Numeric.AD

Mixed-Mode Automatic Differentiation.

Each combinator exported from this module chooses an appropriate AD mode. The following basic operations are supported, modified as appropriate by the suffixes below:

• grad computes the gradient (partial derivatives) of a function at a point
• jacobian computes the Jacobian matrix of a function at a point
• diff computes the derivative of a function at a point
• du computes a directional derivative of a function at a point
• hessian compute the Hessian matrix (matrix of second partial derivatives) of a function at a point

The suffixes have the following meanings:

• ' -- also return the answer
• With lets the user supply a function to blend the input with the output
• F is a version of the base function lifted to return a Traversable (or Functor) result
• s means the function returns all higher derivatives in a list or f-branching Stream
• T means the result is transposed with respect to the traditional formulation.
• 0 means that the resulting derivative list is padded with 0s at the end.

AD modes

class (Num t, Num (Scalar t)) => Mode t where

Methods

auto :: Scalar t -> t

Embed a constant

Instances

Mode (ForwardDouble s)

(Mode t, Mode (Scalar t)) => Mode (On t)

Num a => Mode (Tower a s)

Num a => Mode (Id a s)

Num a => Mode (Sparse a s)

(Num a, Reifies * s Tape) => Mode (Reverse a s)

Num a => Mode (Kahn a s)

Num a => Mode (Forward a s)

(Mode a, Mode b, Chosen s, ~ * (Scalar a) (Scalar b)) => Mode (Or a b s)

(Num a, Traversable f) => Mode (Dense f a s)

type family Scalar t :: *

Gradients (Reverse Mode)

grad :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> f a

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

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

grad' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> (a, f a)

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

>>> grad' (\[x,y,z] -> x*y+z) [1,2,3]
(5,[2,1,1])

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> f b

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. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> (a, f b)

grad' g f calculates the result and 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)

Higher Order Gradients (Sparse-on-Reverse)

grads :: (Traversable f, Num a) => (forall s. f (Sparse a s) -> Sparse a s) -> f a -> Cofree f a

Variadic Gradients (Sparse or Kahn)

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!

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

Instances

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

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

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

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

class Num a => Grads i o a | i -> a o, o -> a i

Instances

Grads i o a => Grads (Sparse a () -> i) (a -> o) a

Num a => Grads (Sparse a ()) (Cofree [] a) a

vgrads :: Grads i o a => i -> o

Jacobians (Sparse or Reverse)

jacobian :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse a s) -> g (Reverse a s)) -> f a -> g (f a)

Calculate the Jacobian of a non-scalar-to-non-scalar function, automatically choosing between sparse and Reverse mode AD.

If you know that you have relatively many outputs per input, consider using jacobian.

>>> jacobian (\[x,y] -> [y,x,x+y,x*y,exp x * sin y]) [pi,1]
[[0.0,1.0],[1.0,0.0],[1.0,1.0],[1.0,3.141592653589793],[19.472221418841606,12.502969588876512]]

jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse a s) -> g (Reverse a s)) -> f a -> g (a, f a)

Calculate both the answer and Jacobian of a non-scalar-to-non-scalar function, using reverse-mode AD.

If you have relatively many outputs per input, consider using jacobian'.

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse a s) -> g (Reverse a s)) -> f a -> g (f b)

jacobianWith g f calculates the Jacobian of a non-scalar-to-non-scalar function, using Reverse mode AD.

The resulting Jacobian matrix is then recombined element-wise with the input using g.

If you know that you have relatively many outputs per input, consider using jacobianWith.

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse a s) -> g (Reverse a s)) -> f a -> g (a, f b)

jacobianWith' g f calculates the answer and Jacobian of a non-scalar-to-non-scalar function, using Reverse mode AD.

The resulting Jacobian matrix is then recombined element-wise with the input using g.

If you know that you have relatively many outputs per input, consider using jacobianWith'.

Higher Order Jacobian (Sparse-on-Reverse)

jacobians :: (Traversable f, Functor g, Num a) => (forall s. f (Sparse a s) -> g (Sparse a s)) -> f a -> g (Cofree f a)

Transposed Jacobians (Forward Mode)

jacobianT :: (Traversable f, Functor g, Num a) => (forall s. f (Forward a s) -> g (Forward a s)) -> f a -> f (g a)

A fast, simple, transposed Jacobian computed with forward-mode AD.

jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (Forward a s) -> g (Forward a s)) -> f a -> f (g b)

A fast, simple, transposed Jacobian computed with Forward mode AD that combines the output with the input.

Hessian (Sparse-On-Reverse)

hessian :: (Traversable f, Num a) => (forall s s'. Reifies s Tape => f (On (Reverse (Sparse a s') s)) -> On (Reverse (Sparse a s') s)) -> f a -> f (f a)

Compute the Hessian via the Jacobian of the gradient. gradient is computed in reverse mode and then the Jacobian is computed in sparse (forward) mode.

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

hessian' :: (Traversable f, Num a) => (forall s. f (Sparse a s) -> Sparse a s) -> f a -> (a, f (a, f a))

Hessian Tensors (Sparse or Sparse-On-Reverse)

hessianF :: (Traversable f, Functor g, Num a) => (forall s s'. Reifies s Tape => f (On (Reverse (Sparse a s') s)) -> g (On (Reverse (Sparse a s') s))) -> f a -> g (f (f a))

Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function using 'Sparse'-on-'Reverse'

>>> 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]]]

Hessian Tensors (Sparse)

hessianF' :: (Traversable f, Functor g, Num a) => (forall s. f (Sparse a s) -> g (Sparse a s)) -> f a -> g (a, f (a, f a))

Hessian Vector Products (Forward-On-Reverse)

hessianProduct :: (Traversable f, Num a) => (forall s s'. Reifies s Tape => f (On (Reverse (Forward a s') s)) -> On (Reverse (Forward a s') s)) -> f (a, a) -> f a

hessianProduct f wv computes the product of the hessian H of a non-scalar-to-scalar function f at w = fst $ wv with a vector v = snd $ wv using "Pearlmutter's method" from http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.6143, which states:

 H v = (d/dr) grad_w (w + r v) | r = 0

Or in other words, we take the directional derivative of the gradient. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.

hessianProduct' :: (Traversable f, Num a) => (forall s s'. Reifies s Tape => f (On (Reverse (Forward a s') s)) -> On (Reverse (Forward a s') s)) -> f (a, a) -> f (a, a)

hessianProduct' f wv computes both the gradient of a non-scalar-to-scalar f at w = fst $ wv and the product of the hessian H at w with a vector v = snd $ wv using "Pearlmutter's method". The outputs are returned wrapped in the same functor.

 H v = (d/dr) grad_w (w + r v) | r = 0

Or in other words, we return the gradient and the directional derivative of the gradient. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.

Derivatives (Forward Mode)

diff :: Num a => (forall s. Forward a s -> Forward a s) -> a -> a

The diff function calculates the first derivative of a scalar-to-scalar function by forward-mode AD

>>> diff sin 0
1.0

diffF :: (Functor f, Num a) => (forall s. Forward a s -> f (Forward a s)) -> a -> f a

The diffF function calculates the first derivatives of scalar-to-nonscalar function by Forward mode AD

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

diff' :: Num a => (forall s. Forward a s -> Forward a s) -> a -> (a, a)

The diff' function calculates the result and first derivative of scalar-to-scalar function by Forward mode AD

 diff' sin == sin &&& cos
 diff' f = f &&& d f

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

>>> diff' exp 0
(1.0,1.0)

diffF' :: (Functor f, Num a) => (forall s. Forward a s -> f (Forward a s)) -> a -> f (a, a)

The diffF' function calculates the result and first derivatives of a scalar-to-non-scalar function by Forward mode AD

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

Derivatives (Tower)

diffs :: Num a => (forall s. Tower a s -> Tower a s) -> a -> [a]

diffsF :: (Functor f, Num a) => (forall s. Tower a s -> f (Tower a s)) -> a -> f [a]

diffs0 :: Num a => (forall s. Tower a s -> Tower a s) -> a -> [a]

diffs0F :: (Functor f, Num a) => (forall s. Tower a s -> f (Tower a s)) -> a -> f [a]

Directional Derivatives (Forward Mode)

du :: (Functor f, Num a) => (forall s. f (Forward a s) -> Forward a s) -> f (a, a) -> a

Compute the directional derivative of a function given a zipped up Functor of the input values and their derivatives

du' :: (Functor f, Num a) => (forall s. f (Forward a s) -> Forward a s) -> f (a, a) -> (a, a)

Compute the answer and directional derivative of a function given a zipped up Functor of the input values and their derivatives

duF :: (Functor f, Functor g, Num a) => (forall s. f (Forward a s) -> g (Forward a s)) -> f (a, a) -> g a

Compute a vector of directional derivatives for a function given a zipped up Functor of the input values and their derivatives.

duF' :: (Functor f, Functor g, Num a) => (forall s. f (Forward a s) -> g (Forward a s)) -> f (a, a) -> g (a, a)

Compute a vector of answers and directional derivatives for a function given a zipped up Functor of the input values and their derivatives.

Directional Derivatives (Tower)

dus :: (Functor f, Num a) => (forall s. f (Tower a s) -> Tower a s) -> f [a] -> [a]

dus0 :: (Functor f, Num a) => (forall s. f (Tower a s) -> Tower a s) -> f [a] -> [a]

dusF :: (Functor f, Functor g, Num a) => (forall s. f (Tower a s) -> g (Tower a s)) -> f [a] -> g [a]

dus0F :: (Functor f, Functor g, Num a) => (forall s. f (Tower a s) -> g (Tower a s)) -> f [a] -> g [a]

Taylor Series (Tower)

taylor :: Fractional a => (forall s. Tower a s -> Tower a s) -> a -> a -> [a]

taylor0 :: Fractional a => (forall s. Tower a s -> Tower a s) -> a -> a -> [a]

Maclaurin Series (Tower)

maclaurin :: Fractional a => (forall s. Tower a s -> Tower a s) -> a -> [a]

maclaurin0 :: Fractional a => (forall s. Tower a s -> Tower a s) -> a -> [a]

Gradient Descent

gradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> [f a]

The gradientDescent 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.)

It uses reverse mode automatic differentiation to compute the gradient.

gradientAscent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> [f a]

Perform a gradient descent using reverse mode automatic differentiation to compute the gradient.

conjugateGradientDescent :: (Traversable f, Ord a, Fractional a) => (forall s1 s2 s3 s4. Chosen s4 => f (Or (On (Forward (Forward a s1) s2)) (Kahn a s3) s4) -> Or (On (Forward (Forward a s1) s2)) (Kahn a s3) s4) -> f a -> [f a]

Perform a conjugate gradient descent using reverse mode automatic differentiation to compute the gradient, and using forward-on-forward mode for computing extrema.

>>> let sq x = x * x
>>> let rosenbrock [x,y] = sq (1 - x) + 100 * sq (y - sq x)
>>> rosenbrock [0,0]
1
>>> rosenbrock (conjugateGradientDescent rosenbrock [0, 0] !! 5) < 0.1
True

conjugateGradientAscent :: (Traversable f, Ord a, Fractional a) => (forall s1 s2 s3 s4. Chosen s4 => f (Or (On (Forward (Forward a s1) s2)) (Kahn a s3) s4) -> Or (On (Forward (Forward a s1) s2)) (Kahn a s3) s4) -> f a -> [f a]

Perform a conjugate gradient ascent using reverse mode automatic differentiation to compute the gradient.