () ': cs)) -> Rec ZeroFunc (i1 () ': (i2 () ': cs)) -> (o1 :*: o2) () -> BVar s ((i1 :*: i2) ())

Op

newtype Op as a

An Op as a describes a differentiable function from as to a.

For example, a value of type

Op '[Int, Bool] Double

is a function from an Int and a Bool, returning a Double. It can be differentiated to give a gradient of an Int and a Bool if given a total derivative for the Double. If we call Bool \(2\), then, mathematically, it is akin to a:

\[ f : \mathbb{Z} \times 2 \rightarrow \mathbb{R} \]

See runOp, gradOp, and gradOpWith for examples on how to run it, and Op for instructions on creating it.

It is simpler to not use this type constructor directly, and instead use the op2, op1, op2, and op3 helper smart constructors.

See Numeric.Backprop.Op for a mini-tutorial on using Rec and 'Rec Identity'.

To use an Op with the backprop library, see liftOp, liftOp1, liftOp2, and liftOp3.

Constructors

Op

Construct an Op by giving a function creating the result, and also a continuation on how to create the gradient, given the total derivative of a.

See the module documentation for Numeric.Backprop.Op for more details on the function that this constructor and Op expect.

Fields

• runOpWith :: Rec Identity as -> (a, a -> Rec Identity as)

  Run the function that the Op encodes, returning a continuation to compute the gradient, given the total derivative of a. See documentation for Numeric.Backprop.Op for more information.

Instances

(RPureConstrained Num as, Floating a) => Floating (Op as a)

Defined in Numeric.Backprop.Op

Methods

pi :: Op as a

exp :: Op as a -> Op as a

log :: Op as a -> Op as a

sqrt :: Op as a -> Op as a

(**) :: Op as a -> Op as a -> Op as a

logBase :: Op as a -> Op as a -> Op as a

sin :: Op as a -> Op as a

cos :: Op as a -> Op as a

tan :: Op as a -> Op as a

asin :: Op as a -> Op as a

acos :: Op as a -> Op as a

atan :: Op as a -> Op as a

sinh :: Op as a -> Op as a

cosh :: Op as a -> Op as a

tanh :: Op as a -> Op as a

asinh :: Op as a -> Op as a

acosh :: Op as a -> Op as a

atanh :: Op as a -> Op as a

log1p :: Op as a -> Op as a

expm1 :: Op as a -> Op as a

log1pexp :: Op as a -> Op as a

log1mexp :: Op as a -> Op as a

(RPureConstrained Num as, Fractional a) => Fractional (Op as a)

Defined in Numeric.Backprop.Op

Methods

(/) :: Op as a -> Op as a -> Op as a

recip :: Op as a -> Op as a

fromRational :: Rational -> Op as a

(RPureConstrained Num as, Num a) => Num (Op as a)

Defined in Numeric.Backprop.Op

Methods

(+) :: Op as a -> Op as a -> Op as a

(-) :: Op as a -> Op as a -> Op as a

(*) :: Op as a -> Op as a -> Op as a

negate :: Op as a -> Op as a

abs :: Op as a -> Op as a

signum :: Op as a -> Op as a

fromInteger :: Integer -> Op as a

Creation

op0 :: a -> Op '[] a

Create an Op that takes no inputs and always returns the given value.

There is no gradient, of course (using gradOp will give you an empty tuple), because there is no input to have a gradient of.

>>> runOp (op0 10) RNil
(10, RNil)

For a constant Op that takes input and ignores it, see opConst and opConst'.

opConst :: forall as a. RPureConstrained Num as => a -> Op as a

An Op that ignores all of its inputs and returns a given constant value.

>>> gradOp' (opConst 10) (1 :& 2 :& 3 :& RNil)
(10, 0 :& 0 :& 0 :& RNil)

idOp :: Op '[a] a

An Op that just returns whatever it receives. The identity function.

idOp = opIso id id

bpOp :: Rec ZeroFunc as -> (forall s. Reifies s W => Rec (BVar s) as -> BVar s b) -> Op as b

bpOp with explicit zero.

Giving gradients directly

op1 :: (a -> (b, b -> a)) -> Op '[a] b

Create an Op of a function taking one input, by giving its explicit derivative. The function should return a tuple containing the result of the function, and also a function taking the derivative of the result and return the derivative of the input.

If we have

\[ \eqalign{ f &: \mathbb{R} \rightarrow \mathbb{R}\cr y &= f(x)\cr z &= g(y) } \]

Then the derivative \( \frac{dz}{dx} \), it would be:

\[ \frac{dz}{dx} = \frac{dz}{dy} \frac{dy}{dx} \]

If our Op represents \(f\), then the second item in the resulting tuple should be a function that takes \(\frac{dz}{dy}\) and returns \(\frac{dz}{dx}\).

As an example, here is an Op that squares its input:

square :: Num a => Op '[a] a
square = op1 $ \x -> (x*x, \d -> 2 * d * x
                     )

Remember that, generally, end users shouldn't directly construct Ops; they should be provided by libraries or generated automatically.

op2 :: (a -> b -> (c, c -> (a, b))) -> Op '[a, b] c

Create an Op of a function taking two inputs, by giving its explicit gradient. The function should return a tuple containing the result of the function, and also a function taking the derivative of the result and return the derivative of the input.

If we have

\[ \eqalign{ f &: \mathbb{R}^2 \rightarrow \mathbb{R}\cr z &= f(x, y)\cr k &= g(z) } \]

Then the gradient \( \left< \frac{\partial k}{\partial x}, \frac{\partial k}{\partial y} \right> \) would be:

\[ \left< \frac{\partial k}{\partial x}, \frac{\partial k}{\partial y} \right> = \left< \frac{dk}{dz} \frac{\partial z}{dx}, \frac{dk}{dz} \frac{\partial z}{dy} \right> \]

If our Op represents \(f\), then the second item in the resulting tuple should be a function that takes \(\frac{dk}{dz}\) and returns \( \left< \frac{\partial k}{dx}, \frac{\partial k}{dx} \right> \).

As an example, here is an Op that multiplies its inputs:

mul :: Num a => Op '[a, a] a
mul = op2' $ \x y -> (x*y, \d -> (d*y, x*d)
                     )

Remember that, generally, end users shouldn't directly construct Ops; they should be provided by libraries or generated automatically.

op3 :: (a -> b -> c -> (d, d -> (a, b, c))) -> Op '[a, b, c] d

Create an Op of a function taking three inputs, by giving its explicit gradient. See documentation for op2 for more details.

From Isomorphisms

opCoerce :: Coercible a b => Op '[a] b

An Op that coerces an item into another item whose type has the same runtime representation.

>>> gradOp' opCoerce (Identity 5) :: (Int, Identity Int)
(5, Identity 1)

opCoerce = opIso coerced coerce

opTup :: Op as (Rec Identity as)

An Op that takes as and returns exactly the input tuple.

>>> gradOp' opTup (1 :& 2 :& 3 :& RNil)
(1 :& 2 :& 3 :& RNil, 1 :& 1 :& 1 :& RNil)

opIso :: (a -> b) -> (b -> a) -> Op '[a] b

An Op that runs the input value through an isomorphism.

Warning: This is unsafe! It assumes that the isomorphisms themselves have derivative 1, so will break for things like exp & log. Basically, don't use this for any "numeric" isomorphisms.

opIsoN :: (Rec Identity as -> b) -> (b -> Rec Identity as) -> Op as b

An Op that runs the input value through an isomorphism between a tuple of values and a value. See opIso for caveats.

In Numeric.Backprop.Op since version 0.1.2.0, but only exported from Numeric.Backprop since version 0.1.3.0.

Since: 0.1.2.0

opLens :: Num a => Lens' a b -> Op '[a] b

An Op that extracts a value from an input value using a Lens'.

Warning: This is unsafe! It assumes that it extracts a specific value unchanged, with derivative 1, so will break for things that numerically manipulate things before returning them.

No gradients

noGrad1 :: (a -> b) -> Op '[a] b

Create an Op with no gradient. Can be evaluated with evalOp, but will throw a runtime exception when asked for the gradient.

Can be used with BVar with liftOp1, and evalBP will work fine. gradBP and backprop will also work fine if the result is never used in the final answer, but will throw a runtime exception if the final answer depends on the result of this operation.

Useful if your only API is exposed through backprop. Just be sure to tell your users that this will explode when finding the gradient if the result is used in the final result.

Since: 0.1.3.0

noGrad :: (Rec Identity as -> b) -> Op as b

Create an Op with no gradient. Can be evaluated with evalOp, but will throw a runtime exception when asked for the gradient.

Can be used with BVar with liftOp, and evalBP will work fine. gradBP and backprop will also work fine if the result is never used in the final answer, but will throw a runtime exception if the final answer depends on the result of this operation.

Useful if your only API is exposed through backprop. Just be sure to tell your users that this will explode when finding the gradient if the result is used in the final result.

Since: 0.1.3.0

Utility

class Reifies (s :: k) a | s -> a

Minimal complete definition

reflect

Instances

KnownNat n => Reifies (n :: Nat) Integer

Defined in Data.Reflection

Methods

reflect :: proxy n -> Integer

KnownSymbol n => Reifies (n :: Symbol) String

Defined in Data.Reflection

Methods

reflect :: proxy n -> String

Reifies Z Int

Defined in Data.Reflection

Methods

reflect :: proxy Z -> Int

Reifies n Int => Reifies (D n :: Type) Int

Defined in Data.Reflection

Methods

reflect :: proxy (D n) -> Int

Reifies n Int => Reifies (SD n :: Type) Int

Defined in Data.Reflection

Methods

reflect :: proxy (SD n) -> Int

Reifies n Int => Reifies (PD n :: Type) Int

Defined in Data.Reflection

Methods

reflect :: proxy (PD n) -> Int

(B b0, B b1, B b2, B b3, B b4, B b5, B b6, B b7, w0 ~ W b0 b1 b2 b3, w1 ~ W b4 b5 b6 b7) => Reifies (Stable w0 w1 a :: Type) a

Defined in Data.Reflection

Methods

reflect :: proxy (Stable w0 w1 a) -> a