Basic tutorial

We present the main features of DifferentiationInterface.jl.

using DifferentiationInterface

Computing a gradient

A common use case of automatic differentiation (AD) is optimizing real-valued functions with first- or second-order methods. Let's define a simple objective (the squared norm) and a random input vector

f(x) = sum(abs2, x)

f (generic function with 1 method)

x = collect(1.0:5.0)

5-element Vector{Float64}:
 1.0
 2.0
 3.0
 4.0
 5.0

To compute its gradient, we need to choose a "backend", i.e. an AD package to call under the hood. Most backend types are defined by ADTypes.jl and re-exported by DifferentiationInterface.jl.

ForwardDiff.jl is very generic and efficient for low-dimensional inputs, so it's a good starting point:

using ForwardDiff: ForwardDiff

backend = AutoForwardDiff()

AutoForwardDiff()

Now you can use the following syntax to compute the gradient:

gradient(f, backend, x)

5-element Vector{Float64}:
  2.0
  4.0
  6.0
  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 197 evaluations per sample.
 Range (min … max):  405.376 ns … 160.512 μs  ┊ GC (min … max):  0.00% … 99.48%
 Time  (median):     620.193 ns               ┊ GC (median):     0.00%
 Time  (mean ± σ):   627.258 ns ±   2.898 μs  ┊ GC (mean ± σ):  11.02% ±  2.62%

   ▂█▃                                    ▄▆▃                    
  ▄███▆▄▃▂▂▂▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▂▁▁▁▂▃▄▆▇███▇█▆▅▄▃▄▄▃▃▃▃▂▂▂▂▂▂ ▃
  405 ns           Histogram: frequency by time          732 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)
grad  # has been mutated

5-element Vector{Float64}:
  2.0
  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 202 evaluations per sample.
 Range (min … max):  385.767 ns … 132.827 μs  ┊ GC (min … max): 0.00% … 99.43%
 Time  (median):     590.559 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   588.304 ns ±   2.439 μs  ┊ GC (mean ± σ):  9.77% ±  2.61%

   ▇▄                                          ▃█▇▅              
  ▅███▅▄▃▂▂▂▂▃▃▃▂▂▂▂▂▂▂▂▂▁▂▂▂▂▂▂▂▂▁▁▂▁▁▂▂▁▁▃▅▆▆████▆▃▃▂▃▂▃▄▄▄▃▂ ▃
  386 ns           Histogram: frequency by time          660 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:

using Random
typical_x = randn!(similar(x))
prep = prepare_gradient(f, backend, typical_x)

DifferentiationInterfaceForwardDiffExt.ForwardDiffGradientPrep{Tuple{typeof(Main.f), AutoForwardDiff{nothing, Nothing}, Vector{Float64}, Tuple{}}, ForwardDiff.GradientConfig{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 5, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 5}}}, Tuple{}}(Val{Tuple{typeof(Main.f), AutoForwardDiff{nothing, Nothing}, Vector{Float64}, Tuple{}}}(), 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 then reuse the prep for different values of the input.

grad = similar(x)
gradient!(f, grad, prep, backend, x)
grad  # has been mutated

5-element Vector{Float64}:
  2.0
  4.0
  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 994 evaluations per sample.
 Range (min … max):  28.151 ns … 51.294 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     28.725 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   29.004 ns ±  1.444 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

    ▃██▆▄▂ ▁                                             ▁▁▁  ▂
  ▇▇█████████▇▆▃▃▃▃▄▆▆▆▃▁▄▁▁▁▁▁▁▃▁▁▃▁▁▃▁▁▁▁▁▃▃▁▁▁▁▁▁▃▃▁▁▆████ █
  28.2 ns      Histogram: log(frequency) by time      36.1 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!

using Zygote: Zygote

backend2 = AutoZygote()

AutoZygote()

Once the backend is created, things run smoothly with exactly the same syntax as before:

gradient(f, backend2, x)

5-element Vector{Float64}:
  2.0
  4.0
  6.0
  8.0
 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, randn!(similar(x)))

@benchmark gradient!($f, $grad, $prep2, $backend2, $x)

BenchmarkTools.Trial: 10000 samples with 996 evaluations per sample.
 Range (min … max):  26.102 ns …  36.446 μs  ┊ GC (min … max):  0.00% … 99.78%
 Time  (median):     51.663 ns               ┊ GC (median):     0.00%
 Time  (mean ± σ):   56.277 ns ± 600.370 ns  ┊ GC (mean ± σ):  18.42% ±  1.73%

    █▂                               ▃   ▃▁                     
  ▃███▄▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▂▂▂▂▂▂▂▂▄█▇███▇██▄▂▂▂▂▂▃▃▃▃▃▃▃▂▂▂▂▂▂ ▃
  26.1 ns         Histogram: frequency by time         69.4 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.