This document describes several techniques and features that can be used in conjunction ForwardDiff's basic API in order to fine-tune calculations and increase performance.

## Retrieving Lower-Order Results

Let's say you want to calculate the value, gradient, and Hessian of some function f at an input x. You could execute f(x), ForwardDiff.gradient(f, x) and ForwardDiff.hessian(f, x), but that would be a horribly redundant way to accomplish this task!

In the course of calculating higher-order derivatives, ForwardDiff ends up calculating all the lower-order derivatives and primal value f(x). To retrieve these results in one fell swoop, you can utilize the DiffResults API.

All mutating ForwardDiff API methods support the DiffResults API. In other words, API methods of the form ForwardDiff.method!(out, args...) will work appropriately if isa(out, DiffResults.DiffResult).

## Configuring Chunk Size

ForwardDiff performs partial derivative evaluation on one "chunk" of the input vector at a time. Each differentiation of a chunk requires a call to the target function as well as additional memory proportional to the square of the chunk's size. Thus, a smaller chunk size makes better use of memory bandwidth at the cost of more calls to the target function, while a larger chunk size reduces calls to the target function at the cost of more memory bandwidth.

For example:

julia> using ForwardDiff: GradientConfig, Chunk, gradient!

# let's use a Rosenbrock function as our target function
julia> function rosenbrock(x)
a = one(eltype(x))
b = 100 * a
result = zero(eltype(x))
for i in 1:length(x)-1
result += (a - x[i])^2 + b*(x[i+1] - x[i]^2)^2
end
return result
end
rosenbrock (generic function with 1 method)

# input vector
julia> x = rand(10000);

# output buffer
julia> out = similar(x);

# construct GradientConfig with chunk size of 1
julia> cfg1 = GradientConfig(rosenbrock, x, Chunk{1}());

# construct GradientConfig with chunk size of 4
julia> cfg4 = GradientConfig(rosenbrock, x, Chunk{4}());

# construct GradientConfig with chunk size of 10
julia> cfg10 = GradientConfig(rosenbrock, x, Chunk{10}());

# (input length of 10000) / (chunk size of 1) = (10000 1-element chunks)
julia> @time gradient!(out, rosenbrock, x, cfg1);
0.775139 seconds (4 allocations: 160 bytes)

# (input length of 10000) / (chunk size of 4) = (2500 4-element chunks)
julia> @time gradient!(out, rosenbrock, x, cfg4);
0.386459 seconds (4 allocations: 160 bytes)

# (input length of 10000) / (chunk size of 10) = (1000 10-element chunks)
julia> @time gradient!(out, rosenbrock, x, cfg10);
0.282529 seconds (4 allocations: 160 bytes)

If you do not explicity provide a chunk size, ForwardDiff will try to guess one for you based on your input vector:

# The GradientConfig constructor will automatically select a
# chunk size in one is not explicitly provided

julia> @time ForwardDiff.gradient!(out, rosenbrock, x, cfg);
0.281853 seconds (4 allocations: 160 bytes)

The underlying heuristic will compute a suitable chunk size smaller or equal to the ForwardDiff.DEFAULT_CHUNK_THRESHOLD constant. As of ForwardDiff v0.10.32 and Julia 1.6, this constant can be configured at load time via Preferences.jl by setting the default_chunk_threshold value.

If your input dimension is constant across calls, you should explicitly select a chunk size rather than relying on ForwardDiff's heuristic. There are two reasons for this. The first is that ForwardDiff's heuristic depends only on the input dimension, whereas in reality the optimal chunk size will also depend on the target function. The second is that ForwardDiff's heuristic is inherently type-unstable, which can cause the entire call to be type-unstable.

If your input dimension is a runtime variable, you can rely on ForwardDiff's heuristic without sacrificing type stability by manually asserting the output type, or - even better - by using the in-place API functions:

# will be type-stable since you're asserting the output type

# will be type-stable since out is returned, and Julia knows the type of out
ForwardDiff.gradient!(out, rosenbrock, x)

One final question remains: How should you select a chunk size? The answer is essentially "perform your own benchmarks and see what works best for your use case." As stated before, the optimal chunk size is heavily dependent on the target function and length of the input vector.

Note that it is usually best to pick a chunk size which divides evenly into the input dimension. Otherwise, ForwardDiff has to construct and utilize an extra "remainder" chunk to complete the calculation.

## Fixing NaN/Inf Issues

ForwardDiff's default behavior is to return NaN for undefined derivatives (or otherwise mirror the behavior of the function in Base, if it would return an error). This is usually the correct thing to do, but in some cases can erroneously "poison" values which aren't sensitive to the input and thus cause ForwardDiff to incorrectly return NaN or Inf derivatives. For example:

# the dual number's perturbation component is zero, so this
# variable should not propagate derivative information
julia> log(ForwardDiff.Dual{:tag}(0.0, 0.0))
Dual{:tag}(-Inf,NaN) # oops, this NaN should be 0.0

Here, ForwardDiff computes the derivative of log(0.0) as NaN and then propagates this derivative by multiplying it by the perturbation component. Usually, ForwardDiff can rely on the identity x * 0.0 == 0.0 to prevent the derivatives from propagating when the perturbation component is 0.0. However, this identity doesn't hold if isnan(y) or isinf(y), in which case a NaN derivative will be propagated instead.

It is possible to fix this behavior by checking that the perturbation component is zero before attempting to propagate derivative information, but this check can noticeably decrease performance (~5%-10% on our benchmarks).

In order to preserve performance in the majority of use cases, ForwardDiff disables this check by default. If your code is affected by this NaN behavior, you can enable ForwardDiff's NaN-safe mode by using the Preferences.jl API to set the nansafe_mode preference to true, for example via:

julia> using ForwardDiff, Preferences

julia> set_preferences!(ForwardDiff, "nansafe_mode" => true)

In the future, we plan on allowing users and downstream library authors to dynamically enable NaN-safe mode via theAbstractConfig API.

## Hessian of a vector-valued function

While ForwardDiff does not have a built-in function for taking Hessians of vector-valued functions, you can easily compose calls to ForwardDiff.jacobian to accomplish this. For example:

julia> ForwardDiff.jacobian(x -> ForwardDiff.jacobian(cumprod, x), [1,2,3])
9×3 Array{Int64,2}:
0  0  0
0  1  0
0  3  2
0  0  0
1  0  0
3  0  1
0  0  0
0  0  0
2  1  0

Since this functionality is composed from ForwardDiff's existing API rather than built into it, you're free to construct a vector_hessian function which suits your needs. For example, if you require the shape of the output to be a tensor rather than a block matrix, you can do so with a reshape (note that reshape does not copy data, so it's not an expensive operation):

julia> function vector_hessian(f, x)
n = length(x)
out = ForwardDiff.jacobian(x -> ForwardDiff.jacobian(f, x), x)
return reshape(out, n, n, n)
end
vector_hessian (generic function with 1 method)

julia> vector_hessian(cumprod, [1, 2, 3])
3×3×3 Array{Int64,3}:
[:, :, 1] =
0  0  0
0  1  0
0  3  2

[:, :, 2] =
0  0  0
1  0  0
3  0  1

[:, :, 3] =
0  0  0
0  0  0
2  1  0

Likewise, you could write a version of vector_hessian which supports functions of the form f!(y, x), or perhaps an in-place Jacobian with ForwardDiff.jacobian!.

## Custom tags and tag checking

The Dual type includes a "tag" parameter indicating the particular function call to which it belongs. This is to avoid a problem known as perturbation confusion which can arise when there are nested differentiation calls. Tags are automatically generated as part of the appropriate config object, and the tag is checked when the config is used as part of a differentiation call (derivative, gradient, etc.): an InvalidTagException will be thrown if the incorrect config object is used.

This checking can sometimes be inconvenient, and there are certain cases where you may want to disable this checking.

Warning

Disabling tag checking should only be done with caution, especially if the code itself could be used inside another differentiation call.

1. (preferred) Provide an extra Val{false}() argument to the differentiation function, e.g.

cfg = ForwarDiff.GradientConfig(g, x)
ForwarDiff.gradient(f, x, cfg, Val{false}())

If using as part of a library, the tag can be checked manually via

ForwardDiff.checktag(cfg, g, x)
2. (discouraged) Construct the config object with nothing instead of a function argument, e.g.

cfg = GradientConfig(nothing, x)
gradient(f, x, cfg)`