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 (\[x,y] -> x**y) [0,2]
[0.0,NaN]

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])

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

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)

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

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'

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

'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 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)

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

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

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]]

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 :: RDouble]
[[[0.0,1.0],[1.0,0.0]],[[0.0,0.0],[0.0,0.0]],[[-1.131204383757,-2.471726672005],[-2.471726672005,1.131204383757]]]

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' 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.

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

>>> diff sin 0
1.0

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]

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)

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)]

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

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

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

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

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.

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

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

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

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

A Jet is a tower of all (higher order) partial derivatives of a function

At each step, a Jet f is wrapped in another layer worth of f.

a :- f a :- f (f a) :- f (f (f a)) :- ...

Take the head of a Jet.

Take the tail of a Jet.

Construct a Jet by unzipping the layers of a Cofree Comonad.