DifferentiableFrankWolfe

DifferentiableFrankWolfe

Differentiable wrapper for FrankWolfe.jl convex optimization routines.

DiffFW{F,G,M,A,I}

Callable parametrized wrapper for the Frank-Wolfe algorithm θ -> argmin_{x ∈ C} f(x, θ), which can be differentiated implicitly wrt θ.

Reference: https://arxiv.org/abs/2105.15183 (section 2 + end of appendix A).

Fields

• f::F: function f(x, θ) to minimize wrt x
• f_grad1::G: gradient ∇ₓf(x, θ) of f wrt x
• lmo::M: linear minimization oracle θ -> argmin_{x ∈ C} θᵀx from [FrankWolfe.jl], implicitly defines the convex set C
• alg::A: optimization algorithm from FrankWolfe.jl
• implicit::I: implicit function from ImplicitDifferentiation.jl

simplex_projection(z)

Compute the Euclidean projection of the vector z onto the probability simplex.

This function is differentiable thanks to a custom chain rule.

Reference: https://arxiv.org/abs/1602.02068.

simplex_projection_and_support(z)

Compute the Euclidean projection p of z on the probability simplex as well as the indicators s of its support, which are useful for differentiation.

Reference: https://arxiv.org/abs/1602.02068.

ForwardFW{F,G,M,A}

Underlying solver for DiffFW, which relies on a variant of Frank-Wolfe.

ConditionsFW{F,G,M}

Differentiable optimality conditions for DiffFW, which rely on a custom simplex_projection implementation.