ImplicitDifferentiation.jl

ImplicitDifferentiation.jl is a package for automatic differentiation of functions defined implicitly, i.e., forward mappings

$$x \in \mathbb{R}^n \longmapsto y(x) \in \mathbb{R}^m$$

whose output is defined by conditions

$$c(x,y(x)) = 0 \in \mathbb{R}^m$$

Background

Implicit differentiation is useful to differentiate through two types of functions:

Those for which automatic differentiation fails. Reasons can vary depending on your backend, but the most common include calls to external solvers, mutating operations or type restrictions.
Those for which automatic differentiation is very slow. A common example is iterative procedures like fixed point equations or optimization algorithms.

If you just need a quick overview, check out our JuliaCon 2022 talk. If you want a deeper dive into the theory, you can refer to the paper Efficient and modular implicit differentiation by Blondel et al. (2022).

To install the stable version, open a Julia REPL and run:

using Pkg; Pkg.add("ImplicitDifferentiation")

For the latest version, run this instead:

using Pkg; Pkg.add(url="https://github.com/JuliaDecisionFocusedLearning/ImplicitDifferentiation.jl")

Please read the documentation, especially the examples and FAQ.

In Julia:

SciML ecosystem, especially LinearSolve.jl, NonlinearSolve.jl and Optimization.jl
jump-dev/DiffOpt.jl: differentiation of convex optimization problems
axelparmentier/InferOpt.jl: approximate differentiation of combinatorial optimization problems
JuliaNonconvex/NonconvexUtils.jl: contains the original implementation from which this package drew inspiration

In Python:

google/jaxopt: hardware accelerated, batchable and differentiable optimizers in JAX