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Advanced Usage Guide

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.

Accessing 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 DiffResult API provided by the DiffBase package. To learn how to use this functionality, please consult the relevant documentation.

Note that running using ForwardDiff will automatically bring the DiffBase module into scope, and that all mutating ForwardDiff API methods support the DiffResult API. In other words, API methods of the form ForwardDiff.method!(out, args...) will work appropriately if out is a DiffResult.

Configuring Chunk Size

ForwardDiff performs partial derivative evaluation on one "chunk" of the input vector at a time. Each differentation 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> import ForwardDiff

# 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 Config with chunk size of 1
julia> cfg1 = ForwardDiff.Config{1}(x);

# construct Config with chunk size of 4
julia> cfg4 = ForwardDiff.Config{4}(x);

# construct Config with chunk size of 10
julia> cfg10 = ForwardDiff.Config{10}(x);

# (input length of 10000) / (chunk size of 1) = (10000 1-element chunks)
julia> @time ForwardDiff.gradient!(out, rosenbrock, x, cfg1);
  0.408305 seconds (4 allocations: 160 bytes)

# (input length of 10000) / (chunk size of 4) = (2500 4-element chunks)
julia> @time ForwardDiff.gradient!(out, rosenbrock, x, cfg4);
  0.295764 seconds (4 allocations: 160 bytes)

# (input length of 10000) / (chunk size of 10) = (1000 10-element chunks)
julia> @time ForwardDiff.gradient!(out, rosenbrock, x, cfg10);
  0.267396 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 Config constructor will automatically select a
# chunk size in one is not explicitly provided
julia> cfg = ForwardDiff.Config(x);

julia> @time ForwardDiff.gradient!(out, rosenbrock, x, cfg);
0.266920 seconds (4 allocations: 160 bytes)

If your input dimension is a constant, 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
ForwardDiff.gradient(rosenbrock, x)::Vector{Float64}

# 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.

When selecting a chunk size, keep in mind that the maximum allowed size is 10 (to change this, you can alter the MAX_CHUNK_SIZE constant in ForwardDiff's source and reload the package). Also, it is usually best to pick a chunk sizes which divides evenly into the input dimension. Otherwise, ForwardDiff has to construct and utilize an extra "remainder" chunk to complete the calculation.

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(sin, x), [1,2,3])
9×3 Array{Float64,2}:
 -0.841471   0.0        0.0
 -0.0       -0.0       -0.0
 -0.0       -0.0       -0.0
 0.0        0.0        0.0
 -0.0       -0.909297  -0.0
 -0.0       -0.0       -0.0
 0.0        0.0        0.0
 -0.0       -0.0       -0.0
 -0.0       -0.0       -0.14112

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(sin, [1, 2, 3])
3×3×3 Array{Float64,3}:
[:, :, 1] =
 -0.841471   0.0   0.0
 -0.0       -0.0  -0.0
 -0.0       -0.0  -0.0

[:, :, 2] =
  0.0   0.0        0.0
 -0.0  -0.909297  -0.0
 -0.0  -0.0       -0.0

[:, :, 3] =
  0.0   0.0   0.0
 -0.0  -0.0  -0.0
 -0.0  -0.0  -0.14112

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!.

SIMD Vectorization

Many operations on ForwardDiff's dual numbers are amenable to SIMD vectorization. For some ForwardDiff benchmarks, we've seen SIMD vectorization yield speedups of almost 3x.

To enable SIMD optimizations, start your Julia process with the -O3 flag. This flag enables LLVM's SLPVectorizerPass during compilation, which attempts to automatically insert SIMD instructions where possible for certain arithmetic operations.

Here's an example of LLVM bitcode generated for an addition of two Dual numbers without SIMD instructions (i.e. not starting Julia with -O3):

julia> using ForwardDiff: Dual

julia> a = Dual(1., 2., 3., 4.)
Dual(1.0,2.0,3.0,4.0)

julia> b = Dual(5., 6., 7., 8.)
Dual(5.0,6.0,7.0,8.0)

julia> @code_llvm a + b

define void @"julia_+_70852"(%Dual* noalias sret, %Dual*, %Dual*) #0 {
top:
  %3 = getelementptr inbounds %Dual, %Dual* %1, i64 0, i32 1, i32 0, i64 0
  %4 = load double, double* %3, align 8
  %5 = getelementptr inbounds %Dual, %Dual* %2, i64 0, i32 1, i32 0, i64 0
  %6 = load double, double* %5, align 8
  %7 = fadd double %4, %6
  %8 = getelementptr inbounds %Dual, %Dual* %1, i64 0, i32 1, i32 0, i64 1
  %9 = load double, double* %8, align 8
  %10 = getelementptr inbounds %Dual, %Dual* %2, i64 0, i32 1, i32 0, i64 1
  %11 = load double, double* %10, align 8
  %12 = fadd double %9, %11
  %13 = getelementptr inbounds %Dual, %Dual* %1, i64 0, i32 1, i32 0, i64 2
  %14 = load double, double* %13, align 8
  %15 = getelementptr inbounds %Dual, %Dual* %2, i64 0, i32 1, i32 0, i64 2
  %16 = load double, double* %15, align 8
  %17 = fadd double %14, %16
  %18 = getelementptr inbounds %Dual, %Dual* %1, i64 0, i32 0
  %19 = load double, double* %18, align 8
  %20 = getelementptr inbounds %Dual, %Dual* %2, i64 0, i32 0
  %21 = load double, double* %20, align 8
  %22 = fadd double %19, %21
  %23 = getelementptr inbounds %Dual, %Dual* %0, i64 0, i32 0
  store double %22, double* %23, align 8
  %24 = getelementptr inbounds %Dual, %Dual* %0, i64 0, i32 1, i32 0, i64 0
  store double %7, double* %24, align 8
  %25 = getelementptr inbounds %Dual, %Dual* %0, i64 0, i32 1, i32 0, i64 1
  store double %12, double* %25, align 8
  %26 = getelementptr inbounds %Dual, %Dual* %0, i64 0, i32 1, i32 0, i64 2
  store double %17, double* %26, align 8
  ret void
}

If we start up Julia with -O3 instead, the call to @code_llvm will show that LLVM can SIMD-vectorize the addition:

julia> @code_llvm a + b

define void @"julia_+_70842"(%Dual* noalias sret, %Dual*, %Dual*) #0 {
top:
  %3 = bitcast %Dual* %1 to <4 x double>*            # cast the Dual to a SIMD-able LLVM vector
  %4 = load <4 x double>, <4 x double>* %3, align 8
  %5 = bitcast %Dual* %2 to <4 x double>*
  %6 = load <4 x double>, <4 x double>* %5, align 8
  %7 = fadd <4 x double> %4, %6                      # SIMD add
  %8 = bitcast %Dual* %0 to <4 x double>*
  store <4 x double> %7, <4 x double>* %8, align 8
  ret void
}

Note that whether or not SIMD instructions can actually be used will depend on your machine and Julia build. For example, pre-built Julia binaries might not emit vectorized LLVM bitcode. To overcome this specific issue, you can locally rebuild Julia's system image.