From 9c3d8cdd82a775efc90876be46f4b31e95d82dcc Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 6 Jan 2025 23:19:24 +0000 Subject: [PATCH] build based on 9b17e3e --- .../dev/.documenter-siteinfo.json | 2 +- DifferentiationInterface/dev/api/index.html | 66 +++++++++---------- .../dev/dev_guide/index.html | 2 +- .../dev/explanation/advanced/index.html | 2 +- .../dev/explanation/backends/index.html | 2 +- .../dev/explanation/operators/index.html | 2 +- .../dev/faq/differentiability/index.html | 2 +- .../dev/faq/limitations/index.html | 2 +- DifferentiationInterface/dev/index.html | 2 +- .../dev/tutorials/advanced/index.html | 52 +++++++-------- .../dev/tutorials/basic/index.html | 54 +++++++-------- 11 files changed, 94 insertions(+), 94 deletions(-) diff --git a/DifferentiationInterface/dev/.documenter-siteinfo.json b/DifferentiationInterface/dev/.documenter-siteinfo.json index fad60e3c7..ef87d896f 100644 --- a/DifferentiationInterface/dev/.documenter-siteinfo.json +++ b/DifferentiationInterface/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-05T05:19:22","documenter_version":"1.8.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-06T23:19:18","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/DifferentiationInterface/dev/api/index.html b/DifferentiationInterface/dev/api/index.html index ef96cfac0..c88e5aa8f 100644 --- a/DifferentiationInterface/dev/api/index.html +++ b/DifferentiationInterface/dev/api/index.html @@ -1,5 +1,5 @@ -API · DifferentiationInterface.jl
GitHub

API

Argument wrappers

DifferentiationInterface.Constant — Type
Constant

Concrete type of Context argument which is kept constant during differentiation.

Note that an operator can be prepared with an arbitrary value of the constant. However, same-point preparation must occur with the exact value that will be reused later.

Example

julia> using DifferentiationInterface
+API · DifferentiationInterface.jl
GitHub
API
Argument wrappers
DifferentiationInterface.Constant — Type
Constant
Concrete type of Context argument which is kept constant during differentiation.
Note that an operator can be prepared with an arbitrary value of the constant. However, same-point preparation must occur with the exact value that will be reused later.
Example
julia> using DifferentiationInterface
 
 julia> import ForwardDiff
 
@@ -13,31 +13,31 @@
 julia> gradient(f, AutoForwardDiff(), [1.0, 2.0], Constant(100))
 2-element Vector{Float64}:
  200.0
- 400.0
source
First order
Pushforward
DifferentiationInterface.prepare_pushforward — Function
prepare_pushforward(f,     backend, x, tx, [contexts...]) -> prep
-prepare_pushforward(f!, y, backend, x, tx, [contexts...]) -> prep
Create a prep object that can be given to pushforward and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.prepare_pushforward_same_point — Function
prepare_pushforward_same_point(f,     backend, x, tx, [contexts...]) -> prep_same
-prepare_pushforward_same_point(f!, y, backend, x, tx, [contexts...]) -> prep_same
Create an prep_same object that can be given to pushforward and its variants if they are applied at the same point x and with the same contexts.
Warning
If the function or the point changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.pushforward — Function
pushforward(f,     [prep,] backend, x, tx, [contexts...]) -> ty
-pushforward(f!, y, [prep,] backend, x, tx, [contexts...]) -> ty
Compute the pushforward of the function f at point x with a tuple of tangents tx.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named jvp.
source
DifferentiationInterface.pushforward! — Function
pushforward!(f,     dy, [prep,] backend, x, tx, [contexts...]) -> ty
-pushforward!(f!, y, dy, [prep,] backend, x, tx, [contexts...]) -> ty
Compute the pushforward of the function f at point x with a tuple of tangents tx, overwriting ty.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named jvp!.
source
DifferentiationInterface.value_and_pushforward — Function
value_and_pushforward(f,     [prep,] backend, x, tx, [contexts...]) -> (y, ty)
-value_and_pushforward(f!, y, [prep,] backend, x, tx, [contexts...]) -> (y, ty)
Compute the value and the pushforward of the function f at point x with a tuple of tangents tx.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named value_and_jvp.
Info
Required primitive for forward mode backends.
source
DifferentiationInterface.value_and_pushforward! — Function
value_and_pushforward!(f,     dy, [prep,] backend, x, tx, [contexts...]) -> (y, ty)
-value_and_pushforward!(f!, y, dy, [prep,] backend, x, tx, [contexts...]) -> (y, ty)
Compute the value and the pushforward of the function f at point x with a tuple of tangents tx, overwriting ty.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named value_and_jvp!.
source
Pullback
DifferentiationInterface.prepare_pullback — Function
prepare_pullback(f,     backend, x, ty, [contexts...]) -> prep
-prepare_pullback(f!, y, backend, x, ty, [contexts...]) -> prep
Create a prep object that can be given to pullback and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.prepare_pullback_same_point — Function
prepare_pullback_same_point(f,     backend, x, ty, [contexts...]) -> prep_same
-prepare_pullback_same_point(f!, y, backend, x, ty, [contexts...]) -> prep_same
Create an prep_same object that can be given to pullback and its variants if they are applied at the same point x and with the same contexts.
Warning
If the function or the point changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.pullback — Function
pullback(f,     [prep,] backend, x, ty, [contexts...]) -> tx
-pullback(f!, y, [prep,] backend, x, ty, [contexts...]) -> tx
Compute the pullback of the function f at point x with a tuple of tangents ty.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named vjp.
source
DifferentiationInterface.pullback! — Function
pullback!(f,     dx, [prep,] backend, x, ty, [contexts...]) -> tx
-pullback!(f!, y, dx, [prep,] backend, x, ty, [contexts...]) -> tx
Compute the pullback of the function f at point x with a tuple of tangents ty, overwriting dx.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named vjp!.
source
DifferentiationInterface.value_and_pullback — Function
value_and_pullback(f,     [prep,] backend, x, ty, [contexts...]) -> (y, tx)
-value_and_pullback(f!, y, [prep,] backend, x, ty, [contexts...]) -> (y, tx)
Compute the value and the pullback of the function f at point x with a tuple of tangents ty.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named value_and_vjp.
Info
Required primitive for reverse mode backends.
source
DifferentiationInterface.value_and_pullback! — Function
value_and_pullback!(f,     dx, [prep,] backend, x, ty, [contexts...]) -> (y, tx)
-value_and_pullback!(f!, y, dx, [prep,] backend, x, ty, [contexts...]) -> (y, tx)
Compute the value and the pullback of the function f at point x with a tuple of tangents ty, overwriting dx.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named value_and_vjp!.
source
Derivative
DifferentiationInterface.prepare_derivative — Function
prepare_derivative(f,     backend, x, [contexts...]) -> prep
-prepare_derivative(f!, y, backend, x, [contexts...]) -> prep
Create a prep object that can be given to derivative and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.derivative — Function
derivative(f,     [prep,] backend, x, [contexts...]) -> der
-derivative(f!, y, [prep,] backend, x, [contexts...]) -> der
Compute the derivative of the function f at point x.
To improve performance via operator preparation, refer to prepare_derivative.
source
DifferentiationInterface.derivative! — Function
derivative!(f,     der, [prep,] backend, x, [contexts...]) -> der
-derivative!(f!, y, der, [prep,] backend, x, [contexts...]) -> der
Compute the derivative of the function f at point x, overwriting der.
To improve performance via operator preparation, refer to prepare_derivative.
source
DifferentiationInterface.value_and_derivative — Function
value_and_derivative(f,     [prep,] backend, x, [contexts...]) -> (y, der)
-value_and_derivative(f!, y, [prep,] backend, x, [contexts...]) -> (y, der)
Compute the value and the derivative of the function f at point x.
To improve performance via operator preparation, refer to prepare_derivative.
source
DifferentiationInterface.value_and_derivative! — Function
value_and_derivative!(f,     der, [prep,] backend, x, [contexts...]) -> (y, der)
-value_and_derivative!(f!, y, der, [prep,] backend, x, [contexts...]) -> (y, der)
Compute the value and the derivative of the function f at point x, overwriting der.
To improve performance via operator preparation, refer to prepare_derivative.
source
Gradient
DifferentiationInterface.prepare_gradient — Function
prepare_gradient(f, backend, x, [contexts...]) -> prep
Create a prep object that can be given to gradient and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
Jacobian
DifferentiationInterface.prepare_jacobian — Function
prepare_jacobian(f,     backend, x, [contexts...]) -> prep
-prepare_jacobian(f!, y, backend, x, [contexts...]) -> prep
Create a prep object that can be given to jacobian and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.jacobian — Function
jacobian(f,     [prep,] backend, x, [contexts...]) -> jac
-jacobian(f!, y, [prep,] backend, x, [contexts...]) -> jac
Compute the Jacobian matrix of the function f at point x.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.jacobian! — Function
jacobian!(f,     jac, [prep,] backend, x, [contexts...]) -> jac
-jacobian!(f!, y, jac, [prep,] backend, x, [contexts...]) -> jac
Compute the Jacobian matrix of the function f at point x, overwriting jac.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.value_and_jacobian — Function
value_and_jacobian(f,     [prep,] backend, x, [contexts...]) -> (y, jac)
-value_and_jacobian(f!, y, [prep,] backend, x, [contexts...]) -> (y, jac)
Compute the value and the Jacobian matrix of the function f at point x.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.value_and_jacobian! — Function
value_and_jacobian!(f,     jac, [prep,] backend, x, [contexts...]) -> (y, jac)
-value_and_jacobian!(f!, y, jac, [prep,] backend, x, [contexts...]) -> (y, jac)
Compute the value and the Jacobian matrix of the function f at point x, overwriting jac.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.MixedMode — Type
MixedMode
Combination of a forward and a reverse mode backend for mixed-mode Jacobian computation.
Danger
MixedMode backends only support jacobian and its variants.
Constructor
MixedMode(forward_backend, reverse_backend)
source
Second order
DifferentiationInterface.SecondOrder — Type
SecondOrder
Combination of two backends for second-order differentiation.
Danger
SecondOrder backends do not support first-order operators.
Constructor
SecondOrder(outer_backend, inner_backend)
Fields
  • outer::AbstractADType: backend for the outer differentiation
  • inner::AbstractADType: backend for the inner differentiation
source
Second derivative
Hessian-vector product
DifferentiationInterface.prepare_hvp — Function
prepare_hvp(f, backend, x, tx, [contexts...]) -> prep
Create a prep object that can be given to hvp and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
DifferentiationInterface.prepare_hvp_same_point — Function
prepare_hvp_same_point(f, backend, x, tx, [contexts...]) -> prep_same
Create an prep_same object that can be given to hvp and its variants if they are applied at the same point x and with the same contexts.
Warning
If the function or the point changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
Hessian
DifferentiationInterface.prepare_hessian — Function
prepare_hessian(f, backend, x, [contexts...]) -> prep
Create a prep object that can be given to hessian and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
Utilities
Backend queries
DifferentiationInterface.outer — Function
outer(backend::SecondOrder)
-outer(backend::AbstractADType)
Return the outer backend of a SecondOrder object, tasked with differentiation at the second order.
For any other backend type, this function acts like the identity.
source
DifferentiationInterface.inner — Function
inner(backend::SecondOrder)
-inner(backend::AbstractADType)
Return the inner backend of a SecondOrder object, tasked with differentiation at the first order.
For any other backend type, this function acts like the identity.
source
Backend switch
DifferentiationInterface.DifferentiateWith — Type
DifferentiateWith
Function wrapper that enforces differentiation with a "substitute" AD backend, possible different from the "true" AD backend that is called.
For instance, suppose a function f is not differentiable with Zygote because it involves mutation, but you know that it is differentiable with Enzyme. Then f2 = DifferentiateWith(f, AutoEnzyme()) is a new function that behaves like f, except that f2 is differentiable with Zygote (thanks to a chain rule which calls Enzyme under the hood). Moreover, any larger algorithm alg that calls f2 instead of f will also be differentiable with Zygote (as long as f was the only Zygote blocker).
Tip
This is mainly relevant for package developers who want to produce differentiable code at low cost, without writing the differentiation rules themselves. If you sprinkle a few DifferentiateWith in places where some AD backends may struggle, end users can pick from a wider variety of packages to differentiate your algorithms.
Warning
DifferentiateWith only supports out-of-place functions y = f(x) without additional context arguments. It only makes these functions differentiable if the true backend is either ForwardDiff or compatible with ChainRules. For any other true backend, the differentiation behavior is not altered by DifferentiateWith (it becomes a transparent wrapper).
Fields
  • f: the function in question, with signature f(x)
  • backend::AbstractADType: the substitute backend to use for differentiation
Note
For the substitute AD backend to be called under the hood, its package needs to be loaded in addition to the package of the true AD backend.
Constructor
DifferentiateWith(f, backend)
Example
julia> using DifferentiationInterface
+ 400.0
source
First order
Pushforward
DifferentiationInterface.prepare_pushforward — Function
prepare_pushforward(f,     backend, x, tx, [contexts...]) -> prep
+prepare_pushforward(f!, y, backend, x, tx, [contexts...]) -> prep
Create a prep object that can be given to pushforward and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.prepare_pushforward_same_point — Function
prepare_pushforward_same_point(f,     backend, x, tx, [contexts...]) -> prep_same
+prepare_pushforward_same_point(f!, y, backend, x, tx, [contexts...]) -> prep_same
Create an prep_same object that can be given to pushforward and its variants if they are applied at the same point x and with the same contexts.
Warning
If the function or the point changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.pushforward — Function
pushforward(f,     [prep,] backend, x, tx, [contexts...]) -> ty
+pushforward(f!, y, [prep,] backend, x, tx, [contexts...]) -> ty
Compute the pushforward of the function f at point x with a tuple of tangents tx.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named jvp.
source
DifferentiationInterface.pushforward! — Function
pushforward!(f,     dy, [prep,] backend, x, tx, [contexts...]) -> ty
+pushforward!(f!, y, dy, [prep,] backend, x, tx, [contexts...]) -> ty
Compute the pushforward of the function f at point x with a tuple of tangents tx, overwriting ty.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named jvp!.
source
DifferentiationInterface.value_and_pushforward — Function
value_and_pushforward(f,     [prep,] backend, x, tx, [contexts...]) -> (y, ty)
+value_and_pushforward(f!, y, [prep,] backend, x, tx, [contexts...]) -> (y, ty)
Compute the value and the pushforward of the function f at point x with a tuple of tangents tx.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named value_and_jvp.
Info
Required primitive for forward mode backends.
source
DifferentiationInterface.value_and_pushforward! — Function
value_and_pushforward!(f,     dy, [prep,] backend, x, tx, [contexts...]) -> (y, ty)
+value_and_pushforward!(f!, y, dy, [prep,] backend, x, tx, [contexts...]) -> (y, ty)
Compute the value and the pushforward of the function f at point x with a tuple of tangents tx, overwriting ty.
To improve performance via operator preparation, refer to prepare_pushforward and prepare_pushforward_same_point.
Tip
Pushforwards are also commonly called Jacobian-vector products or JVPs. This function could have been named value_and_jvp!.
source
Pullback
DifferentiationInterface.prepare_pullback — Function
prepare_pullback(f,     backend, x, ty, [contexts...]) -> prep
+prepare_pullback(f!, y, backend, x, ty, [contexts...]) -> prep
Create a prep object that can be given to pullback and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.prepare_pullback_same_point — Function
prepare_pullback_same_point(f,     backend, x, ty, [contexts...]) -> prep_same
+prepare_pullback_same_point(f!, y, backend, x, ty, [contexts...]) -> prep_same
Create an prep_same object that can be given to pullback and its variants if they are applied at the same point x and with the same contexts.
Warning
If the function or the point changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.pullback — Function
pullback(f,     [prep,] backend, x, ty, [contexts...]) -> tx
+pullback(f!, y, [prep,] backend, x, ty, [contexts...]) -> tx
Compute the pullback of the function f at point x with a tuple of tangents ty.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named vjp.
source
DifferentiationInterface.pullback! — Function
pullback!(f,     dx, [prep,] backend, x, ty, [contexts...]) -> tx
+pullback!(f!, y, dx, [prep,] backend, x, ty, [contexts...]) -> tx
Compute the pullback of the function f at point x with a tuple of tangents ty, overwriting dx.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named vjp!.
source
DifferentiationInterface.value_and_pullback — Function
value_and_pullback(f,     [prep,] backend, x, ty, [contexts...]) -> (y, tx)
+value_and_pullback(f!, y, [prep,] backend, x, ty, [contexts...]) -> (y, tx)
Compute the value and the pullback of the function f at point x with a tuple of tangents ty.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named value_and_vjp.
Info
Required primitive for reverse mode backends.
source
DifferentiationInterface.value_and_pullback! — Function
value_and_pullback!(f,     dx, [prep,] backend, x, ty, [contexts...]) -> (y, tx)
+value_and_pullback!(f!, y, dx, [prep,] backend, x, ty, [contexts...]) -> (y, tx)
Compute the value and the pullback of the function f at point x with a tuple of tangents ty, overwriting dx.
To improve performance via operator preparation, refer to prepare_pullback and prepare_pullback_same_point.
Tip
Pullbacks are also commonly called vector-Jacobian products or VJPs. This function could have been named value_and_vjp!.
source
Derivative
DifferentiationInterface.prepare_derivative — Function
prepare_derivative(f,     backend, x, [contexts...]) -> prep
+prepare_derivative(f!, y, backend, x, [contexts...]) -> prep
Create a prep object that can be given to derivative and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.derivative — Function
derivative(f,     [prep,] backend, x, [contexts...]) -> der
+derivative(f!, y, [prep,] backend, x, [contexts...]) -> der
Compute the derivative of the function f at point x.
To improve performance via operator preparation, refer to prepare_derivative.
source
DifferentiationInterface.derivative! — Function
derivative!(f,     der, [prep,] backend, x, [contexts...]) -> der
+derivative!(f!, y, der, [prep,] backend, x, [contexts...]) -> der
Compute the derivative of the function f at point x, overwriting der.
To improve performance via operator preparation, refer to prepare_derivative.
source
DifferentiationInterface.value_and_derivative — Function
value_and_derivative(f,     [prep,] backend, x, [contexts...]) -> (y, der)
+value_and_derivative(f!, y, [prep,] backend, x, [contexts...]) -> (y, der)
Compute the value and the derivative of the function f at point x.
To improve performance via operator preparation, refer to prepare_derivative.
source
DifferentiationInterface.value_and_derivative! — Function
value_and_derivative!(f,     der, [prep,] backend, x, [contexts...]) -> (y, der)
+value_and_derivative!(f!, y, der, [prep,] backend, x, [contexts...]) -> (y, der)
Compute the value and the derivative of the function f at point x, overwriting der.
To improve performance via operator preparation, refer to prepare_derivative.
source
Gradient
DifferentiationInterface.prepare_gradient — Function
prepare_gradient(f, backend, x, [contexts...]) -> prep
Create a prep object that can be given to gradient and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
Jacobian
DifferentiationInterface.prepare_jacobian — Function
prepare_jacobian(f,     backend, x, [contexts...]) -> prep
+prepare_jacobian(f!, y, backend, x, [contexts...]) -> prep
Create a prep object that can be given to jacobian and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again. For in-place functions, y is mutated by f! during preparation.
source
DifferentiationInterface.jacobian — Function
jacobian(f,     [prep,] backend, x, [contexts...]) -> jac
+jacobian(f!, y, [prep,] backend, x, [contexts...]) -> jac
Compute the Jacobian matrix of the function f at point x.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.jacobian! — Function
jacobian!(f,     jac, [prep,] backend, x, [contexts...]) -> jac
+jacobian!(f!, y, jac, [prep,] backend, x, [contexts...]) -> jac
Compute the Jacobian matrix of the function f at point x, overwriting jac.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.value_and_jacobian — Function
value_and_jacobian(f,     [prep,] backend, x, [contexts...]) -> (y, jac)
+value_and_jacobian(f!, y, [prep,] backend, x, [contexts...]) -> (y, jac)
Compute the value and the Jacobian matrix of the function f at point x.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.value_and_jacobian! — Function
value_and_jacobian!(f,     jac, [prep,] backend, x, [contexts...]) -> (y, jac)
+value_and_jacobian!(f!, y, jac, [prep,] backend, x, [contexts...]) -> (y, jac)
Compute the value and the Jacobian matrix of the function f at point x, overwriting jac.
To improve performance via operator preparation, refer to prepare_jacobian.
source
DifferentiationInterface.MixedMode — Type
MixedMode
Combination of a forward and a reverse mode backend for mixed-mode Jacobian computation.
Danger
MixedMode backends only support jacobian and its variants.
Constructor
MixedMode(forward_backend, reverse_backend)
source
Second order
DifferentiationInterface.SecondOrder — Type
SecondOrder
Combination of two backends for second-order differentiation.
Danger
SecondOrder backends do not support first-order operators.
Constructor
SecondOrder(outer_backend, inner_backend)
Fields
  • outer::AbstractADType: backend for the outer differentiation
  • inner::AbstractADType: backend for the inner differentiation
source
Second derivative
Hessian-vector product
DifferentiationInterface.prepare_hvp — Function
prepare_hvp(f, backend, x, tx, [contexts...]) -> prep
Create a prep object that can be given to hvp and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
DifferentiationInterface.prepare_hvp_same_point — Function
prepare_hvp_same_point(f, backend, x, tx, [contexts...]) -> prep_same
Create an prep_same object that can be given to hvp and its variants if they are applied at the same point x and with the same contexts.
Warning
If the function or the point changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
Hessian
DifferentiationInterface.prepare_hessian — Function
prepare_hessian(f, backend, x, [contexts...]) -> prep
Create a prep object that can be given to hessian and its variants.
Warning
If the function changes in any way, the result of preparation will be invalidated, and you will need to run it again.
source
Utilities
Backend queries
DifferentiationInterface.outer — Function
outer(backend::SecondOrder)
+outer(backend::AbstractADType)
Return the outer backend of a SecondOrder object, tasked with differentiation at the second order.
For any other backend type, this function acts like the identity.
source
DifferentiationInterface.inner — Function
inner(backend::SecondOrder)
+inner(backend::AbstractADType)
Return the inner backend of a SecondOrder object, tasked with differentiation at the first order.
For any other backend type, this function acts like the identity.
source
Backend switch
DifferentiationInterface.DifferentiateWith — Type
DifferentiateWith
Function wrapper that enforces differentiation with a "substitute" AD backend, possible different from the "true" AD backend that is called.
For instance, suppose a function f is not differentiable with Zygote because it involves mutation, but you know that it is differentiable with Enzyme. Then f2 = DifferentiateWith(f, AutoEnzyme()) is a new function that behaves like f, except that f2 is differentiable with Zygote (thanks to a chain rule which calls Enzyme under the hood). Moreover, any larger algorithm alg that calls f2 instead of f will also be differentiable with Zygote (as long as f was the only Zygote blocker).
Tip
This is mainly relevant for package developers who want to produce differentiable code at low cost, without writing the differentiation rules themselves. If you sprinkle a few DifferentiateWith in places where some AD backends may struggle, end users can pick from a wider variety of packages to differentiate your algorithms.
Warning
DifferentiateWith only supports out-of-place functions y = f(x) without additional context arguments. It only makes these functions differentiable if the true backend is either ForwardDiff or compatible with ChainRules. For any other true backend, the differentiation behavior is not altered by DifferentiateWith (it becomes a transparent wrapper).
Fields
  • f: the function in question, with signature f(x)
  • backend::AbstractADType: the substitute backend to use for differentiation
Note
For the substitute AD backend to be called under the hood, its package needs to be loaded in addition to the package of the true AD backend.
Constructor
DifferentiateWith(f, backend)
Example
julia> using DifferentiationInterface
 
 julia> import FiniteDiff, ForwardDiff, Zygote
 
@@ -62,7 +62,7 @@
 julia> Zygote.gradient(alg, [3.0, 5.0])[1]
 2-element Vector{Float64}:
  42.0
- 70.0
source
Sparsity detection
DifferentiationInterface.DenseSparsityDetector — Type
DenseSparsityDetector
Sparsity pattern detector satisfying the detection API of ADTypes.jl.
The nonzeros in a Jacobian or Hessian are detected by computing the relevant matrix with dense AD, and thresholding the entries with a given tolerance (which can be numerically inaccurate). This process can be very slow, and should only be used if its output can be exploited multiple times to compute many sparse matrices.
Danger
In general, the sparsity pattern you obtain can depend on the provided input x. If you want to reuse the pattern, make sure that it is input-agnostic.
Warning
DenseSparsityDetector functionality is now located in a package extension, please load the SparseArrays.jl standard library before you use it.
Fields
  • backend::AbstractADType is the dense AD backend used under the hood
  • atol::Float64 is the minimum magnitude of a matrix entry to be considered nonzero
Constructor
DenseSparsityDetector(backend; atol, method=:iterative)
The keyword argument method::Symbol can be either:
  • :iterative: compute the matrix in a sequence of matrix-vector products (memory-efficient)
  • :direct: compute the matrix all at once (memory-hungry but sometimes faster).
Note that the constructor is type-unstable because method ends up being a type parameter of the DenseSparsityDetector object (this is not part of the API and might change).
Examples
using ADTypes, DifferentiationInterface, SparseArrays
+ 70.0
source
Sparsity detection
DifferentiationInterface.DenseSparsityDetector — Type
DenseSparsityDetector
Sparsity pattern detector satisfying the detection API of ADTypes.jl.
The nonzeros in a Jacobian or Hessian are detected by computing the relevant matrix with dense AD, and thresholding the entries with a given tolerance (which can be numerically inaccurate). This process can be very slow, and should only be used if its output can be exploited multiple times to compute many sparse matrices.
Danger
In general, the sparsity pattern you obtain can depend on the provided input x. If you want to reuse the pattern, make sure that it is input-agnostic.
Warning
DenseSparsityDetector functionality is now located in a package extension, please load the SparseArrays.jl standard library before you use it.
Fields
  • backend::AbstractADType is the dense AD backend used under the hood
  • atol::Float64 is the minimum magnitude of a matrix entry to be considered nonzero
Constructor
DenseSparsityDetector(backend; atol, method=:iterative)
The keyword argument method::Symbol can be either:
  • :iterative: compute the matrix in a sequence of matrix-vector products (memory-efficient)
  • :direct: compute the matrix all at once (memory-hungry but sometimes faster).
Note that the constructor is type-unstable because method ends up being a type parameter of the DenseSparsityDetector object (this is not part of the API and might change).
Examples
using ADTypes, DifferentiationInterface, SparseArrays
 import ForwardDiff
 
 detector = DenseSparsityDetector(AutoForwardDiff(); atol=1e-5, method=:direct)
@@ -85,13 +85,13 @@
 # output
 
 1×2 SparseMatrixCSC{Bool, Int64} with 1 stored entry:
- 1  ⋅
source
Internals
The following is not part of the public API.
DifferentiationInterface.AutoSimpleFiniteDiff — Type
AutoSimpleFiniteDiff <: ADTypes.AbstractADType
Forward mode backend based on the finite difference (f(x + ε) - f(x)) / ε, with artificial chunk size to mimick ForwardDiff.
Constructor
AutoSimpleFiniteDiff(ε=1e-5; chunksize=nothing)
source
DifferentiationInterface.BatchSizeSettings — Type
BatchSizeSettings{B,singlebatch,aligned}
Configuration for the batch size deduced from a backend and a sample array of length N.
Type parameters
  • B::Int: batch size
  • singlebatch::Bool: whether B == N (B > N is not allowed)
  • aligned::Bool: whether N % B == 0
Fields
  • N::Int: array length
  • A::Int: number of batches A = div(N, B, RoundUp)
  • B_last::Int: size of the last batch (if aligned is false)
source
ADTypes.mode — Method
mode(backend::SecondOrder)
Return the outer mode of the second-order backend.
source
DifferentiationInterface.basis — Method
basis(backend, a::AbstractArray, i)
Construct the i-th standard basis array in the vector space of a with element type eltype(a).
Note
If an AD backend benefits from a more specialized basis array implementation, this function can be extended on the backend type.
source
DifferentiationInterface.multibasis — Method
multibasis(backend, a::AbstractArray, inds::AbstractVector)
Construct the sum of the i-th standard basis arrays in the vector space of a with element type eltype(a), for all i ∈ inds.
Note
If an AD backend benefits from a more specialized basis array implementation, this function can be extended on the backend type.
source
DifferentiationInterface.prepare!_derivative — Function
prepare!_derivative(f,     prep, backend, x, [contexts...]) -> new_prep
-prepare!_derivative(f!, y, prep, backend, x, [contexts...]) -> new_prep
Same behavior as prepare_derivative but can modify an existing prep object to avoid some allocations.
There is no guarantee that prep will be mutated, or that performance will be improved compared to preparation from scratch.
Danger
For efficiency, this function needs to rely on backend package internals, therefore it not protected by semantic versioning.
source
DifferentiationInterface.prepare!_gradient — Function
prepare!_gradient(f, prep, backend, x, [contexts...]) -> new_prep
Same behavior as prepare_gradient but can modify an existing prep object to avoid some allocations.
There is no guarantee that prep will be mutated, or that performance will be improved compared to preparation from scratch.
Danger
For efficiency, this function needs to rely on backend package internals, therefore it not protected by semantic versioning.
source
DifferentiationInterface.prepare!_hessian — Function
prepare!_hessian(f, backend, x, [contexts...]) -> new_prep
Same behavior as prepare_hessian but can modify an existing prep object to avoid some allocations.
There is no guarantee that prep will be mutated, or that performance will be improved compared to preparation from scratch.
Danger
For efficiency, this function needs to rely on backend package internals, therefore it not protected by semantic versioning.
source
DifferentiationInterface.prepare!_hvp — Function
prepare!_hvp(f, backend, x, tx, [contexts...]) -> new_prep
Same behavior as prepare_hvp but can modify an existing prep object to avoid some allocations.
There is no guarantee that prep will be mutated, or that performance will be improved compared to preparation from scratch.
Danger
For efficiency, this function needs to rely on backend package internals, therefore it not protected by semantic versioning.
source
DifferentiationInterface.prepare!_jacobian — Function
prepare!_jacobian(f,     prep, backend, x, [contexts...]) -> new_prep
-prepare!_jacobian(f!, y, prep, backend, x, [contexts...]) -> new_prep
Same behavior as prepare_jacobian but can modify an existing prep object to avoid some allocations.
There is no guarantee that prep will be mutated, or that performance will be improved compared to preparation from scratch.
Danger
For efficiency, this function needs to rely on backend package internals, therefore it not protected by semantic versioning.
source
DifferentiationInterface.prepare!_pullback — Function
prepare!_pullback(f,     prep, backend, x, ty, [contexts...]) -> new_prep
-prepare!_pullback(f!, y, prep, backend, x, ty, [contexts...]) -> new_prep
Same behavior as prepare_pullback but can modify an existing prep object to avoid some allocations.
There is no guarantee that prep will be mutated, or that performance will be improved compared to preparation from scratch.
Danger
For efficiency, this function needs to rely on backend package internals, therefore it not protected by semantic versioning.
source
DifferentiationInterface.prepare!_pushforward — Function
prepare!_pushforward(f,     prep, backend, x, tx, [contexts...]) -> new_prep
-prepare!_pushforward(f!, y, prep, backend, x, tx, [contexts...]) -> new_prep
Same behavior as prepare_pushforward but can modify an existing prep object to avoid some allocations.
There is no guarantee that prep will be mutated, or that performance will be improved compared to preparation from scratch.
Danger
For efficiency, this function needs to rely on backend package internals, therefore it not protected by semantic versioning.
source

+
diff --git a/DifferentiationInterface/dev/dev_guide/index.html b/DifferentiationInterface/dev/dev_guide/index.html index 0cbaa5b4e..5faaa75f0 100644 --- a/DifferentiationInterface/dev/dev_guide/index.html +++ b/DifferentiationInterface/dev/dev_guide/index.html @@ -4,4 +4,4 @@ startOnLoad: true, theme: "neutral" }); - + diff --git a/DifferentiationInterface/dev/explanation/advanced/index.html b/DifferentiationInterface/dev/explanation/advanced/index.html index 9778e4152..0d0e81664 100644 --- a/DifferentiationInterface/dev/explanation/advanced/index.html +++ b/DifferentiationInterface/dev/explanation/advanced/index.html @@ -5,4 +5,4 @@ startOnLoad: true, theme: "neutral" }); - + diff --git a/DifferentiationInterface/dev/explanation/backends/index.html b/DifferentiationInterface/dev/explanation/backends/index.html index 4940628d4..49fb7a8f4 100644 --- a/DifferentiationInterface/dev/explanation/backends/index.html +++ b/DifferentiationInterface/dev/explanation/backends/index.html @@ -4,4 +4,4 @@ startOnLoad: true, theme: "neutral" }); - + diff --git a/DifferentiationInterface/dev/explanation/operators/index.html b/DifferentiationInterface/dev/explanation/operators/index.html index 8b7c89130..e08013036 100644 --- a/DifferentiationInterface/dev/explanation/operators/index.html +++ b/DifferentiationInterface/dev/explanation/operators/index.html @@ -5,4 +5,4 @@ startOnLoad: true, theme: "neutral" }); - + diff --git a/DifferentiationInterface/dev/faq/differentiability/index.html b/DifferentiationInterface/dev/faq/differentiability/index.html index ce1b07025..998ae9f14 100644 --- a/DifferentiationInterface/dev/faq/differentiability/index.html +++ b/DifferentiationInterface/dev/faq/differentiability/index.html @@ -7,4 +7,4 @@ startOnLoad: true, theme: "neutral" }); - + diff --git a/DifferentiationInterface/dev/faq/limitations/index.html b/DifferentiationInterface/dev/faq/limitations/index.html index a4bfe8c82..0b4258d41 100644 --- a/DifferentiationInterface/dev/faq/limitations/index.html +++ b/DifferentiationInterface/dev/faq/limitations/index.html @@ -4,4 +4,4 @@ startOnLoad: true, theme: "neutral" }); - + diff --git a/DifferentiationInterface/dev/index.html b/DifferentiationInterface/dev/index.html index 15210697b..ad69dcd3e 100644 --- a/DifferentiationInterface/dev/index.html +++ b/DifferentiationInterface/dev/index.html @@ -20,4 +20,4 @@ startOnLoad: true, theme: "neutral" }); - + diff --git a/DifferentiationInterface/dev/tutorials/advanced/index.html b/DifferentiationInterface/dev/tutorials/advanced/index.html index 5836dae9a..c5ecfccac 100644 --- a/DifferentiationInterface/dev/tutorials/advanced/index.html +++ b/DifferentiationInterface/dev/tutorials/advanced/index.html @@ -88,34 +88,34 @@ [2, 4] [5, 7] [6, 8]

Sparsity speedup

When preparation is used, the speedup due to sparsity becomes very visible in large dimensions.

xbig = rand(1000)
jac_prep_dense = prepare_jacobian(f_sparse_vector, dense_first_order_backend, zero(xbig))
-@benchmark jacobian($f_sparse_vector, $jac_prep_dense, $dense_first_order_backend, $xbig)
BenchmarkTools.Trial: 422 samples with 1 evaluation.
- Range (min … max):   4.883 ms … 184.315 ms  ┊ GC (min … max):  9.93% … 93.98%
- Time  (median):      5.719 ms               ┊ GC (median):    17.60%
- Time  (mean ± σ):   11.824 ms ±  27.026 ms  ┊ GC (mean ± σ):  48.45% ± 19.62%
+@benchmark jacobian($f_sparse_vector, $jac_prep_dense, $dense_first_order_backend, $xbig)
BenchmarkTools.Trial: 460 samples with 1 evaluation.
+ Range (min … max):   4.788 ms … 169.192 ms  ┊ GC (min … max):  9.35% … 96.98%
+ Time  (median):      5.390 ms               ┊ GC (median):    11.86%
+ Time  (mean ± σ):   10.940 ms ±  27.290 ms  ┊ GC (mean ± σ):  50.42% ± 17.93%
 
-  █▃     ▁                                                      
-  ███▆▁▅▆█▁▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆▄▄▁▄ ▆
-  4.88 ms       Histogram: log(frequency) by time       172 ms <
+  █▃▁                                                           
+  ███▇▄▁▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄▄▁▁▁▁▁▇ ▆
+  4.79 ms       Histogram: log(frequency) by time       167 ms <
 
  Memory estimate: 57.63 MiB, allocs estimate: 1515.
jac_prep_sparse = prepare_jacobian(f_sparse_vector, sparse_first_order_backend, zero(xbig))
 @benchmark jacobian($f_sparse_vector, $jac_prep_sparse, $sparse_first_order_backend, $xbig)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
- Range (min … max):  24.415 μs …  4.749 ms  ┊ GC (min … max):  0.00% … 97.90%
- Time  (median):     29.134 μs              ┊ GC (median):     0.00%
- Time  (mean ± σ):   38.573 μs ± 91.349 μs  ┊ GC (mean ± σ):  13.52% ±  7.18%
+ Range (min … max):  21.900 μs …  2.963 ms  ┊ GC (min … max):  0.00% … 96.98%
+ Time  (median):     28.253 μs              ┊ GC (median):     0.00%
+ Time  (mean ± σ):   33.511 μs ± 65.192 μs  ┊ GC (mean ± σ):  12.55% ±  7.18%
 
-  ▆██▇▄▂▂▂▁                                           ▂▁      ▂
-  ██████████▇▇▆▆▄▃▁▁▁▄▃▁▁▃▃▁▁▃▃▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆███▇▇▇▆ █
-  24.4 μs      Histogram: log(frequency) by time       150 μs <
+   ▂▂█                                                         
+  ▄███▆▃▃▂▃▂▂▂▂▂▂▂▂▂▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▂▁▂▁▁▁▁▁▁▁▁▁▁▂▂▂ ▂
+  21.9 μs         Histogram: frequency by time         139 μs <
 
  Memory estimate: 305.31 KiB, allocs estimate: 27.

Better memory use can be achieved by pre-allocating the matrix from the preparation result (so that it has the correct structure).

jac_buffer = similar(sparsity_pattern(jac_prep_sparse), eltype(xbig))
 @benchmark jacobian!($f_sparse_vector, $jac_buffer, $jac_prep_sparse, $sparse_first_order_backend, $xbig)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
- Range (min … max):  20.588 μs …   8.042 ms  ┊ GC (min … max):  0.00% … 10.75%
- Time  (median):     24.877 μs               ┊ GC (median):     0.00%
- Time  (mean ± σ):   33.824 μs ± 117.083 μs  ┊ GC (mean ± σ):  11.45% ±  5.81%
+ Range (min … max):  18.745 μs …  3.222 ms  ┊ GC (min … max): 0.00% … 98.66%
+ Time  (median):     24.456 μs              ┊ GC (median):    0.00%
+ Time  (mean ± σ):   28.777 μs ± 56.848 μs  ┊ GC (mean ± σ):  9.92% ±  5.86%
 
-  ▄▇█▇▃▁▁▁▂▂                                           ▂▂▁     ▂
-  ███████████▇▆▆▆▃▄▁▃▁▃▁▁▁▁▁▃▁▁▄▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▅█████▆▆▇ █
-  20.6 μs       Histogram: log(frequency) by time       118 μs <
+  ▂▄▇█▇▃▂▁▁▁▁                                                 ▂
+  ████████████▇▇▇▇▄▅▃▁▃▁▁▁▃▁▁▃▃▃▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▃▁▃▁▁▁▁▁▃▁▃▇██ █
+  18.7 μs      Histogram: log(frequency) by time       111 μs <
 
  Memory estimate: 234.75 KiB, allocs estimate: 18.

And for optimal speed, one should write non-allocating and type-stable functions.

function f_sparse_vector!(y::AbstractVector, x::AbstractVector)
     n = length(x)
@@ -130,17 +130,17 @@
 ybig ≈ f_sparse_vector(xbig)
true

In this case, the sparse Jacobian should also become non-allocating (for our specific choice of backend).

jac_prep_sparse_nonallocating = prepare_jacobian(f_sparse_vector!, zero(ybig), sparse_first_order_backend, zero(xbig))
 jac_buffer = similar(sparsity_pattern(jac_prep_sparse_nonallocating), eltype(xbig))
 @benchmark jacobian!($f_sparse_vector!, $ybig, $jac_buffer, $jac_prep_sparse_nonallocating, $sparse_first_order_backend, $xbig)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
- Range (min … max):  12.944 μs … 51.286 μs  ┊ GC (min … max): 0.00% … 0.00%
- Time  (median):     13.204 μs              ┊ GC (median):    0.00%
- Time  (mean ± σ):   13.345 μs ±  1.100 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
+ Range (min … max):  13.555 μs … 43.441 μs  ┊ GC (min … max): 0.00% … 0.00%
+ Time  (median):     13.915 μs              ┊ GC (median):    0.00%
+ Time  (mean ± σ):   14.089 μs ±  1.206 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
 
-  ▅█▇▂     ▁                                                  ▂
-  ████▆▆▆▆▆█▇▅▅▅▄▇▅▅▅▅▃▃▃▁▄▁▄▃▄▃▁▁▁▁▁▁▃▄▃▁▁▁▁▁▄▄▁▁▁▁▁▁▁▁▁▃▃▆▇ █
-  12.9 μs      Histogram: log(frequency) by time      21.2 μs <
+  ▁▆█▆                                                        ▁
+  █████▄▅▇██▇▇▄▄▅▁▅▆▆▄▆▄▄▄▆▄▁▅▆▅▅▁▄▄▄▄▄▄▁▃▃▅▁▃▃▁▁▁▁▃▁▁▃▁▁▃▄▆▇ █
+  13.6 μs      Histogram: log(frequency) by time      21.9 μs <
 
  Memory estimate: 0 bytes, allocs estimate: 0.
+ diff --git a/DifferentiationInterface/dev/tutorials/basic/index.html b/DifferentiationInterface/dev/tutorials/basic/index.html index df0d4e3c8..f54f64136 100644 --- a/DifferentiationInterface/dev/tutorials/basic/index.html +++ b/DifferentiationInterface/dev/tutorials/basic/index.html @@ -13,14 +13,14 @@ 8.0 10.0

Was that fast? BenchmarkTools.jl helps you answer that question.

using BenchmarkTools
 
-@benchmark gradient($f, $backend, $x)
BenchmarkTools.Trial: 10000 samples with 219 evaluations.
- Range (min … max):  338.804 ns … 124.334 μs  ┊ GC (min … max):  0.00% … 99.56%
- Time  (median):     415.272 ns               ┊ GC (median):     0.00%
- Time  (mean ± σ):   523.934 ns ±   2.206 μs  ┊ GC (mean ± σ):  10.91% ±  3.49%
+@benchmark gradient($f, $backend, $x)
BenchmarkTools.Trial: 10000 samples with 199 evaluations.
+ Range (min … max):  413.935 ns … 134.223 μs  ┊ GC (min … max): 0.00% … 99.55%
+ Time  (median):     612.065 ns               ┊ GC (median):    0.00%
+ Time  (mean ± σ):   599.706 ns ±   2.288 μs  ┊ GC (mean ± σ):  9.24% ±  3.08%
 
-   ▅█                                                            
-  ▃██▅▃▃▂▂▂▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▂▂▂▂▂▂▃▅▆▆▅▅▅▅▅▃▃▃▃▃▂▂▂▂▂▂▂▂▂▂ ▃
-  339 ns           Histogram: frequency by time          673 ns <
+  ▁▆█▇▄▂▁  ▁▃▃▁                       ▂▅▆▆▅▅▅▅▅▄▃▃▂▂▂▂▂▁        ▂
+  ███████████████▆▇▇▇▅▄▆▅▇▆▆▅▁▁▄▅▃▁▄▄███████████████████▆▇▇▅▆▆▇ █
+  414 ns        Histogram: log(frequency) by time        749 ns <
 
  Memory estimate: 624 bytes, allocs estimate: 5.

Not bad, but you can do better.

Overwriting a gradient

Since you know how much space your gradient will occupy (the same as your input x), you can pre-allocate that memory and offer it to AD. Some backends get a speed boost from this trick.

grad = similar(x)
 gradient!(f, grad, backend, x)
@@ -29,14 +29,14 @@
   4.0
   6.0
   8.0
- 10.0

The bang indicates that one of the arguments of gradient! might be mutated. More precisely, our convention is that every positional argument between the function and the backend is mutated.

@benchmark gradient!($f, $grad, $backend, $x)
BenchmarkTools.Trial: 10000 samples with 225 evaluations.
- Range (min … max):  330.524 ns … 111.968 μs  ┊ GC (min … max): 0.00% … 99.40%
- Time  (median):     534.660 ns               ┊ GC (median):    0.00%
- Time  (mean ± σ):   509.024 ns ±   1.990 μs  ┊ GC (mean ± σ):  9.41% ±  3.33%
+ 10.0

The bang indicates that one of the arguments of gradient! might be mutated. More precisely, our convention is that every positional argument between the function and the backend is mutated.

@benchmark gradient!($f, $grad, $backend, $x)
BenchmarkTools.Trial: 10000 samples with 200 evaluations.
+ Range (min … max):  402.945 ns … 122.134 μs  ┊ GC (min … max): 0.00% … 99.40%
+ Time  (median):     602.520 ns               ┊ GC (median):    0.00%
+ Time  (mean ± σ):   583.018 ns ±   2.146 μs  ┊ GC (mean ± σ):  8.75% ±  3.04%
 
-  ▄█▇▄▂     ▂▃▁                             ▁▄▅▇▇▆▅▃▁ ▁▂▃▃▃▂    ▂
-  ██████▆▄▄▇█████▇▇▆▅▄▅▅▄▅▅▄▇▆▆▅▅▃▄▃▁▃▁▄▃▄▁▅██████████████████▇ █
-  331 ns        Histogram: log(frequency) by time        610 ns <
+   █▁                                           ▁▁               
+  ▄██▃▂▂▂▂▁▂▂▃▃▂▂▂▂▂▂▁▂▂▂▂▂▂▂▂▂▂▁▂▂▂▁▁▁▂▂▂▁▁▂▃▅▇██▇▅▃▂▂▂▂▃▃▃▃▃▂ ▃
+  403 ns           Histogram: frequency by time          672 ns <
 
  Memory estimate: 528 bytes, allocs estimate: 3.

For some reason the in-place version is not much better than your first attempt. However, it makes fewer allocations, thanks to the gradient vector you provided. Don't worry, you can get even more performance.

Preparing for multiple gradients

Internally, ForwardDiff.jl creates some data structures to keep track of things. These objects can be reused between gradient computations, even on different input values. We abstract away the preparation step behind a backend-agnostic syntax:

prep = prepare_gradient(f, backend, zero(x))
DifferentiationInterfaceForwardDiffExt.ForwardDiffGradientPrep{ForwardDiff.GradientConfig{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 5, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 5}}}}(ForwardDiff.GradientConfig{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 5, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 5}}}((Partials(1.0, 0.0, 0.0, 0.0, 0.0), Partials(0.0, 1.0, 0.0, 0.0, 0.0), Partials(0.0, 0.0, 1.0, 0.0, 0.0), Partials(0.0, 0.0, 0.0, 1.0, 0.0), Partials(0.0, 0.0, 0.0, 0.0, 1.0)), ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 5}[Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(1.0,1.0,0.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(2.0,0.0,1.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(3.0,0.0,0.0,1.0,0.0,0.0), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(4.0,0.0,0.0,0.0,1.0,0.0), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(5.0,0.0,0.0,0.0,0.0,1.0)]))

You don't need to know what this object is, you just need to pass it to the gradient operator. Note that preparation does not depend on the actual components of the vector x, just on its type and size. You can thus reuse the prep for different values of the input.

grad = similar(x)
 gradient!(f, grad, prep, backend, x)
@@ -46,13 +46,13 @@
   6.0
   8.0
  10.0

Preparation makes the gradient computation much faster, and (in this case) allocation-free.

@benchmark gradient!($f, $grad, $prep, $backend, $x)
BenchmarkTools.Trial: 10000 samples with 995 evaluations.
- Range (min … max):  27.498 ns … 54.585 ns  ┊ GC (min … max): 0.00% … 0.00%
- Time  (median):     27.922 ns              ┊ GC (median):    0.00%
- Time  (mean ± σ):   28.163 ns ±  1.349 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
+ Range (min … max):  28.807 ns … 55.611 ns  ┊ GC (min … max): 0.00% … 0.00%
+ Time  (median):     29.321 ns              ┊ GC (median):    0.00%
+ Time  (mean ± σ):   29.979 ns ±  2.445 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
 
-  ▄▇██▇▆▃▁                                                ▁▁▁ ▂
-  █████████▇▆▆▃▁▄▄▆▄▆▅▁▃▁▃▁▃▁▃▃▃▁▁▃▁▁▁▁▁▁▁▁▃▃▁▁▁▃▁▁▁▁▁▃▅▇████ █
-  27.5 ns      Histogram: log(frequency) by time      35.3 ns <
+  ▃▇█▇▅▃▁                                 ▁▁▁             ▁▁▁ ▂
+  ████████▆▃▄▅▆▆▅▅▄▃▄▃▁▁▁▃▁▃▁▃▁▃▁▃▁▃▄▁▃▁▃▇████▇▇▅▇▆▇▇▇▆██████ █
+  28.8 ns      Histogram: log(frequency) by time      39.3 ns <
 
  Memory estimate: 0 bytes, allocs estimate: 0.

Beware that the prep object is nearly always mutated by differentiation operators, even though it is given as the last positional argument.

Switching backends

The whole point of DifferentiationInterface.jl is that you can easily experiment with different AD solutions. Typically, for gradients, reverse mode AD might be a better fit, so let's try Zygote.jl!

import Zygote
 
@@ -64,17 +64,17 @@
  10.0

And you can run the same benchmarks to see what you gained (although such a small input may not be realistic):

prep2 = prepare_gradient(f, backend2, zero(x))
 
 @benchmark gradient!($f, $grad, $prep2, $backend2, $x)
BenchmarkTools.Trial: 10000 samples with 994 evaluations.
- Range (min … max):  32.133 ns …  30.331 μs  ┊ GC (min … max):  0.00% … 99.65%
- Time  (median):     57.754 ns               ┊ GC (median):     0.00%
- Time  (mean ± σ):   62.328 ns ± 503.373 ns  ┊ GC (mean ± σ):  13.94% ±  1.73%
+ Range (min … max):  32.061 ns …  29.728 μs  ┊ GC (min … max):  0.00% … 99.70%
+ Time  (median):     57.834 ns               ┊ GC (median):     0.00%
+ Time  (mean ± σ):   62.103 ns ± 486.137 ns  ┊ GC (mean ± σ):  13.52% ±  1.73%
 
-   █▄                            ▃▄▆                            
-  ▄██▄▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆███▆▃▂▁▁▁▁▁▂▄▅▇▆▆▅▂▂▁▁▁▁▁▁▁▁▁ ▂
-  32.1 ns         Histogram: frequency by time         79.2 ns <
+   █                               ▄                            
+  ▅█▇▂▂▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂█▇█▆▃▂▁▂▂▂▁▂▂▄▅▅▅▅▃▂▁▁▁▁▁▁▁▁▁ ▂
+  32.1 ns         Histogram: frequency by time         78.6 ns <
 
  Memory estimate: 96 bytes, allocs estimate: 2.

In short, DifferentiationInterface.jl allows for easy testing and comparison of AD backends. If you want to go further, check out the documentation of DifferentiationInterfaceTest.jl. This related package provides benchmarking utilities to compare backends and help you select the one that is best suited for your problem.

+