NonconvexUtils

CI Coverage

Useful hacks for use in Nonconvex.jl.

Hack #1: AbstractDiffFunction and ForwardDiffFunction

Nonconvex.jl uses Zygote.jl for automatic differentiation (AD). In order to force the use of another AD package for a function f, one can specify any AD backend from AbstractDifferentiation.jl in the following way:

g = AbstractDiffFunction(f, backend)

If you want to use ForwardDiff.jl to differentiate the function f, you can also use

g = ForwardDiffFunction(f)

which is short for:

AbstractDiffFunction(f, AbstractDifferentiation.ForwardDiffBackend())

Hack #2: TraceFunction

Often one may want to store intermediate solutions, function values and gradients for visualisation or post-processing. This is currently not possible with Nonconvex.jl as not all solvers support a callback mechanism. To workround this, TraceFunction can be used to store input, output and optionally gradient values during the optimization:

g = TraceFunction(f; on_call = false, on_grad = true)

If the on_call keyword argument is set to true, the input and output values are stored every time the function g is called. If the on_grad keyword argument is set to true, the input, output and gradient values are stored every time the function g is differentiated with a ChainRules-compatible AD package such as Zygote.jl which is used by Nonconvex.jl. The history is stored in f.trace.

Hack #3: CustomGradFunction

Often a function f can have analytic an gradient function ∇f that is more efficient than using AD on f. The way to make use of this gradient function in Nonconvex.jl has been to define an rrule for the function f. Now the following can be used instead. This will work for scalar-valued or vector-valued functions f where ∇f is either the gradient function or Jacobian function respectively.

g = CustomGradFunction(f, ∇f)

Hack #4: CustomHessianFunction and Hessian-vector products

Similar to CustomGradFunction if a function f has a custom gradient function ∇f and a custom Hessian function ∇²f, they can be used to force Zygote to use them in the following code:

g = CustomHessianFunction(f, ∇f, ∇²f)
Zygote.gradient(f, x)
Zygote.jacobian(x -> Zygote.gradient(f, x)[1], x)

It is on the user to ensure that the custom Hessian is always a symmetric matrix.

Note that one has to use Zygote for both levels of differentiation for this to work which makes it currently impossible to use in Nonconvex.jl directly, e.g. with IPOPT, because Nonconvex.jl uses ForwardDiff.jl for the second order differentiation, but this will be fixed soon by making more use of AbstractDifferentiation when it gets a ZygoteBackend implemented.

If instead of ∇²f, you only have access to a Hessian-vector product function hvp which takes 2 inputs: x (the input to f) and v (the vector to multiply the Hessian H by), and returns H * v, you can use this as follows:

g = CustomHessianFunction(f, ∇f, hvp; hvp = true)

Hack #5: ImplicitFunction

Differentiating implicit functions efficiently using the implicit function theorem has many applications in nonlinear partial differential equation constrained optimization, differentiable optimization layers in deep learning (aka deep declarative layers), differentiable fixed point iteration algorithms for optimal transport (e.g. the Sinkhorn methods), gradient-based bi-level and robust optimization (aka anti-optimization) and multi-parameteric programming.

There are 4 components to any implicit function:

  1. The parameters p
  2. The variables x
  3. The residual f(p, x) which is used to define x(p) as the x which satisfies f(p, x) == 0 for a given value p
  4. The algorithm used to evaluate x(p) satisfying the condition f(p, x)

In order to define a differentiable implicit function using NonconvexUtils, you have to specify the "forward" algorithm which finds x(p). For instance, consider the following example:

using SparseArrays, NLsolve, Zygote, NonconvexUtils

N = 10
A = spdiagm(0 => fill(10.0, N), 1 => fill(-1.0, N-1), -1 => fill(-1.0, N-1))
p0 = randn(N)

f(p, x) = A * x + 0.1 * x.^2 - p
function forward(p)
  # Solving nonlinear system of equations
  sol = nlsolve(x -> f(p, x), zeros(N), method = :anderson, m = 10)
  # Return the zero found (ignore the second returned value for now)
  return sol.zero, nothing
end

forward above solves for x in the nonlinear system of equations f(p, x) given the value of p. In this case, the residual function is the same as the function f used in the forward pass. One can then use the 2 functions forward and f to define an implicit function using:

imf = ImplicitFunction(forward, f)
xstar = imf(p0)

where imf(p0) solves the nonlinear system for p = p0 and returns the zero xstar of the nonlinear system. This function can now be part of any arbitrary Julia function differentiated by Zygote, e.g. it can be part of an objective function in an optimization problem using gradient-based optimization:

obj(p) = sum(imf(p))
g = Zygote.gradient(obj, p0)[1]

In the implicit function's adjoint rule definition the partial Jacobian ∂f/∂x is used according to the implicit function theorem. Often this Jacobian or a good approximation of it might be a by-product of the forward function. For example when the forward function does an optimization using a BFGS-based approximation of the Hessian of the Lagrangian function, the final BFGS approximation can be a good approximation of ∂f/∂x where the residual f is the gradient of the Lagrangian function wrt x. In those cases, this Jacobian by-product can be returned as the second argument from forward instead of nothing.