8.12
6 Automatic Differentiation
Automatic differentiation functions are described in terms of these types, that are described
in Overview.
dual? - Duals
- link? - Links included in a dual. Defined as the type
(-> dual? tensor? gradient-state? gradient-state?) gradient-state? - A hashtable from dual? to tensor?
differentiable? - Either a dual?, or a (listof differentiable?). In the learner representation (vectorof differentiable?) is also considered to be differentiable?, but not in other representations.
Constructs a dual with ρ as its real part, and κ as its link.
Returns #t if x is a dual.
If d is a tensor?, returns d. Otherwise, returns the real part of the dual.
If d is a tensor?, returns the function end-of-chain. Otherwise, returns the link of the dual.
Returns #t if x is a number? or x is a dual? and (ρ x) is a number?
The default link that terminates gradient computation for any dual. It returns a
new gradient-state? that includes the mapping of d to the addition of z
and the mapping of d in σ, if it exists, or 0.0 otherwise. If z is not a number?,
+-ρ is used for the addition.
In the learner representation, z can only be a scalar?.
procedure
f : (-> differentiable? ... differentiable?) t0 : differentiable? tn : differentiable?
Returns a list of gradients (list g0 ... gn) where gi is the gradient of (f t0 ... tn) with
respect to ti. If (f t0 ... tn) is not a scalar?, then gi is the sum of the gradients
of each scalar in (f t0 ... tn) with respect to ti.
This is equivalent to
(ref (∇¹ f θ) 0)
procedure
f : (-> (listof tensor?) tensor?) θ : (listof tensor?)
This is also equivalent to
(ref (∇¹ f θ) 0)