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

Stability | experimental |

Maintainer | ekmett@gmail.com |

Safe Haskell | None |

Mixed-Mode Automatic Differentiation.

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.

- grad :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f a
- grad' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f a)
- gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f b
- gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f b)
- jacobian :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f a)
- jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f a)
- jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f b)
- jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f b)
- hessian :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f (f a)
- hessianF :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f (f a))
- diff :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> a
- diff' :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a)
- diffF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f a
- diffF' :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f (a, a)
- 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

# Gradient

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

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f bSource

function calculates the gradient of a non-scalar-to-scalar function `grad`

g f`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. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f b)Source

# Jacobian

jacobian :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD 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.

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

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD 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. Mode s => f (AD s a) -> g (AD 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)

# Hessian

hessian :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s 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 reverse 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`

.

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

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

Less efficient than `hessianF`

.

# Derivatives

diff' :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a)Source

The `d'`

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