Copyright	(c) Edward Kmett 2010-2014
License	BSD3
Maintainer	ekmett@gmail.com
Stability	experimental
Portability	GHC only
Safe Haskell	None
Language	Haskell2010

Numeric.AD.Rank1.Kahn

Contents

Gradient
Jacobian
Hessian
Derivatives
Unsafe Variadic Gradient

Description

This module provides reverse-mode Automatic Differentiation using post-hoc linear time topological sorting.

For reverse mode AD we use StableName to recover sharing information from the tape to avoid combinatorial explosion, and thus run asymptotically faster than it could without such sharing information, but the use of side-effects contained herein is benign.

Synopsis

Documentation

data Kahn a Source

Kahn is a Mode using reverse-mode automatic differentiation that provides fast diffFU, diff2FU, grad, grad2 and a fast jacobian when you have a significantly smaller number of outputs than inputs.

Instances

(Num a, Bounded a) => Bounded (Kahn a)
(Num a, Enum a) => Enum (Kahn a)
(Num a, Eq a) => Eq (Kahn a)
Floating a => Floating (Kahn a)
Fractional a => Fractional (Kahn a)
Num a => Num (Kahn a)
(Num a, Ord a) => Ord (Kahn a)
Real a => Real (Kahn a)
RealFloat a => RealFloat (Kahn a)
RealFrac a => RealFrac (Kahn a)
Show a => Show (Kahn a)
MuRef (Kahn a)
Erf a => Erf (Kahn a)
InvErf a => InvErf (Kahn a)
Num a => Mode (Kahn a)
Num a => Jacobian (Kahn a)
Num a => Grad (Kahn a) [a] (a, [a]) a
Typeable (* -> *) Kahn
Grad i o o' a => Grad (Kahn a -> i) (a -> o) (a -> o') a
type DeRef (Kahn a) = Tape a
type Scalar (Kahn a) = a
type D (Kahn a) = Id a

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

Embed a constant

Gradient

grad :: (Traversable f, Num a) => (f (Kahn a) -> Kahn a) -> f a -> f a Source

The grad function calculates the gradient of a non-scalar-to-scalar function with kahn-mode AD in a single pass.

>>> grad (\[x,y,z] -> x*y+z) [1,2,3]
[2,1,1]

grad' :: (Traversable f, Num a) => (f (Kahn a) -> Kahn a) -> f a -> (a, f a) Source

The grad' function calculates the result and gradient of a non-scalar-to-scalar function with kahn-mode AD in a single pass.

>>> grad' (\[x,y,z] -> 4*x*exp y+cos z) [1,2,3]
(28.566231899122155,[29.5562243957226,29.5562243957226,-0.1411200080598672])

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (f (Kahn a) -> Kahn a) -> f a -> f b Source

grad g f function calculates the gradient of a non-scalar-to-scalar function f with kahn-mode AD in a single pass. The gradient is combined element-wise with the argument using the function g.

grad = gradWith (_ dx -> dx)
id = gradWith const

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (f (Kahn a) -> Kahn a) -> f a -> (a, f b) Source

grad' g f calculates the result and gradient of a non-scalar-to-scalar function f with kahn-mode AD in a single pass the gradient is combined element-wise with the argument using the function g.

grad' == gradWith' (_ dx -> dx)

Jacobian

jacobian :: (Traversable f, Functor g, Num a) => (f (Kahn a) -> g (Kahn a)) -> f a -> g (f a) Source

The jacobian function calculates the jacobian of a non-scalar-to-non-scalar function with kahn AD lazily in m passes for m outputs.

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

>>> jacobian (\[x,y] -> [exp y,cos x,x+y]) [1,2]
[[0.0,7.38905609893065],[-0.8414709848078965,0.0],[1.0,1.0]]

jacobian' :: (Traversable f, Functor g, Num a) => (f (Kahn a) -> g (Kahn 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 kahn AD, where m is the output dimensionality. Applying fmap snd to the result will recover the result of jacobian | An alias for gradF'

ghci> jacobian' ([x,y] -> [y,x,x*y]) [2,1] [(1,[0,1]),(2,[1,0]),(2,[1,2])]

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Kahn a) -> g (Kahn a)) -> f a -> g (f b) Source

'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function f with kahn AD lazily in m passes for m outputs.

Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.

jacobian = jacobianWith (_ dx -> dx)
jacobianWith const = (f x -> const x <$> f x)

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Kahn a) -> g (Kahn 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 kahn AD, where m is the output dimensionality. Applying fmap snd to the result will recover the result of jacobianWith

Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.

jacobian' == jacobianWith' (_ dx -> dx)

Hessian

hessian :: (Traversable f, Num a) => (f (On (Kahn (Kahn a))) -> On (Kahn (Kahn a))) -> f a -> f (f a) Source

Compute the hessian via the jacobian of the gradient. gradient is computed in Kahn mode and then the jacobian is computed in Kahn mode.

However, since the grad f :: f a -> f a is square this is not as fast as using the forward-mode jacobian of a reverse mode gradient provided by hessian.

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

hessianF :: (Traversable f, Functor g, Num a) => (f (On (Kahn (Kahn a))) -> g (On (Kahn (Kahn a)))) -> f a -> g (f (f a)) Source

Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the Kahn-mode Jacobian of the Kahn-mode Jacobian of the function.

Less efficient than hessianF.

>>> hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2]
[[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.1312043837568135,-2.4717266720048188],[-2.4717266720048188,1.1312043837568135]]]

Derivatives

diff :: Num a => (Kahn a -> Kahn a) -> a -> a Source

Compute the derivative of a function.

>>> diff sin 0
1.0

>>> cos 0
1.0

diff' :: Num a => (Kahn a -> Kahn a) -> a -> (a, a) Source

The diff' function calculates the value and derivative, as a pair, of a scalar-to-scalar function.

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

diffF :: (Functor f, Num a) => (Kahn a -> f (Kahn a)) -> a -> f a Source

Compute the derivatives of a function that returns a vector with regards to its single input.

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

diffF' :: (Functor f, Num a) => (Kahn a -> f (Kahn a)) -> a -> f (a, a) Source

Compute the derivatives of a function that returns a vector with regards to its single input as well as the primal answer.

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

Unsafe Variadic Gradient

Variadic combinators for variadic mixed-mode automatic differentiation.

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

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

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

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