Numeric.AD.Rank1.Kahn

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.

Documentation

data Kahn a

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

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

Embed a constant

Gradient

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

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)

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

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)

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)

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)

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)

'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)

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)

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))

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

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)

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

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)

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

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

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

Minimal complete definition

pack, unpack, unpack'

Instances

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

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