What is up with the different symbols?

Δx, ∂x, dx

ChainRules uses these perhaps atypically. As a notation that is the same across propagators, regardless of direction (in contrast see and below).

  • Δx is the input to a propagator, (i.e a seed for a pullback; or a perturbation for a pushforward).
  • ∂x is the output of a propagator.
  • dx could be either input or output.

dots and bars: $\dot{y} = \dfrac{∂y}{∂x} = \overline{x}$

  • is a derivative of the input moving forward: $v̇ = \frac{∂v}{∂x}$ for input $x$, intermediate value $v$.
  • is a derivative of the output moving backward: $v̄ = \frac{∂y}{∂v}$ for output $y$, intermediate value $v$.


  • Ω is often used as the return value of the function. Especially, but not exclusively, for scalar functions.
    • ΔΩ is thus a seed for the pullback.
    • ∂Ω is thus the output of a pushforward.

Why does rrule return the primal function evaluation?

You might wonder why frule(f, x) returns f(x) and the derivative of f at x, and similarly for rrule returning f(x) and the pullback for f at x. Why not just return the pushforward/pullback, and let the user call f(x) to get the answer separately?

There are three reasons the rules also calculate the f(x).

  1. For some rules an alternative way of calculating f(x) can give the same answer while also generating intermediate values that can be used in the calculations required to propagate the derivative.
  2. For many rrules the output value is used in the definition of the pullback. For example tan, sigmoid etc.
  3. For some frules there exists a single, non-separable operation that will compute both derivative and primal result. For example, this is the case for many of the methods for differential equation sensitivity analysis.

For more information and examples see the design notes on changing the primal.

Where are the derivatives for keyword arguments?

Pullbacks do not return a sensitivity for keyword arguments; similarly, pushforwards do not accept a perturbation for keyword arguments. This is because in practice functions are very rarely differentiable with respect to keyword arguments.

As a rule, keyword arguments tend to control side-effects, like logging verbosity, or to be functionality-changing to perform a different operation, e.g. dims=3, and thus not differentiable.

To the best of our knowledge no Julia AD system, with support for the definition of custom primitives, supports differentiating with respect to keyword arguments. At some point in the future ChainRules may support these. Maybe.

What is the difference between ZeroTangent and NoTangent ?

ZeroTangent and NoTangent act almost exactly the same in practice: they result in no change whenever added to anything. Odds are if you write a rule that returns the wrong one everything will just work fine. We provide both to allow for clearer writing of rules, and easier debugging.

ZeroTangent() represents the fact that if one perturbs (adds a small change to) the matching primal, there will be no change in the behaviour of the primal function. For example, in fst(x, y) = x, the derivative of fst with respect to y is ZeroTangent(). fst(10, 5) == 10 and if we add 0.1 to 5 we still get fst(10, 5.1) == 10.

NoTangent() represents the fact that if one perturbs the matching primal, the primal function will now error. For example, in access(xs, n) = xs[n], the derivative of access with respect to n is NoTangent(). access([10, 20, 30], 2) == 20, but if we add 0.1 to 2 we get access([10, 20, 30], 2.1) which errors as indexing can't be applied at fractional indexes.

Why do I get an error involving nothing?

When no custom frule or rrule exists, if you try to call one of those, it will return nothing by default. As a result, you may encounter errors like

MethodError: no method matching iterate(::Nothing)

Sometimes you think you have implemented the right rule, but it is called with a slightly different set of arguments than you expected. You can use Cthulhu.jl to dive into the call stack and figure out which method you are missing.

An alternative is to call back into AD: read the documentation on rule configuration to know more.

When to use ChainRules vs ChainRulesCore?

ChainRulesCore.jl is a light-weight dependency for defining rules for functions in your packages, without you needing to depend on ChainRules.jl itself. It has almost no dependencies of its own. If you only want to define rules, not use them, then you probably only want to load ChainRulesCore.jl.

ChainRules.jl provides the full functionality for AD systems. In particular, it has all the rules for Base Julia and the standard libraries. It is thus a much heavier package to load. AD systems making use of frules and rrules should load ChainRules.jl.

Where should I put my rules?

We recommend adding custom rules to your own packages with ChainRulesCore.jl. It is good to have them in the same package that defines the original function. This avoids type-piracy, and makes it easy to keep in-sync. ChainRulesCore is a very light-weight dependency.

How do I test my rules?

You can use ChainRulesTestUtils.jl to test your custom rules. ChainRulesTestUtils.jl has some dependencies, so it is a separate package from ChainRulesCore.jl. This means your package can depend on the light-weight ChainRulesCore.jl, and make ChainRulesTestUtils.jl a test-only dependency.

Remember to read the section On writing good rrule / frule methods.

Is removing a thunk a breaking change?

Removing thunks is not considered a breaking change. This is because (in principle) removing them changes the implementation of the values returned by an rrule, not the value that they represent. This is morally the same as similar issues discussed in ColPrac, such as details of floating point arithmetic changing.

On a practical level, it's important that this is the case because thunks are a bit of a hack, and over time it is hoped that the need for them will reduce, as they increase code-complexity and place additional stress on the compiler.

Where can I learn more about AD ?

There are not so many truly excellent learning resources for autodiff out there in the world, which is a bit sad. The list here is incomplete, but is vetted for quality.