Copyright | (c) Edward Kmett 2010-2015 |
---|---|

License | BSD3 |

Maintainer | ekmett@gmail.com |

Stability | experimental |

Portability | GHC only |

Safe Haskell | None |

Language | Haskell2010 |

- AD modes
- Gradients (Reverse Mode)
- Higher Order Gradients (Sparse-on-Reverse)
- Variadic Gradients (Sparse or Kahn)
- Jacobians (Sparse or Reverse)
- Higher Order Jacobian (Sparse-on-Reverse)
- Transposed Jacobians (Forward Mode)
- Hessian (Sparse-On-Reverse)
- Hessian Tensors (Sparse or Sparse-On-Reverse)
- Hessian Tensors (Sparse)
- Hessian Vector Products (Forward-On-Reverse)
- Derivatives (Forward Mode)
- Derivatives (Tower)
- Directional Derivatives (Forward Mode)
- Directional Derivatives (Tower)
- Taylor Series (Tower)
- Maclaurin Series (Tower)
- Gradient Descent

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.

- data AD s a
- class (Num t, Num (Scalar t)) => Mode t where
- grad :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f a
- grad' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f a)
- gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f b
- gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> (a, f b)
- grads :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> Cofree f a
- class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o
- 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
- vgrads :: Grads i o a => i -> o
- jacobian :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f a)
- jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f a)
- jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (f b)
- jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Reifies s Tape => f (Reverse s a) -> g (Reverse s a)) -> f a -> g (a, f b)
- jacobians :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (Cofree f a)
- jacobianT :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g a)
- jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g b)
- hessian :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (On (Reverse s (Sparse a))) -> On (Reverse s (Sparse a))) -> f a -> f (f a)
- hessian' :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> (a, f (a, f a))
- hessianF :: (Traversable f, Functor g, Num a) => (forall s. Reifies s Tape => f (On (Reverse s (Sparse a))) -> g (On (Reverse s (Sparse a)))) -> f a -> g (f (f a))
- hessianF' :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Sparse a)) -> g (AD s (Sparse a))) -> f a -> g (a, f (a, f a))
- hessianProduct :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (On (Reverse s (Forward a))) -> On (Reverse s (Forward a))) -> f (a, a) -> f a
- hessianProduct' :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (On (Reverse s (Forward a))) -> On (Reverse s (Forward a))) -> f (a, a) -> f (a, a)
- diff :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> a
- diffF :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f a
- diff' :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> (a, a)
- diffF' :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f (a, a)
- diffs :: Num a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]
- diffsF :: (Functor f, Num a) => (forall s. AD s (Tower a) -> f (AD s (Tower a))) -> a -> f [a]
- diffs0 :: Num a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]
- diffs0F :: (Functor f, Num a) => (forall s. AD s (Tower a) -> f (AD s (Tower a))) -> a -> f [a]
- du :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> a
- du' :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> (a, a)
- duF :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g a
- duF' :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g (a, a)
- dus :: (Functor f, Num a) => (forall s. f (AD s (Tower a)) -> AD s (Tower a)) -> f [a] -> [a]
- dus0 :: (Functor f, Num a) => (forall s. f (AD s (Tower a)) -> AD s (Tower a)) -> f [a] -> [a]
- dusF :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Tower a)) -> g (AD s (Tower a))) -> f [a] -> g [a]
- dus0F :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Tower a)) -> g (AD s (Tower a))) -> f [a] -> g [a]
- taylor :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> a -> [a]
- taylor0 :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> a -> [a]
- maclaurin :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]
- maclaurin0 :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]
- gradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> [f a]
- gradientAscent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> [f a]
- conjugateGradientDescent :: (Traversable f, Ord a, Fractional a) => (forall s. Chosen s => f (Or s (On (Forward (Forward a))) (Kahn a)) -> Or s (On (Forward (Forward a))) (Kahn a)) -> f a -> [f a]
- conjugateGradientAscent :: (Traversable f, Ord a, Fractional a) => (forall s. Chosen s => f (Or s (On (Forward (Forward a))) (Kahn a)) -> Or s (On (Forward (Forward a))) (Kahn a)) -> f a -> [f a]
- stochasticGradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Scalar a) -> f (Reverse s a) -> Reverse s a) -> [f (Scalar a)] -> f a -> [f a]

# Documentation

Bounded a => Bounded (AD s a) | |

Enum a => Enum (AD s a) | |

Eq a => Eq (AD s a) | |

Floating a => Floating (AD s a) | |

Fractional a => Fractional (AD s a) | |

Num a => Num (AD s a) | |

Ord a => Ord (AD s a) | |

Read a => Read (AD s a) | |

Real a => Real (AD s a) | |

RealFloat a => RealFloat (AD s a) | |

RealFrac a => RealFrac (AD s a) | |

Show a => Show (AD s a) | |

Erf a => Erf (AD s a) | |

InvErf a => InvErf (AD s a) | |

Mode a => Mode (AD s a) | |

Typeable (* -> * -> *) AD | |

type Scalar (AD s a) = Scalar a |

# AD modes

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

Mode Double | |

Mode Float | |

Mode Int | |

Mode Int8 | |

Mode Int16 | |

Mode Int32 | |

Mode Int64 | |

Mode Integer | |

Mode Word | |

Mode Word8 | |

Mode Word16 | |

Mode Word32 | |

Mode Word64 | |

Mode Natural | |

Mode ForwardDouble | |

Integral a => Mode (Ratio a) | |

RealFloat a => Mode (Complex a) | |

Num a => Mode (Id a) | |

Num a => Mode (Tower a) | |

Num a => Mode (Sparse a) | |

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

Num a => Mode (Kahn a) | |

Num a => Mode (Forward a) | |

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

Mode a => Mode (AD s a) | |

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

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

# Gradients (Reverse Mode)

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

The `grad`

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

`>>>`

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

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

The `grad'`

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

`>>>`

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

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

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

# Higher Order Gradients (Sparse-on-Reverse)

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

# 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!

# Jacobians (Sparse or Reverse)

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

The `jacobian`

function calculates the jacobian of a non-scalar-to-non-scalar function with reverse AD lazily in `m`

passes for `m`

outputs.

`>>>`

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

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

is the output dimensionality. Applying `fmap snd`

to the result will recover the result of `jacobian`

| An alias for `gradF'`

`>>>`

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

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

'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. Reifies s Tape => f (Reverse s a) -> g (Reverse s 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 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)

# Higher Order Jacobian (Sparse-on-Reverse)

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

# Transposed Jacobians (Forward Mode)

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

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

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

# Hessian (Sparse-On-Reverse)

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

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.

`>>>`

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

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

# Hessian Tensors (Sparse or Sparse-On-Reverse)

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

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

`>>>`

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

# Hessian Tensors (Sparse)

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

# Hessian Vector Products (Forward-On-Reverse)

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

computes the product of the hessian `hessianProduct`

f wv`H`

of a non-scalar-to-scalar function `f`

at `w = `

with a vector `fst`

$ wv`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. Reifies s Tape => f (On (Reverse s (Forward a))) -> On (Reverse s (Forward a))) -> f (a, a) -> f (a, a) Source

computes both the gradient of a non-scalar-to-scalar `hessianProduct'`

f wv`f`

at `w = `

and the product of the hessian `fst`

$ wv`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)

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

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

# Derivatives (Tower)

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

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

# Directional Derivatives (Forward Mode)

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

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 (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> (a, a) Source

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 (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g a Source

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 (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g (a, a) Source

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 (AD s (Tower a)) -> AD s (Tower a)) -> f [a] -> [a] Source

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

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

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

# Taylor Series (Tower)

# Maclaurin Series (Tower)

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

# Gradient Descent

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

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 s a) -> Reverse s a) -> f a -> [f a] Source

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

conjugateGradientDescent :: (Traversable f, Ord a, Fractional a) => (forall s. Chosen s => f (Or s (On (Forward (Forward a))) (Kahn a)) -> Or s (On (Forward (Forward a))) (Kahn a)) -> f a -> [f a] Source

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)`

`>>>`

1`rosenbrock [0,0]`

`>>>`

True`rosenbrock (conjugateGradientDescent rosenbrock [0, 0] !! 5) < 0.1`

conjugateGradientAscent :: (Traversable f, Ord a, Fractional a) => (forall s. Chosen s => f (Or s (On (Forward (Forward a))) (Kahn a)) -> Or s (On (Forward (Forward a))) (Kahn a)) -> f a -> [f a] Source

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

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

The `stochasticGradientDescent`

function approximates
the true gradient of the constFunction by a gradient at
a single example. As the algorithm sweeps through the training
set, it performs the update for each training example.

It uses reverse mode automatic differentiation to compute the gradient The learning rate is constant through out, and is set to 0.001