Portability | GHC only? |
---|---|

Stability | experimental |

Maintainer | bjorn.buckwalter@gmail.com |

Forward Automatic Differentiation via overloading to perform nonstandard interpretation that replaces original numeric type with corresponding generalized dual number type.

Credits:

Authors: Copyright 2008, Barak A. Pearlmutter (barak@cs.nuim.ie) & Jeffrey Mark Siskind (qobi@purdue.edu)

Work started as stripped-down version of higher-order tower code published by Jerzy Karczmarczuk (jerzy.karczmarczuk@info.unicaen.fr) which used a non-standard standard prelude.

Initial perturbation-confusing code is a modified version of http://cdsmith.wordpress.com/2007/11/29/some-playing-with-derivatives/

Tag trick, called "branding" in the Haskell community, from Björn Buckwalter (bjorn.buckwalter@gmail.com) http://thread.gmane.org/gmane.comp.lang.haskell.cafe/22308/

Notes:

Each invocation of the differentiation function introduces a distinct perturbation, which requires a distinct dual number type. In order to prevent these from being confused, tagging, called branding in the Haskell community, is used. This seems to prevent perturbation confusion, although it would be nice to have an actual proof of this. The technique does require adding invocations of lift at appropriate places when nesting is present.

- data Dual tag a
- lift :: Num a => a -> Dual tag a
- diffUU :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> b
- diffUF :: (Num a, Num b, Functor f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> f b
- diffMU :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [a] -> [a] -> b
- diffMF :: (Num a, Num b, Functor f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [a] -> [a] -> f b
- diff2UU :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> (b, b)
- diff2UF :: (Num a, Num b, Functor f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> (f b, f b)
- diff2MU :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [a] -> [a] -> (b, b)
- diff2MF :: (Num a, Num b, Functor f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [a] -> [a] -> (f b, f b)
- diffsUU :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> [b]
- diffsUF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> [f b]
- diffsMU :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [[a]] -> [b]
- diffsMF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [[a]] -> [f b]
- diffs0UU :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> [b]
- diffs0UF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> [f b]
- diffs0MU :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [[a]] -> [b]
- diffs0MF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [[a]] -> [f b]
- diff :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> b
- diff2 :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> (b, b)
- diffs :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> [b]
- diffs0 :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> [b]
- grad :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [a] -> [b]
- jacobian :: (Num a, Num b) => (forall tag. [Dual tag a] -> [Dual tag b]) -> [a] -> [[b]]
- zeroNewton :: Fractional a => (forall tag. Dual tag a -> Dual tag a) -> a -> [a]
- inverseNewton :: Fractional a => (forall tag. Dual tag a -> Dual tag a) -> a -> a -> [a]
- fixedPointNewton :: Fractional a => (forall tag. Dual tag a -> Dual tag a) -> a -> [a]
- extremumNewton :: Fractional a => (forall tag tag1. Dual tag1 (Dual tag a) -> Dual tag1 (Dual tag a)) -> a -> [a]
- argminNaiveGradient :: (Fractional a, Ord a) => (forall tag. [Dual tag a] -> Dual tag a) -> [a] -> [[a]]
- taylor :: Fractional a => (forall tag. Dual tag a -> Dual tag a) -> a -> a -> [a]

# Higher-Order Dual Numbers

The `Dual`

type is a concrete representation of a higher-order
Dual number, meaning a number augmented with a tower of
derivatives. These generalize the Dual numbers of Clifford (1873),
which hold only a first derivative. They can be converted to
formal power series via division by the sequence of factorials.

(Enum a, Num a) => Enum (Dual tag a) | |

(Eq a, Num a) => Eq (Dual tag a) | |

Floating a => Floating (Dual tag a) | |

Fractional a => Fractional (Dual tag a) | |

Num a => Num (Dual tag a) | |

(Ord a, Num a) => Ord (Dual tag a) | |

Real a => Real (Dual tag a) | |

(RealFloat a, RealFrac a) => RealFloat (Dual tag a) | |

RealFrac a => RealFrac (Dual tag a) | |

Show a => Show (Dual tag a) |

# First-Order Differentiation Operators

These have two-letter suffices for the arity of the input and output of the passed functions: U for univariate, meaning a number, M for multivariate, meaning a list of numbers.

When the input is multivariate a directional derivative is calculated; this requires an additional "direction" parameter. The multivariate case is treated as a list (on input) and as a functor of arbitrary shape, which includes lists as a special case, on output.

Naming convention:

`diff{U/M}{U/F}`

- Derivative-taking operators that return a primal/first-derivative pair, for all combinations of scalar/nonscalar input & output
`diff2{U/M}{U/F}`

- Derivative-taking operators that calculate a (primal, first-derivative) pair, for all combinations of scalar/nonscalar input & output

diffUU :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> bSource

The `diffUU`

function calculates the first derivative of a
scalar-to-scalar function.

diffUF :: (Num a, Num b, Functor f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> f bSource

The `diffUF`

function calculates the first derivative of
scalar-to-nonscalar function.

diffMU :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [a] -> [a] -> bSource

The `diffMU`

function calculate the product of the Jacobian of a
nonscalar-to-scalar function with a given vector. Aka: directional
derivative.

diffMF :: (Num a, Num b, Functor f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [a] -> [a] -> f bSource

The `diffMF`

function calculates the product of the Jacobian of a
nonscalar-to-nonscalar function with a given vector. Aka:
directional derivative.

diff2UU :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> (b, b)Source

The `diff2UU`

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

diff2UF :: (Num a, Num b, Functor f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> (f b, f b)Source

The `diffUF2`

function calculates the value and derivative, as a
pair, of a scalar-to-nonscalar function.

diff2MU :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [a] -> [a] -> (b, b)Source

The `diffMU2`

function calculates the value and directional
derivative, as a pair, of a nonscalar-to-scalar function.

diff2MF :: (Num a, Num b, Functor f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [a] -> [a] -> (f b, f b)Source

The `diffMF2`

function calculates the value and directional
derivative, as a pair, of a nonscalar-to-nonscalar function.

# Higher-Order Differentiation Operators

Naming convention:

`diffs{U/M}{U/F}`

- : Derivative-taking operators that return a list [primal, first-derivative, 2nd-derivative, ...], for all combinations of scalar/nonscalar input & output.

diffsUU :: (Num a, Num b) => (forall tag. Dual tag a -> Dual tag b) -> a -> [b]Source

The `diffsUU`

function calculates a list of derivatives of a
scalar-to-scalar function. The 0-th element of the list is the
primal value, the 1-st element is the first derivative, etc.

diffsUF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> [f b]Source

The `diffsUF`

function calculates an infinite list of derivatives
of a scalar-to-nonscalar function. The 0-th element of the list is
the primal value, the 1-st element is the first derivative, etc.

diffsMU :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [[a]] -> [b]Source

The `diffsMU`

function calculates an infinite list of derivatives
of a nonscalar-to-scalar function. The 0-th element of the list is
the primal value, the 1-st element is the first derivative, etc.
The input is a (possibly truncated) list of the primal, first
derivative, etc, of the input.

diffsMF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [[a]] -> [f b]Source

The `diffsMF`

function calculates an infinite list of derivatives
of a nonscalar-to-nonscalar function. The 0-th element of the list
is the primal value, the 1-st element is the first derivative, etc.
The input is a (possibly truncated) list of the primal, first
derivative, etc, of the input.

diffs0UF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. Dual tag a -> f (Dual tag b)) -> a -> [f b]Source

diffs0MF :: (Num a, Num b, Functor f, Foldable f) => (forall tag. [Dual tag a] -> f (Dual tag b)) -> [[a]] -> [f b]Source

# Common access patterns

grad :: (Num a, Num b) => (forall tag. [Dual tag a] -> Dual tag b) -> [a] -> [b]Source

The `grad`

function calculates the gradient of a
nonscalar-to-scalar function, using n invocations of forward AD,
where n is the input dimmensionality. NOTE: this is O(n)
inefficient as compared to reverse AD.

jacobian :: (Num a, Num b) => (forall tag. [Dual tag a] -> [Dual tag b]) -> [a] -> [[b]]Source

The `jacobian`

function calcualtes the Jacobian of a
nonscalar-to-nonscalar function, using n invocations of forward AD,
where n is the input dimmensionality.

# Optimization Routines

zeroNewton :: Fractional a => (forall tag. Dual tag a -> Dual tag a) -> a -> [a]Source

The `zeroNewton`

function finds a zero of a scalar function using
Newton's method; its output is a stream of increasingly accurate
results. (Modulo the usual caveats.)

TEST CASE:
`take 10 $ zeroNewton (\x->x^2-4) 1 -- converge to 2.0`

TEST CASE
:module Data.Complex Numeric.FAD
`take 10 $ zeroNewton ((+1).(^2)) (1 :+ 1) -- converge to (0 :+ 1)`

inverseNewton :: Fractional a => (forall tag. Dual tag a -> Dual tag a) -> a -> a -> [a]Source

The `inverseNewton`

function inverts a scalar function using
Newton's method; its output is a stream of increasingly accurate
results. (Modulo the usual caveats.)

TEST CASE:
`take 10 $ inverseNewton sqrt 1 (sqrt 10) -- converge to 10`

fixedPointNewton :: Fractional a => (forall tag. Dual tag a -> Dual tag a) -> a -> [a]Source

The `fixedPointNewton`

function find a fixedpoint of a scalar
function using Newton's method; its output is a stream of
increasingly accurate results. (Modulo the usual caveats.)

extremumNewton :: Fractional a => (forall tag tag1. Dual tag1 (Dual tag a) -> Dual tag1 (Dual tag a)) -> a -> [a]Source

The `extremumNewton`

function finds an extremum of a scalar
function using Newton's method; produces a stream of increasingly
accurate results. (Modulo the usual caveats.)

argminNaiveGradient :: (Fractional a, Ord a) => (forall tag. [Dual tag a] -> Dual tag a) -> [a] -> [[a]]Source

The `argminNaiveGradient`

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.) The
gradient is calculated using Forward AD, which is O(n) inefficient
as compared to Reverse AD, where n is the input dimensionality.