{-# LANGUAGE Rank2Types, TemplateHaskell, BangPatterns, MultiParamTypeClasses, FunctionalDependencies, FlexibleInstances, UndecidableInstances, ScopedTypeVariables #-} ----------------------------------------------------------------------------- -- | -- Module : Numeric.AD.Mode.Reverse -- Copyright : (c) Edward Kmett 2010 -- License : BSD3 -- Maintainer : ekmett@gmail.com -- Stability : experimental -- Portability : GHC only -- -- Mixed-Mode Automatic Differentiation. -- -- For reverse mode AD we use 'System.Mem.StableName.StableName' to recover sharing information from -- the tape to avoid combinatorial explosion, and thus run asymptotically faster -- than it could without such sharing information, but the use of side-effects -- contained herein is benign. -- ----------------------------------------------------------------------------- module Numeric.AD.Mode.Reverse ( -- * Gradient grad , grad' , gradWith , gradWith' -- * Jacobian , jacobian , jacobian' , jacobianWith , jacobianWith' -- * Hessian , hessian , hessianF -- * Derivatives , diff , diff' , diffF , diffF' -- * Unsafe Variadic Gradient , vgrad, vgrad' , Grad ) where import Control.Applicative ((<$>)) import Data.Traversable (Traversable) import Numeric.AD.Types import Numeric.AD.Internal.Classes import Numeric.AD.Internal.Composition import Numeric.AD.Internal.Reverse -- | The 'grad' function calculates the gradient of a non-scalar-to-scalar function with 'Reverse' AD in a single pass. grad :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f a grad f as = unbind vs (partialArray bds $ f vs) where (vs,bds) = bind as {-# INLINE grad #-} -- | The 'grad'' function calculates the result and gradient of a non-scalar-to-scalar function with 'Reverse' AD in a single pass. grad' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f a) grad' f as = (primal r, unbind vs $ partialArray bds r) where (vs, bds) = bind as r = f vs {-# INLINE grad' #-} -- | @'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. Mode s => f (AD s a) -> AD s a) -> f a -> f b gradWith g f as = unbindWith g vs (partialArray bds $ f vs) where (vs,bds) = bind as {-# INLINE gradWith #-} -- | @'grad'' g f@ calculates the result and gradient of a non-scalar-to-scalar function @f@ with 'Reverse' AD in a single pass -- the gradient is combined element-wise with the argument using the function @g@. -- -- > grad' == gradWith' (\_ dx -> dx) gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f b) gradWith' g f as = (primal r, unbindWith g vs $ partialArray bds r) where (vs, bds) = bind as r = f vs {-# INLINE gradWith' #-} -- | 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 :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f a) jacobian f as = unbind vs . partialArray bds <$> f vs where (vs, bds) = bind as {-# INLINE jacobian #-} -- | The 'jacobian'' function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using @m@ invocations of reverse AD, -- where @m@ is the output dimensionality. Applying @fmap snd@ to the result will recover the result of 'jacobian' -- | An alias for 'gradF'' jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f a) jacobian' f as = row <$> f vs where (vs, bds) = bind as row a = (primal a, unbind vs (partialArray bds a)) {-# INLINE jacobian' #-} -- | '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 <$> f x) -- jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f b) jacobianWith g f as = unbindWith g vs . partialArray bds <$> f vs where (vs, bds) = bind as {-# INLINE jacobianWith #-} -- | '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) -- jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f b) jacobianWith' g f as = row <$> f vs where (vs, bds) = bind as row a = (primal a, unbindWith g vs (partialArray bds a)) {-# INLINE jacobianWith' #-} diff :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> a diff f a = derivative $ f (var a 0) {-# INLINE diff #-} -- | The 'd'' function calculates the value and derivative, as a -- pair, of a scalar-to-scalar function. diff' :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a) diff' f a = derivative' $ f (var a 0) {-# INLINE diff' #-} diffF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f a diffF f a = derivative <$> f (var a 0) {-# INLINE diffF #-} diffF' :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f (a, a) diffF' f a = derivative' <$> f (var a 0) {-# INLINE diffF' #-} -- | Compute the hessian via the jacobian of the gradient. gradient is computed in reverse mode and then the jacobian is computed in reverse mode. -- -- However, since the @'grad f :: f a -> f a'@ is square this is not as fast as using the forward-mode Jacobian of a reverse mode gradient provided by 'Numeric.AD.hessian'. hessian :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f (f a) hessian f = jacobian (grad (decomposeMode . f . fmap composeMode)) -- | Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the reverse-mode Jacobian of the reverse-mode Jacobian of the function. -- -- Less efficient than 'Numeric.AD.Mode.Mixed.hessianF'. hessianF :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f (f a)) hessianF f = decomposeFunctor . jacobian (ComposeFunctor . jacobian (fmap decomposeMode . f . fmap composeMode))